Masaryk UniversityFaculty of Science

Algorithmic construction of thePostnikov tower for diagrams of

simplicial sets

Doctoral Thesis

Marek Filakovský

Brno, 2015

Declaration

Hereby I declare, that this paper is my original authorial work, which I have worked outby my own. All sources, references and literature used or excerpted during elaborationof this work are properly cited and listed in complete reference to the due source.

Marek Filakovský

Advisor: doc.RNDr. Martin Čadek, CSc.

iii

Acknowledgement

I would like to thank my supervisor M. Čadek for his support and many useful discus-sions, suggestions and comments. The course in algebraic topology he taught inspiredme to continue in this field. I can hardly imagine a teacher that could be more generouswith his time and knowledge.

I am also indebted to L. Vokřínek, who took the unpaid job of being my unofficialsecondary supervisor. Many of the results of this thesis follow from ideas originallydeveloped by him. Trying to keep up with his thoughts encouraged me to learn moreabout model categories, homotopy theory and simplicial sets.

This work also benefited from results achieved by M. Krčál, J. Matoušek, F. Ser-geraert and U. Wagner.

Finally, I would like to thank my family for their endless material and emotionalsupport. I dedicate this thesis to my wife, Martina.

v

Abstract

The aim of the thesis is to provide an algorithm that given a nonnegative integer 𝑛and a finite diagram of simplicial sets 𝑌 : ℐ → sSet, where 𝑌 (𝑖) is simply connectedfor all 𝑖 ∈ ℐ, constructs the 𝑛-stage Postnikov tower for 𝑌 .

Given a finite simplicial set 𝑌 with an action of a finite group 𝐺, the Elmendorf’stheorem provides a finite diagram of simplicial sets 𝑌 : 𝒪op

𝐺 → sSet, where the spacesare fixed points 𝑌 𝐻 for subgroups𝐻 ≤ 𝐺. The diagram 𝑌 further reflects the homotopyproperties of space 𝑌 . Therefore, in the case the set of fixed points 𝑌 𝐻 is simplyconnected for every subgroup 𝐻 ≤ 𝐺, the algorithm constructs the 𝑛-stage Postnikovtower for 𝑌 , which, informally speaking, represents the 𝑛-stage Postnikov tower for 𝑌as a 𝐺-simplicial set.

Further, we present an algorithm that decides if a simplicial map 𝑓 : 𝑋 → 𝑌 betweenfinite simplicial sets 𝑋, 𝑌 is homotopic to a trivial map under the assumption that 𝑌is simply connected.

Keywords

simplicial set, Postnikov tower, chain complex, effective homology, equivariant algebraictopology, model category

Abstrakt

Hlavním cílem této práce je popis algoritmu, který pro konečný diagram simpliciálníchmnožin 𝑌 : ℐ → sSet, kde 𝑌 (𝑖) je jednoduše souvislý prostor pro každé 𝑖 ∈ ℐ, a prolibovolné nezáporné 𝑛 ∈ Z, zkonstruuje 𝑛-patrovou Postnikovovu věž pro diagram 𝑌 .

Podle Elmendorfovy věty, lze každé konečné simpliciální množině 𝑌 s akcí grupy𝐺 přiřadit diagram simpliciálních množin 𝑌 : 𝒪op

𝐺 → sSet. Prvky v tomto diagramujsou prostory pevných bodů 𝑌 𝐻 pro podgrupy 𝐻 ≤ 𝐺. Diagram 𝑌 dále zachováváhomotopické vlastnosti prostoru 𝑌 . Proto v případě kdy je každý prostor 𝑌 𝐻 , 𝐻 ≤ 𝐺jednoduše souvislý, algoritmus konstruuje 𝑛-patrovou Postnikovovu věž pro diagram𝑌 , která, neformálně řečeno, zodpovídá 𝑛-patrové Postnikovově věži pro 𝐺-simpliciálnímnožinu 𝑌 .

Dále uvádíme algoritmus, který pro dané simpliciální zobrazení 𝑓 : 𝑋 → 𝑌 mezikonečnými simpliciálními prostory, kde 𝑌 je jednoduše souvislý prostor, rozhoduje,zda je 𝑓 homotopické s triviálním zobrazením.

Klíčová slova

simpliciální množina, Postnikovova věž, řetězcový komplex, efektivní homologie, ekvi-variantní algebraická topologie, modelová kategorie

vii

Contents

Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Representing simplicial sets and simplicial maps in a computer . . . . . 3Finite simplicial sets . . . . . . . . . . . . . . . . . . . . . . . . . . . 3Locally effective simplicial sets . . . . . . . . . . . . . . . . . . . . . 4

1.2 Postnikov tower for simplicial sets . . . . . . . . . . . . . . . . . . . . . 41.3 Effective homology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.4 Our motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.5 Category of orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

Simplicial sets with a group action . . . . . . . . . . . . . . . . . . . 8Effective homology for diagrams . . . . . . . . . . . . . . . . . . . . . 9

1.6 Postnikov tower for diagrams . . . . . . . . . . . . . . . . . . . . . . . 102 Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.1 Simplicial sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13Fibrations, cofibrations and weak equivalences . . . . . . . . . . . . . 14Twisted products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2 Diagrams of simplicial sets . . . . . . . . . . . . . . . . . . . . . . . . . 15Homotopy and homology . . . . . . . . . . . . . . . . . . . . . . . . 15Model structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16Homotopy left Kan extension . . . . . . . . . . . . . . . . . . . . . . 17

2.3 Effective homology of chain complexes . . . . . . . . . . . . . . . . . . 19Effective chain complexes, reductions and strong equivalences . . . . 20Perturbation Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.4 Effective homology of twisted products . . . . . . . . . . . . . . . . . . 22Effective chain complex for twisted product . . . . . . . . . . . . . . 24Vector fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26Twisted division . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.5 Effective homology for diagrams . . . . . . . . . . . . . . . . . . . . . . 28Constructions with effective homology . . . . . . . . . . . . . . . . . 31Perturbation lemmas for diagrams . . . . . . . . . . . . . . . . . . . 32Filtrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.6 Homotopy colimit and cofibrant replacement have effective homology . 35Functorial cofibrant replacement . . . . . . . . . . . . . . . . . . . . 36

2.7 Effective abelian groups . . . . . . . . . . . . . . . . . . . . . . . . . . 372.8 Polycyclic groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

Computations with fully effective polycyclic groups . . . . . . . . . . 402.9 Eilenberg–MacLane spaces and diagrams . . . . . . . . . . . . . . . . . 42

Evaluation maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43Simplicial maps to 𝐸(𝜋, 𝑘) and 𝐾(𝜋, 𝑘) . . . . . . . . . . . . . . . . . 45Representing a map of diagrams by an effective cocycle . . . . . . . . 48A pullback from a fibration of Eilenberg–MacLane diagrams . . . . . 48Effective homology for 𝐸(𝜋, 𝑛) and 𝐾(𝜋, 𝑛) . . . . . . . . . . . . . . 49

3 Postnikov tower for diagrams . . . . . . . . . . . . . . . . . . . . . . . . 513.1 Reformulation of Theorem A . . . . . . . . . . . . . . . . . . . . . . . . 513.2 Description of the algorithm . . . . . . . . . . . . . . . . . . . . . . . . 523.3 Correctness of the algorithm . . . . . . . . . . . . . . . . . . . . . . . . 53

ix

The cochain 𝜅ef𝑘−1 is a cocycle. . . . . . . . . . . . . . . . . . . . . . 54The map 𝜙′𝑘 takes values in 𝑃 ′𝑘 . . . . . . . . . . . . . . . . . . . . . 54𝑃𝑘 and 𝜙𝑘 satisfy the properties of the Postnikov system . . . . . . . 55

3.4 Computing equivariant cohomology operations . . . . . . . . . . . . . . 564 How to decide if a map is homotopically trivial . . . . . . . . . . . . 59

Relative statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594.1 Computations with Postnikov towers . . . . . . . . . . . . . . . . . . . 604.2 Maps out of suspensions . . . . . . . . . . . . . . . . . . . . . . . . . . 61

Homotopy concatenation . . . . . . . . . . . . . . . . . . . . . . . . . 614.3 Deciding the existence of a homotopy . . . . . . . . . . . . . . . . . . . 62

An exact sequence associated with a fibration . . . . . . . . . . . . . 62Proof of Theorem D . . . . . . . . . . . . . . . . . . . . . . . . . . . 62Proof of (poly)𝑛−1 + (null)𝑛−1 → (poly)𝑛 . . . . . . . . . . . . . . 63

x

Foreword

This thesis contains the results of my research during my PhD studies.My initial assignment was to deal with the effective homology of twisted cartesian

products, namely to generalize F Seregraert’s previous results [39]. The generalizationwas needed for the paper [8].

The work on this issue turned to be relatively straightforward and after a year, Ipublished my results in [16]. In this thesis these results are contained in Section 2.4and the main result is stated as Corollary 2.21.

Together with L. Vokřínek, we used the methods introduced in [8] to give an al-gorithm that decides whether two simplicial maps are homotopic. Our result can befound in [15] and is contained here in a simplified version as Chapter 4.

Afterwards, my advisor and L. Vokřínek suggested a particular road map thatwould lead us to generalize one of the main results of [8] - an algorithm that for givenfinite simplicial sets 𝑋, 𝑌 with an action of a finite group 𝐺 computes the set [𝑋, 𝑌 ]𝐺of equivariant homotopy classes of maps, whereas [8] deals with the situation wherethe group 𝐺 acts only freely. Our general aim was to use Elmendorf’s theorem on anequivalence of the category of 𝐺-simplicial sets with a certain category of diagrams ofsimplicial sets. Hence our attention turned to working with diagrams of simplicial sets.

Following an idea of L. Vokřínek, I summed up some introductory technical resultsin article [17]. These are also utilized here in Section 2.5 and Section 3.4.

The main result of this thesis describes an algorithm that given a finite diagram of1-connected simplicial sets 𝑌 and a positive integer 𝑛, constructs the 𝑛-stage Postnikovsystem for 𝑌 . This serves as a generalization of [7] and is proved in Chapter 3.

xi

1 Motivation

In this introductory chapter, we will focus on algorithms that compute solutions ofclassical problems in algebraic topology. We will mainly concentrate on the followingproblems: decide whether topological spaces 𝑋, 𝑌 are homotopy equivalent, describethe structure of the set of homotopy classes [𝑋, 𝑌 ] of maps from 𝑋 to 𝑌 and given thefollowing diagram of spaces 𝐴,𝐵,𝑋, 𝑌 and maps 𝑖, 𝑝, 𝑓, 𝑔,

𝐴𝑔

//

𝑖��

𝑌

𝑝��

𝑋𝑓

//

𝑓 88

𝐵

(1.1)

determine whether there is a lift, i.e. the dotted arrow 𝑓 making the diagram commut-ative and classify all such lifts up to homotopy.

The last problem is known as the lifting–extension problem. If we set𝐵 = * (a point)then this is an extension problem and if 𝐴 = ∅, this problem is called a lifting problem.We will also deal with corresponding equivariant versions of tasks listed above, wherethe spaces are topological spaces with an action of a group 𝐺 and maps are equivariant.

Classical approach of algebraic topology is to solve these problems using algebraicinvariants such as homology and cohomology groups, K–theory, homotopy groups etc.

However, we can look at these problems from a computational and algorithmicperspective: Given a description of spaces 𝑋, 𝑌 , we ask whether there is an algorithmthat computes [𝑋, 𝑌 ] and similarly for the lifting–extension problem, we ask whetherthere exists an algorithm that for given 𝑓, 𝑔, 𝑖, 𝑝 decides the existence of an extension𝑓 and that computes all such extensions up to homotopy.

The first paper with this point of view was the paper [2] by E. H. Brown jr. In hiswork, he assumed that the spaces 𝑋, 𝑌 are represented as finite simplicial complexesand he then provided the following algorithms:

∙ Given 𝑋, 𝑌 simply connected simplicial complexes with finite homology groups,an algorithm decides whether they are homotopy equivalent.

∙ Given a finite subcomplex 𝐴 ⊆ 𝑋 and a map 𝑓 : 𝐴 → 𝑌 , where 𝑌 has finitehomology groups an algorithm decides whether 𝑓 can be extended to a map𝑓 : 𝑋 → 𝑌 .

∙ Assuming 𝑌 is simply connected, an algorithm computes 𝜋𝑛(𝑌 ) = [𝑆𝑛, 𝑌 ].

The main tool which Brown used was the construction of the Postnikov tower for thefinite simplicial complex 𝑌 in terms of simplicial sets.

In the construction, the Postnikov stages 𝑃𝑛 of the tower are simplicial complexes.Using the bijection [𝑋, 𝑌 ] ∼= [𝑋,𝑃𝑛] for 𝑋 such that dim𝑋 ≤ 𝑛, the computationof [𝑋,𝑃𝑛] and therefore [𝑋, 𝑌 ] is given by checking inductively which from the finitenumber of maps 𝑓 ∈ [𝑋,𝑃𝑗−1] can be lifted to 𝑃𝑗.

𝑃𝑗

��

𝑋𝑓

//

𝑓 88

𝑃𝑗−1

1

1. Motivation

Although the results of [2] lead for example to an algorithm computing the higherhomotopy groups of spheres, Brown himself remarked that the algorithms are imprac-tical for computations. Another problem of his algorithms lies within the restrictingcondition on the finiteness of the homology and homotopy groups.

Francis Sergeraert [39] introduced the notion of objects with effective homology.We will elaborate on the precise definition later, for now we only remark that it is acollection of algorithms that allow us to compute the homology groups of the spaceeven if the space is e.g. an infinite simplicial complex.

Together with his students and collaborators P. Real, A. Romero and J. Rubio,[33, 34, 36, 38, 39, 37], they further presented a wide range of algorithmic constructionswith such objects. Using these methods, one can for example compute data from someinstances of Serre or Eilenberg–Moore spectral sequences and recently also Bousfield-Kan spectral sequences. Many of the above–mentioned algorithms were implementedin a package for Common Lisp called Kenzo.

In a series of articles [41], R. Schön presented a different approach to algorithmic cal-culations and introduced his own method of computing the algebraic data from certainspectral sequences using calculable sequences of groups. A connection and comparisonbetween his methods and the methods of effective homology as e.g. in [39] is not en-tirely clear and, to the best of our knowledge, the algorithms he presented in [41] werenot implemented.

A. Nabutovsky and S. Weinberger in the article [31] sketched an algorithm that forpiecewise–linear or smooth simply connected manifolds 𝑀𝑛 and 𝑁𝑘, decides whetherthey are homeomorphic, diffeomorphic or piecewise–linear homeomorphic provided thatthe dimension of one of the manifolds is at least five. In [30, 31] they further providedexamples of problems that are not algorithmically solvable. Their result uses argumentfrom surgery theory and rational homotopy theory and is mainly based on an algorithmthat decides whether two 1–connected simplicial complexes have the same homotopytype. However details of this algorithm are not entirely clear.

Subsequent interest in the algorithmic computations was sparked by the group ofauthors J. Matoušek, M. Tancer and U. Wagner in their study [26] of the embeddabilityproblem:

Given a finite 𝑘–dimensional simplicial complex 𝐾, decide algorithmically whetherit can be embedded in R𝑛.

It can be deduced, that if 𝐾 embeds, then there exists a Z2-equivariant map(𝐾 × 𝐾) ∖ Δ𝐾 → 𝑆𝑛−1, where Δ𝐾 is the diagonal and 𝑆𝑛−1 is equipped with anantipodal action of Z2. In the metastable range, i.e. in the case 𝑘 ≤ 2

3𝑛 − 1, the con-

verse also holds. The authors were able to present an algorithmic solution or proveundecidability in many ranges of dimensions, but the metastable range remained anopen question.

We state the problem coming from the above embeddability problem as follows:We want to algorithmically decide whether the set of equivariant homotopy classes ofmaps

[𝐾 ×𝐾 ∖Δ𝐾 , 𝑆𝑛−1]Z2

is nonempty. Further, in the metastable range, there exists an abelian group structureon this set and we aim to describe this structure.

It is thus a special case of computing [𝑋, 𝑌 ]𝐺, i.e. the set of 𝐺-equivariant homotopyclasses of maps 𝑋 → 𝑌 for a finite group 𝐺. This served as a motivation to introducealgorithmic methods of computation in algebraic topology and homotopy theory in

2

1. Motivation

bigger generality than the algorithms described by Brown.A progress in this direction was achieved by a group of authors M. Čadek, M. Krčál,

J. Matoušek, F. Sergeraert, L. Vokřínek and U. Wagner. In paper [5] they presented analgorithm that computes [𝑋, 𝑌 ] for spaces 𝑋, 𝑌 given as finite simplicial sets satisfyingdim𝑋 ≤ 2 conn𝑌 and 1 ≤ conn𝑌 , where dim𝑋 denotes the dimension of 𝑋 andconn𝑌 the connectivity of 𝑌 .

The construction of their algorithm was further detailed in [7], where the compu-tational complexity of the algorithms was discussed.

Similar to the Brown’s result, the computation of [𝑋, 𝑌 ] was done using Postnikovtower {𝑃𝑛}𝑛≥0 for 𝑌 and the bijection [𝑋, 𝑌 ] ∼= [𝑋,𝑃𝑛], which holds for dim𝑋 ≤ 𝑛.The authors namely offered an algorithm that given a 1–connected finite simplicial set𝑌 and some 𝑛 ∈ N computes the Postnikov tower of 𝑌 .

The solution of the embeddability problem from [26] stated above was achievedin the article [8] by M.Čadek, M.Krčál, L.Vokřínek, where the authors extended theresults from [5, 7] to the case when a finite group 𝐺 acts freely on the spaces 𝑋, 𝑌 . Inparticular, they have obtained the following results:

∙ An algorithm, that given spaces 𝑋, 𝑌 as finite simplicial sets with a free actionof a finite group 𝐺 and assuming dim𝑋 ≤ 2 conn𝑌 , computes the group [𝑋, 𝑌 ]𝐺of equivariant homotopy classes.

∙ An algorithm that for the instance of lifting–extension problem (1.1), where thespaces 𝐴,𝐵,𝑋, 𝑌 are specified as finite simplicial sets with a free action of agroup 𝐺 and 𝑖, 𝑝, 𝑓, 𝑔 are equivariant maps, decides, whether a lift 𝑓 exists.

Further, in [15] myself and L.Vokřínek, used the result from paper [8] to derive analgorithm that decides whether two equivariant maps 𝑓, 𝑔 : 𝑋 → 𝑌 of finite simplicialsets with a free action of a finite group 𝐺, are homotopic assuming 𝑌 is simply con-nected. In order to do so, we further desribed an algorithm that computes the group[Σ𝑋, 𝑌 ]* of pointed homotopy classes of maps from a suspension Σ𝑋 to a simply con-nected simplicial set 𝑌 . We remark that this result generalizes the computation ofhomotopy groups of spheres described by Brown [2].

We remark that the algorithms presented in [5, 7, 8] further led to other resultssuch as proving that certain extension problems are undecidable [6], and were used e.g.to describe robust satisfiability of systems of equations [19].

1.1 Representing simplicial sets and simplicial maps in a com-puter

In this section, we outline how simplicial sets and maps are handled in a computer.The method we describe is originally due to Sergeraert [39] and was further used (withpossible slight modifications) in [5, 7, 8, 6, 15] and other articles, as a part of theframework of effective homology. In this thesis, we will utilize this approach as well.

Basic facts and definitions for simplicial sets and simplicial maps can be foundeither in Section 2.1 or in standard textbooks [27, 20].

Finite simplicial sets. We first describe how one can handle finite simplicial sets:Suppose 𝑋 is a simplicial set and that it is finite, i.e. the set of its nondegenerate

3

1. Motivation

simplices𝑋N is finite. Any simplex 𝑥 ∈ 𝑋 can be described as a sequence 𝑥 = 𝑠𝑖1 · · · 𝑠𝑖𝑡𝑦,where 𝑠𝑖𝑘 are the degeneracy operators and 𝑦 ∈ 𝑋N (see chapter 2.1).

To give a complete description of 𝑋, it is enough to describe how the faces of thenondegenerate simplices are glued together. These relations can be given in many ways,for example, they can all be described in a form 𝑑𝑗𝑥 = 𝑠𝑖1 · · · 𝑠𝑖𝑡𝑦, where 𝑥, 𝑦 ∈ 𝑋N and𝑑𝑗 is the face operator. Because there is only finitely many nondegenerate simplices,the list of all relations as above is also finite.

In conclusion, a complete description of a simplicial set 𝑋 can be obtained by afinite table of its nondegenerate simplices 𝑋N together with their dimensions and afinite table specifying the relations between the simplices.

A simplicial map 𝑓 : 𝑋 → 𝑌 between finite simplicial sets can be represented by afinite table that lists the images of the nondegenerate simplices of 𝑋.

Locally effective simplicial sets. In the calculations, we often encounter a situation,when we have to work with infinite simplicial sets such as the Eilenberg–MacLanespaces or e.g. any Kan complexes. In these cases, the simplicial sets are assumed to belocally effective simplicial sets.

The main idea of this concept is to focus on a local description of a simplicial setonly: Let 𝑋 be a simplicial set. We say that 𝑋 is locally effective if we are given aspecified encoding of the simplices of 𝑋 and a collection of algorithms computing theface and degeneracy operations on any simplex of 𝑋. We remark that we have noglobal information here, for example in general we are not able to output a full list ofnondegenerate simplices of a given dimension.

For maps of locally effective simplicial sets, we say that a map 𝑓 : 𝑋 → 𝑌 iscomputable if there is an algorithm that for any simplex 𝑥 ∈ 𝑋 computes the encodingof 𝑓(𝑥).

As a special case, we remark that any finite simplicial set represented by a finitetable as described above can be seen as a locally effective simplicial set. Any sim-plex of 𝑋 is encoded by a list of degeneracies applied to one nondegenerate simplex𝑠𝑖1 · · · 𝑠𝑖𝑡𝑥 ↦→ (𝑖1, · · · , 𝑖𝑡, enc(𝑥)).

1.2 Postnikov tower for simplicial sets

In this section, we give a definition of the Postnikov tower of a simplicial set and wediscuss how this construction is used in the papers [5, 7].

Let 𝑌 be a simplicial set, a (simplicial) Postnikov tower for 𝑌 is the following

4

1. Motivation

commutative diagram of simplicial sets and maps

𝑃𝑛

𝑝𝑛

��

𝑃𝑛−1

𝑌

𝜙𝑛

@@

𝜙𝑛−1

77

𝜙1

//

𝜙0''

𝑃1

𝑝1��

𝑃0

where the following conditions are satisfied:

∙ The induced map 𝜙𝑛* : 𝜋𝑘(𝑌 )→ 𝜋𝑘(𝑃𝑛) is an isomorphism for 0 ≤ 𝑘 ≤ 𝑛.

∙ 𝜋𝑘(𝑃𝑛) ∼= 0 for 𝑘 ≥ 𝑛+ 1

∙ 𝑃𝑛 is the pullback in the diagram

𝑃𝑛//

𝑝𝑛

��

𝐸(𝜋𝑛(𝑌 ), 𝑛)

𝛿��

𝑃𝑛−1𝑘′𝑛

// 𝐾(𝜋𝑛(𝑌 ), 𝑛+ 1)

The symbols 𝜋𝑛(𝑌 ) denote the 𝑛–th homotopy group of the geometric realization ofthe simplicial set 𝑌 , 𝐾(𝜋𝑛(𝑌 ), 𝑛+1) is the Eilenberg–MacLane space and 𝐸(𝜋𝑛(𝑌 ), 𝑛)its path space.

The space 𝑃𝑛, called the 𝑛–th Postnikov stage, can be considered as a homotopyapproximation of 𝑌 up to the dimension 𝑛 since for a finite dimensional simplicial set 𝑋with dim𝑋 ≤ 𝑛 there is a bijection between the sets of homotopy classes of simplicialmaps [𝑋, 𝑌 ] ∼= [𝑋,𝑃𝑛]. Hence the space 𝑌 and maps 𝑋 → 𝑌 can be replaced by aPostnikov stage 𝑃𝑛 and maps 𝑋 → 𝑃𝑛. This has two advantages:

1. Unless the simplicial set 𝑌 is a so–called Kan complex (see [27], Chapter 1 orSection 2.1) it is hard to define homotopy classes of maps: We say simplicial maps𝑓, 𝑔 : 𝑋 → 𝑌 are homotopic and denote this by 𝑓 ∼ 𝑔, if there exists a simplicialmap 𝐻 : 𝑋 ×Δ1 → 𝑌 such that 𝐻|𝑋×{0} = 𝑓 and 𝐻|𝑋×{1} = 𝑔. Further, by thesymbol [𝑋, 𝑌 ] we traditionally mean [|𝑋|, |𝑌 |] i.e. the set of homotopy classes ofmaps of geometric realizations of the simplicial sets 𝑋, 𝑌 (details can be foundin [11] and in Chapter 2).

We can define a map sSet(𝑋, 𝑌 )/∼ → [𝑋, 𝑌 ], but if 𝑌 is not Kan this map ingeneral fails to be a bijection or it even cannot be defined at all. However, thePostnikov stages 𝑃𝑛 are Kan complexes.

2. The algorithm constructing the Postnikov tower, originally presented in [7] andfurther generalized in [8] computes [𝑋,𝑃𝑛] by gradually computing [𝑋,𝑃𝑘], 𝑘 < 𝑛and using the long exact sequence of the fibration 𝐾(𝜋𝑘(𝑌 ), 𝑘)→ 𝑃𝑘 → 𝑃𝑘−1.

5

1. Motivation

We remark, that during the algorithmic construction as in [7], the homotopy groups𝜋𝑘(𝑌 ) are computed. They are obtained from Hurewicz theorem on the mapping coneof the map 𝜙𝑘−1 as the mapping cone is (𝑘 − 1)–connected and 𝐻𝑘(cone𝜙𝑘−1) ∼=𝜋𝑘(cone𝜙𝑘−1) ∼= 𝜋𝑘(𝑌 ). It remains to compute this homology group. Although thismay seem simple in theory, the spaces such as 𝑃𝑘−1 and therefore also cone𝜙𝑘−1 areinfinite, so this computation is not straightforward.

In the papers [5, 7], the authors used Sergeraert’s framework of effective homologyor simplicial sets with effective homology to overcome this obstacle. This allowed themto compute the homology even for the infinite locally effective simplicial sets.

1.3 Effective homology

A finite simplicial set 𝑋 describes also the chain complex 𝐶*(𝑋), generated by thenondegerate 𝑛–simplices. Thus the chain complex 𝐶*(𝑋) consists of finitely many fi-nitely generated abelian groups and we can compute the homology groups 𝐻*(𝑋).

Given a simplicial set 𝑋 and a chain 𝑐 ∈ 𝐶𝑛(𝑋), we want to be able to computethe following tasks:

∙ decide whether 𝑐 is a cycle,

∙ decide whether 𝑐 is a boundary 𝑐 = 𝜕𝑐′ of some 𝑐′ ∈ 𝐶𝑛+1(𝑋) and compute 𝑐′.

For a finite simplicial set this is easy. Otherwise, the groups 𝐶𝑛(𝑋) can have infinitelymany generators, so it is not clear how to compute the desired tasks in a finite time.

We overcome this problem by introducing an effective chain complex 𝐶ef* (𝑋), where

the tasks listed above are computable and a so called strong equivalence between𝐶ef* (𝑋) and 𝐶*(𝑋). This is denoted by 𝐶*(𝑋)⇐⇐⇒⇒ 𝐶ef

* (𝑋). We postpone the definitionof the strong equivalence to Section 2.3. The proper definition of an effective chaincomplex will also be given later. Here, for the purposes of this chapter, we only describethe main point:

∙ A chain complex 𝐶* is called effective if for any 𝑛 ∈ N0, we can compute thefinite list of generators {𝑐𝛼} of 𝐶𝑛, 𝛼 ∈ 𝐴, where 𝐴 is a finite set and every chain𝑐 ∈ 𝐶𝑛 can be expressed uniquely as a combination

𝑐 =∑

𝑘𝛼𝑐𝛼

with integer coefficients 𝑘𝛼 in Z.

We say that a locally effective simplicial set 𝑋 has effective homology if 𝐶*(𝑋) ⇐⇐⇒⇒𝐶ef* (𝑋) and 𝐶ef

* (𝑋) is an effective chain complex.We remark that in order to solve the cycle and boundary problems algorithmically,

it suffices to find a (computable) chain homotopy equivalence 𝐶*(𝑋) ≃ 𝐶ef* (𝑋). We use

strong equivalence, because it enables us to utilize so called perturbation lemmas (seechapter 2.3). This allows us to introduce algorithmic constructions that given simplicialsets with effective homology as inputs, produce simplicial sets with effective homologyon the outputs. The mapping cone or the mapping cylinder are computed this way.

6

1. Motivation

1.4 Our motivation

We are interested in extending the results from [8, 15] to the case where the action ofa finite group 𝐺 on spaces 𝑋, 𝑌 is not free.

This could potentially be applied to provide a solution of the generalization ofthe embeddability problem: To decide if a 𝑘-dimensional simplicial complex 𝐾 can beembedded into R𝑛, where the image of 𝐾 may intersect itself at most 𝑘 times.

Again, in some ranges of dimensions, this corresponds to a problem of the existenceof a Σ𝑘 equivariant map 𝐾 ′ → 𝑆𝑛, where the action of the symmetric group Σ𝑘 is notfree on the space 𝐾 ′, a version of the deleted product for 𝐾.

As the algorithmic construction of the Postnikov tower for finite simplicial sets andthe corresponding version for finite simplicial sets with free action of a finite group isone of the main tools used in [8, 15], the algorithmic construction of the Postnikovtower for finite simplicial sets with a non–free action is the obvious way to generalizeresults in [8, 15] to the equivariant setting.

The main aim of this thesis is therefore to obtain a statement (theorem) that canbe informally written as follows:

Informal statement. Let 𝑌 be a simplicial set with an action of a finite group 𝐺.Then there is an algorithm that computes the equivariant Postnikov tower for 𝑌 .

However, instead of the Informal statement we will prove a result involving finitediagrams of simplicial sets, which will enable us to approach the equivariant Postnikovtowers from a different perspective.

In the rest of the text, by a finite diagram 𝑋 in a category 𝒞 (or a finite diagram ofobjects in that category), we mean a functor 𝑋 : ℐ → 𝒞, where ℐ is a finite category,i.e. a category with finitely many objects and arrows.

For the sake of better readability of the remainder of this thesis, we will stress thefact that an object is a diagram by using the boldface hyphenation. This notation allowscertain inconsistencies which we hope won’t be very confusing: Suppose 𝑋 : ℐ → Topis a diagram of spaces, then 𝑋(𝑖), where 𝑖 ∈ ℐ is a topological space, but it is (partly)highlighted in boldface.

A pedantic reader might also remark that the category of simplicial sets is a categoryof diagrams of sets but the notation is not emphasized. We are aware of this, as isdocumented by this paragraph.

The main effort in this thesis will be concentrated to prove the following theorem

Theorem A. Let ℐ be a finite category and let 𝑌 : ℐ → sSet be a diagram of 1–connected simplicial sets. Given 𝑛 ∈ N, there is an algorithm that computes the 𝑛-stagePostnikov tower for 𝑌 .

In the next section, we will demonstrate that the equivariant Postnikov tower for a𝐺–simplicial set 𝑌 can be replaced by a tower of diagrams for a special diagram ℐ. Thereason why this is true follows from the fact that problems in the homotopy categoryof simplicial sets with an action of 𝐺 can be restated as problems in the homotopycategory of certain finite diagrams of simplicial sets.

1.5 Category of orbits

In this section we will be using some language of model categories. Some details onmodel categories can be found in Sections 2.1 and 2.2. For the full description, we refer

7

1. Motivation

to [11]. For the purposes of this chapter it is enough to say that a model structure ona category 𝒞 allows us to define the set [𝑋,𝑋 ′]𝒞 of homotopy classes of maps from 𝑋to 𝑋 ′ in the category 𝒞, where 𝑋,𝑋 ′ ∈ 𝒞.

Further, for model categories 𝒞,𝒟 one can define special adjunctions, called Quillen

adjunctions. Roughly speaking, these are adjuctions (𝐿 ⊣ 𝑅) : 𝒞𝑅←→𝐿𝒟 respecting the

model category structure. Quillen adjunctions further induce functors L : 𝒞 → 𝒟 andR : 𝒟 → 𝒞. These constitute a Quillen equivalence if for any objects 𝑋,𝑋 ′ ∈ 𝒞 and𝑌, 𝑌 ′ ∈ 𝒟 we have [𝑋,𝑋 ′]𝒞 ∼= [L(𝑋),L(𝑋 ′)]𝒟 and in the opposite direction [𝑌, 𝑌 ′]𝒟 ∼=[R(𝑑),R(𝑌 ′)]𝒞.

Simplicial sets with a group action. Given a finite group 𝐺, simplicial sets witha 𝐺–action and 𝐺–equivariant simplicial maps between them form the category sSet𝐺,which is sometimes called category of 𝐺-simplicial sets. It is a model category, we candefine a notion of homotopy and we denote the set of homotopy classes of maps in thiscategory by [𝑋, 𝑌 ]𝐺. For details, see Chapter 1 in [28].

Further, there is a category of orbits 𝒪𝐺, where the objects are orbit sets 𝐺/𝐻,where 𝐻 ≤ 𝐺 and the morphisms are equivariant maps 𝐺/𝐻1 → 𝐺/𝐻2. As 𝐺 isassumed to be finite, the category 𝒪𝐺 is finite and so is the category 𝒪𝐺

op. An object𝑋 in the category sSet𝒪

op𝐺 of functors 𝒪𝐺

op → sSet is thus a finite diagram of simplicialsets. Any category of diagrams of simplicial sets sSetℐ admits a model structure calledthe projective model structure, which we are going to use. We can define homotopyand denote the set of homotopy classes of maps in the category sSetℐ by [−,−]ℐ .

We define a functor Φ: sSet𝐺 → sSet𝒪op𝐺 called the fixed–point functor by

Φ(𝑋)(𝐺/𝐻) = 𝑋𝐻 = {𝑥 ∈ 𝑋 | ℎ𝑥 = 𝑥,∀ℎ ∈ 𝐻}.

The functor Φ assigns a finite diagram of simplicial sets to every 𝑋 ∈ sSet𝐺.We illustrate this in the following example

Example 1.1. Assume 𝐺 = C2 = {1,−1} i.e. a two–element group. The category 𝒪𝐺

looks as follows:

C2/C2

id

C2/{1}𝜄oo

id

−1

UU

and given 𝑋 ∈ sSet𝐺, Φ(𝑋) ∈ sSet𝒪op𝐺 is the following diagram. Notice that arrows are

reversed.

𝑋C2

id

��Φ(𝜄)

// 𝑋

id

��

−1

EE

By Elemendorf’s theorem [14] the categories sSet𝐺 and sSet𝒪op𝐺 are Quillen equival-

ent, which in particular implies that [𝑋, 𝑌 ]𝐺 ∼= [Φ𝑋,Φ𝑌 ]𝒪𝐺. For details see e.g. chapter

V in [28] or [14, 43].To sum up, many computational problems in 𝐺–simplicial sets can be restated as

computational problems in the category sSet𝒪op𝐺 . Instead of computing the Postnikov

8

1. Motivation

tower for 𝑌 in the category sSet𝐺 as suggested by the Informal statement, we computethe Postnikov tower for Φ(𝑌 ) using Theorem A.

In the following section, we will outline how the construction of the Postnikov towerfor diagrams of simplicial sets differs from the algorithmic construction described in[7].

Effective homology for diagrams. The algorithm that constructs the Postnikovtower as seen in [7, 8] uses the simplicial sets with effective homology. In the nextsection, we will present the main idea of the proof of Theorem A. Because we want tobuild a Postnikov tower for diagrams, we will define effective homology of diagrams ofsimplicial sets.

In fact we introduce two notions of effective homology that generalize the effectivehomology from [39] namely a diagram with pointwise effective homology and a dia-gram with effective homology. As with the effective homology, we postpone the properdefinitions and describe just the main ideas:

∙ We say that a diagram of chain complexes 𝐶 : ℐ → Ch+ has pointwise effectivehomology if for every 𝑖 ∈ ℐ there is given an effective chain complex 𝐶ef(𝑖) anda strong equivalence of chain complexes 𝐶(𝑖)⇐⇐⇒⇒ 𝐶ef(𝑖).

∙ A diagram of simplicial sets 𝑋 : ℐ → sSet has pointwise effective homology iffor every 𝑖 ∈ ℐ there is given an effective chain complex 𝐶ef(𝑖) and a strongequivalence of chain complexes 𝐶(𝑋(𝑖))⇐⇐⇒⇒ 𝐶ef(𝑖)

∙ A diagram of chain complexes 𝐶 : ℐ → Ch+ is effective if for any 𝑛 ∈ N0 we cancompute a finite list of generators {𝑐𝛼 | 𝛼 ∈ 𝐴}, where 𝑐𝛼 ∈ 𝐶(𝑖𝛼) and 𝐴 is afinite set. Further, given an element 𝑐 ∈ 𝐶(𝑖), there is an algorithm that outputsa unique description of 𝑐 as

𝑐 =∑

𝛼,𝑓𝛼 : 𝑖𝛼→𝑖

𝑘𝑓𝛼𝑓𝛼*(𝑐𝛼)

where 𝑓𝛼* : 𝐶(𝑖𝛼)→ 𝐶(𝑖) and 𝑘𝑓𝛼 ∈ Z.

∙ A diagram of chain complexes 𝐶 : ℐ → Ch+ has effective homology if there isgiven a strong equivalence of diagrams 𝐶 ⇐⇐⇒⇒ 𝐶ef where 𝐶ef is an effectivediagram.

∙ A diagram of simplicial sets 𝑋 : ℐ → sSet has effective homology if there is givena strong equivalence 𝐶*(𝑋) ⇐⇐⇒⇒ 𝐶ef

* between the diagram of chain complexesfor 𝑋 and some effective chain complex 𝐶ef

* .

The pointwise effective homology is a weaker notion, because we do not need any in-formation regarding the maps between the chain complexes 𝐶ef(𝑖). In fact, any diagramthat has effective homology has pointwise effective homology.

From a historical standpoint, the effective homology of a diagram was first intro-duced in [8] for two particular diagrams and this served as a motivation for our generaldefinition. We present these diagrams in the following examples.

Example 1.2. Let ℐ be a category with two objects 𝑖0, 𝑖1 and one nonidentity arrow𝑓 : 𝑖0 → 𝑖1. Then a diagram of chain complexes 𝐶 : ℐ → Ch+ is effective, if

9

1. Motivation

∙ The morphism 𝐶(𝑖0)→ 𝐶(𝑖1) is an inclusion.

∙ Chain complexes 𝐶(𝑖1),𝐶(𝑖0) are effective and generators of 𝐶(𝑖0) are generatorsin 𝐶(𝑖1).

Example 1.3. It is a classical example from the category theory that a chain complex𝐶 with an action of a finite group 𝐺 is nothing else than a diagram 𝐶 : 𝐺 → Ch+,where the group 𝐺 is interpreted as a category with one object * and arrows labelledby the elements of 𝐺. Assuming 𝐶 is effective, 𝐶 can be seen as a Z𝐺-module with afinite Z𝐺-basis.

In other words, for each 𝑛 ≥ 0 there is a finite list of distinguished elements {𝑐𝛼}of 𝐶𝑛 such that they are all distinct and each 𝑐 ∈ 𝐶𝑛 has a unique description

𝑐 =∑

𝑘𝛼𝑔𝛼𝑐𝛼

where the coefficients 𝑘𝛼 lie in Z and 𝑔𝛼 ∈ 𝐺.

1.6 Postnikov tower for diagrams

In this section, we summarize the gist of our construction of the Postnikov tower fordiagrams of simplicial sets and therefore also the essence of the proof of Theorem A.

First, we remark that the algorithm that constructs the Postnikov tower for sim-plicial set 𝑌 as seen in [7, 8] is based on repeating one construction step iteratively:

∙ Given a map 𝜙𝑛−1 : 𝑌 → 𝑃𝑛−1 of simplicial sets with effective homology, thealgorithm outputs the following diagram of simplicial sets with effective homology.In particular, we compute 𝜙𝑛 : 𝑌 → 𝑃𝑛.

𝑃𝑛

𝑝𝑛����

// 𝐸(𝜋𝑛, 𝑛)

����

𝑌𝜙𝑛−1

//

𝜙𝑛

::

ℓ𝑛

&&

𝑃𝑛−1𝑘𝑛 // 𝐾(𝜋𝑛, 𝑛+ 1)

Our aim is to generalize this to diagrams.Similar to the construction of the Postnikov tower for simplicial sets our algorithm

will iteratively construct the Postnikov stages 𝑃𝑛 and maps 𝜙𝑛 : 𝑌 → 𝑃𝑛. The situationis, however more complicated and we describe this in more detail:

The algorithm first computes the diagram of homotopy groups 𝜋𝑛(𝑌 ) : ℐ → Abfrom the morphism of diagrams 𝜙𝑛−1 : 𝑌 → 𝑃𝑛−1.

Then we construct a diagram 𝑃 ′𝑛 and a morphism 𝜙′𝑛 : 𝑌 → 𝑃 ′𝑛. We remark thatto do so, we require that 𝑌 and 𝑃𝑛−1 have effective homology. From the homotopystandpoint, the diagram 𝑃 ′𝑛 is the next Postnikov stage, but it has pointwise effectivehomology only.

Fortunately, 𝑃 ′𝑛 can be “upgraded” to a diagram that has effective homology bycomputing a specific replacement 𝑃𝑛 = 𝑃 ′𝑛

cof , introduced in the next chapter, togetherwith a weak equivalence 𝑃𝑛 → 𝑃 ′𝑛. Further, this construction is algorithmic.

What remains is a morphism 𝑌 → 𝑃𝑛. In general, it does not have to exist at all.We overcome this obstacle in the following way: Since our replacement construction is

10

1. Motivation

functorial and computable, we replace 𝜙′𝑛 : 𝑌 → 𝑃 ′𝑛 by a map of diagrams that haseffective homology 𝜙 : 𝑌 cof → 𝑃𝑛.

Using a notation 𝑌0 = 𝑌 and 𝑌𝑛 = 𝑌 cof𝑛−1, we sum up the algorithmic iterative step

of the construction as a succession of the following two computations:

∙ Given 𝜙𝑛−1 : 𝑌𝑛−1 → 𝑃𝑛−1 as a map of diagrams that have effective homology, onealgorithmically constructs a Postnikov stage 𝑃 ′𝑛 as a diagram which has pointwiseeffective homology and a computable map 𝜙′𝑛 : 𝑌𝑛−1 → 𝑃 ′𝑛.

∙ Given a map of diagrams 𝜙′𝑛 : 𝑌𝑛−1 → 𝑃 ′𝑛 that have pointwise effective homology,one algorithmically constructs the map 𝜙𝑛 : 𝑌𝑛 → 𝑃𝑛 between diagrams whichhave effective homology.

We picture the whole construction as follows:

𝑃 ′cof𝑛 = 𝑃𝑛

��

𝑌𝑛 = 𝑌 cof𝑛−1

𝜙𝑛

88

��

𝑃 ′𝑛

𝑝𝑛����

// 𝐸(𝜋𝑛, 𝑛)

����

𝑌𝑛−1𝜙𝑛−1

//

𝜙′𝑛

88

𝑃𝑛−1𝑘𝑛 //𝐾(𝜋𝑛, 𝑛+ 1)

11

2 Tools

2.1 Simplicial sets

In this section, we give a brief introduction to simplicial sets and their homotopy theory.We omit many details and refer the reader to standard textbooks [27, 20].

A simplicial set 𝑋 can be seen as a graded set 𝑋 indexed by the non-negative in-tegers together with a collection of mappings 𝑑𝑖 : 𝑋𝑛 → 𝑋𝑛−1 and 𝑠𝑖 : 𝑋𝑛 → 𝑋𝑛+1, 0 ≤𝑖 ≤ 𝑛 called the face and degeneracy operators. They satisfy the following identities:

𝑑𝑖𝑠𝑖 = 𝑑𝑖+1𝑠𝑖 = id; 𝑑𝑖𝑠𝑗 = 𝑠𝑗𝑑𝑖−1 𝑖 > 𝑗 + 1;𝑑𝑖𝑑𝑗 = 𝑑𝑗−1𝑑𝑖; 𝑑𝑖𝑠𝑗 = 𝑠𝑗−1𝑑𝑖 𝑖 < 𝑗;𝑠𝑖𝑠𝑗 = 𝑠𝑗+1𝑠𝑖; 𝑖 ≤ 𝑗.

Simplicial maps i. e. morphisms of simplicial sets are then defined as maps of gradedsets which commute with the face and degeneracy mappings. The simplicial sets andsimplicial maps form a category that we denote sSet.

The elements of 𝑋𝑛 are called 𝑛-simplices. We say that a simplex 𝑥 ∈ 𝑋𝑛 is nonde-generate if it cannot be expressed as 𝑥 = 𝑠𝑖𝑦 for some 𝑦 ∈ 𝑋𝑛−1. We will denote theset of nondegenerate simplices of 𝑋 by 𝑋N.

Using the relations between the face and degeneracy operators, one can deduce thatany 𝑛-simplex 𝑥 can be described uniquely as 𝑠𝑖1 · · · 𝑠𝑖𝑡𝑦, where 𝑦 is a nondegenerate(𝑛− 𝑡) simplex, 0 ≤ 𝑖1 ≤ 𝑖2 ≤ · · · < 𝑖𝑡.

A typical example of a simplicial set is the standard 𝑛–simplex, denoted Δ𝑛, whichcan be seen as a simplicial set freely generated by a single nondegenerate element𝑒𝑛 ∈ Δ𝑛

𝑛.Erasing the unique 𝑛-simplex 𝑒𝑛 of Δ𝑛 together with its face 𝑑𝑘𝑒𝑛 and all their

degeneracies, produces a new simplicial set, commonly called the 𝑘-th horn of Δ𝑛. Wedenote this simplicial set by 𝑛

𝑘 .Given a simplicial set 𝑋, we define the 𝑛-skeleton sk𝑛𝑋 as the simplicial set gen-

erated by the nondegenerate simplices of 𝑋 of dimension ≤ 𝑛. Alternatively, one canpicture this as removing all nondegenerate simplices of dimension greater than 𝑛 to-gether with all their degeneracies. As an example for the standard 𝑛-simplex Δ𝑛, where𝑛 ≥ 0, we can define its boundary 𝜕Δ𝑛 = sk𝑛−1𝑋. We remark that sk−1𝑋 = ∅.

A (simplicial) nerve of a small category 𝒞 is a simplicial set 𝑁(𝒞) defined in thefollowing way: 𝑁(𝒞)0 consists of the objects of 𝒞 and 𝑁(𝒞)𝑘, 𝑘 ≥ 1 is a collection ofthe 𝑘-tuples of composable morphisms 𝐶0 → 𝐶1 → · · · → 𝐶𝑘 in 𝒞. The degeneracymorphisms 𝑠𝑖 are defined by inserting the identity morphism at the 𝑖-th object. Theface operator 𝑑𝑖, 0 < 𝑖 < 𝑘 is given by composing the 𝑖-th morphism with the (𝑖+1)-stand the outer face maps 𝑑0, 𝑑𝑘 are given by erasing the first and the last morphism,respectively. We will encounter the nerve of a category later.

For a simplicial set 𝑋, we can define chain complexes 𝐶 full* (𝑋) and 𝐶N

* (𝑋). Themain difference between these chain complexes is that the group 𝐶 full

𝑛 (𝑋) is a freeabelian group generated by the elements of 𝑋𝑛, whereas 𝐶N

* (𝑋), called normalizedchain complex, is defined by 𝐶N

* (𝑋) = 𝐶 full* (𝑋)/𝐶D

* (𝑋), where 𝐶D* (𝑋) is chain sub-

complex generated by the degenerate elements). The boundary operator 𝜕 on bothchain complexes, is defined by 𝜕(𝑐) =

∑𝑛𝑖=0(−1)𝑖𝑑𝑖(𝑐). We remark that chain com-

plexes 𝐶 full* (𝑋) and 𝐶N

* (𝑋) are chain homotopy equivalent. In the rest of the text, ifnot stated otherwise, the symbol 𝐶*(𝑋) denotes 𝐶N

* (𝑋).

13

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Fibrations, cofibrations and weak equivalences. We call a simplicial map𝑔 : 𝑋 → 𝑌 fibration or Kan fibration if for any commutative diagram

𝑛𝑘

//

𝜄��

𝑋

𝑔

��

Δ𝑛 //

88

𝑌

there is a lift Δ𝑛 → 𝑋. Here 𝜄 denotes the inclusion of the horn into the standardsimplex.

To every simplicial set 𝑋, there is an associated topological space (in fact a CW–complex) |𝑋| called the geometric realization of 𝑋. The space |𝑋| is constructed byappointing a standard topological 𝑛–simplex to every nondegenerate 𝑛–simplex of 𝑋and gluing the faces according to the face and degeneracy relations specified by 𝑋. Thegeometric realization is a functor | − | : sSet→ Top.

The homotopy groups 𝜋*(𝑋) of a simplicial set 𝑋 are defined as the homotopygroups of the geometric realization 𝜋*(|𝑋|). We remark that for Kan complexes thereis an alternative definition that does not involve the geometric realization, see Chapter1 in [27].

We call a simplicial map 𝑔 : 𝑋 → 𝑌 a weak equivalence if the induced homomorph-isms 𝑔𝑛 : 𝜋𝑛(|𝑋|)→ 𝜋𝑛(|𝑌 |), 𝑛 ≥ 0 are isomorphisms.

We finally say that a simplicial map 𝑓 : 𝐴→ 𝐵 is a cofibration if it has the so–calledleft lifting property (LLP) with respect to maps that are both weak equivalences andfibrations: For every commutative diagram in sSet

𝐴 //

𝑓��

𝑋

𝑔��

𝐵 //

88

𝑌

such that 𝑔 is a weak equivalence and a fibration, there exists a lift in the diagram.We remark without a proof that in this model category structure, the cofibrations

correspond to injective maps and thus for any 𝑋 ∈ sSet the unique inclusion ∅ → 𝑋 isa cofibration (see [20], p.122).

A simplicial set 𝑋 is called a Kan complex or fibrant if the unique simplicial map𝑋 → Δ0 = {*} is a fibration. Not every simplicial set is fibrant, but the theory ofmodel categories tells us that for any 𝑋 ∈ sSet there exists some 𝑋 ′ that is fibrant anda weak equivalence 𝑋 → 𝑋 ′. Such 𝑋 ′ is called a fibrant replacement of 𝑋 and denoted𝑋fib.

We remark that weak equivalence, fibration, cofibration and the fibrant replace-ment are standard notions from the theory of model categories (see [11]) and in ourcontext they describe the Kan model structure on the category sSet. The homotopycategory Ho(sSet) induced by the model structure on sSet then corresponds (is Quillenequivalent) to the classical homotopy category for topological spaces Ho(Top).

Twisted products. We first remind the reader of the Cartesian product 𝑋 × 𝑌 ofsimplicial sets 𝑋, 𝑌 : The set of 𝑛-simplices (𝑋 × 𝑌 )𝑛 consists of tuples (𝑥, 𝑦), where𝑥 ∈ 𝑋𝑛, 𝑦 ∈ 𝑌𝑛. Further the face and degeneracy operators on 𝑋 × 𝑌 are defined by𝑑𝑖(𝑥, 𝑦) = (𝑑𝑖𝑥, 𝑑𝑖𝑦), 𝑠𝑖(𝑥, 𝑦) = (𝑠𝑖𝑥, 𝑠𝑖𝑦).

A simplicial group 𝐺 is a simplicial set such that each 𝐺𝑛 is a group and 𝑑𝑖, 𝑠𝑖 aregroup homomorphisms.

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We will deal with certain fiber bundles where 𝐹 , 𝐵, and 𝐸 are simplicial sets, anda simplicial group 𝐺 acts on 𝐹 . The action i.e. a simplicial map 𝜑 : 𝐹 ×𝐺→ 𝐹 satisfiesthe usual conditions for a (right) action of a group on a set; that is, 𝜑(𝛾𝛾′) = (𝜑𝛾)𝛾′

and 𝜑𝑒𝑛 = 𝜑 (𝜑 ∈ 𝐹𝑛, 𝛾, 𝛾′ ∈ 𝐺𝑛, 𝑒𝑛 the unit element of 𝐺𝑛).We can now define a twisted Cartesian product.

Definition 2.1 (Twisted Cartesian product). Let 𝐵 and 𝐹 be simplicial sets and 𝐺 asimplicial group acting on 𝐹 . Let 𝜏 be a mapping of graded sets 𝐵 → 𝐺 of degree −1,such that

(i) 𝑑0𝜏(𝛽) = 𝜏(𝑑1𝛽)𝜏(𝑑0𝛽)−1;

(ii) 𝑑𝑖𝜏(𝛽) = 𝜏(𝑑𝑖+1𝛽) for 𝑖 ≥ 1;(iii) 𝑠𝑖𝜏(𝛽) = 𝜏(𝑠𝑖+1𝛽) for all 𝑖; and(iv) 𝜏(𝑠0𝛽) = 𝑒𝑛 for all 𝛽 ∈ 𝐵𝑛, where 𝑒𝑛 is the unit element of 𝐺𝑛.

Then 𝜏 is called a twisting operator and the twisted Cartesian product 𝐹 ×𝜏 𝐵 is asimplicial set 𝐸 with 𝐸𝑛 = 𝐹𝑛×𝐵𝑛, i.e., the 𝑛-simplices are the same as in the Cartesianproduct 𝐹 × 𝐵, and the face and degeneracy operators are also as in the Cartesianproduct, i.e. 𝑑𝑖(𝑓, 𝑏) = (𝑑𝑖𝑓, 𝑑𝑖𝑏) , with the sole exception of 𝑑0, which is given by

𝑑0(𝜑, 𝛽) B (𝑑0(𝜑)𝜏(𝛽), 𝑑0𝛽), (𝜑, 𝛽) ∈ 𝐹𝑛 ×𝐵𝑛.

A twisted Cartesian product 𝐹 ×𝜏 𝐵 is called principal if 𝐹 = 𝐺 and the consideredright action of 𝐺 on itself is by (right) multiplication.

Thus, the only way in which 𝐹 ×𝜏 𝐵 differs from the ordinary Cartesian product𝐹 ×𝐵 is in the 0-th face operator. It is not trivial to see why this should be the rightway of representing fiber bundles simplicially, but for us, it is only important that itworks, and we will have explicit formulas available for the twisting operator for all thespecific applications.

2.2 Diagrams of simplicial sets

In this section, we define homotopy, homology and cohomology groups in the categorysSetℐ , i.e. a category of functors (or diagrams) ℐ → sSet for some fixed category ℐ,which we assume to be finite. Then we describe in more details the model categorystructure on sSetℐ and finally we present Bousfield–Kan model of a homotopy left Kanextension, which as a special case gives us models for a cofibrant replacement and ahomotopy colimit.

Homotopy and homology. We aim to define homotopy groups of diagrams 𝑋 : ℐ →sSet in such a way they can be seen as functors 𝜋𝑘(−) : sSetℐ → Grpℐ . To do so, wewill assume that 𝑋(𝑖) are simply connected and thus 𝜋𝑘(𝑋(𝑖)) do not depend onbasepoints.

To define homotopy groups for diagrams of simplicial sets that are not simply con-nected, basepoints have to be introduced, possibly as a subdiagram pt of 𝑋. Howeverin this situation, 𝜋𝑘 in general does not appear as a functor sSetℐ → Grpℐ .

Definition 2.2. Let 𝑋 : ℐ → sSet be a diagram of simply connected simplicial sets.We define the 𝑘-th homotopy group 𝜋𝑘(𝑋) of 𝑋 as a diagram ℐ → Set satisfying

𝜋𝑘(𝑋)(𝑖) = 𝜋𝑘(𝑋(𝑖)), 𝑖 ∈ ℐ

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and the maps in the diagram 𝜋*(𝑋) are given as follows: for any 𝑓 : 𝑖→ 𝑗, 𝑖, 𝑗 ∈ ℐ wehave

𝜋*(𝑋)(𝑓) = 𝑋(𝑓)* : 𝜋*(𝑋(𝑖))→ 𝜋*(𝑋(𝑗)).

Later in the text, we will work with relative homotopy groups for a pair (𝑋,𝐴).Here 𝐴 is a subdiagram of 𝑋, i.e for any 𝑖 ∈ ℐ we have 𝐴(𝑖) ⊆𝑋(𝑖) and for arbitrary𝑓 : 𝑖 → 𝑗 we have 𝐴(𝑓) = 𝑋(𝑓)|𝐴. Given that both 𝑋 and 𝐴 are 1-connected, thehomotopy group 𝜋(𝑋,𝐴) appears as a functor Pair(sSetℐ)→ Grpℐ .

Definition 2.3. For a diagram 𝑋, we define a diagram of chain complexes 𝐶*(𝑋)by setting 𝐶*(𝑋)(𝑖) = 𝐶*(𝑋(𝑖)). Similarly, we define the homology groups 𝐻*(𝑋) of𝐶*(𝑋) as diagrams of abelian groups.

There is another version of homology (and cohomology), namely the Bredon co-homology and homology. It was originally defined for 𝐺–simplicial sets (or, in effectfor sSet𝒪

op𝐺 , see [1]), but it can be easily generalized to any diagrams of simplicial sets:

Definition 2.4. Let 𝑋 : ℐ → sSet be a diagram of simplicial sets and let 𝜌 : ℐ → Abbe a diagram of abelian groups.

We define the cochain complex 𝐶*ℐ(𝑋;𝜌) B Hom(𝐶*(𝑋),𝜌). Its 𝑛-th cohomologygroup 𝐻𝑛

ℐ (𝑋;𝜌) is called the 𝑛-th cohomology group of 𝑋 with coefficients in 𝜌.

As our notation suggests, 𝐶*ℐ(𝑋;𝜌) is not a diagram, but a chain complex only. Inthe literature [28, 1] the diagrams 𝜌 are sometimes called coefficient systems.

To give the Bredon homology with coefficients, we need the coefficient system to bea contravariant functor, so we assume 𝜌 : ℐop → Ab. We define chain complex 𝐶ℐ* (𝑋;𝜌)as a tensor product 𝐶𝑘(𝑋)⊗ℐ 𝜌. For details see [28], Chapter 1 or (2.11) in Section 2.7.

It follows, that 𝐶ℐ𝑘 (𝑋;𝜌) is an abelian group. The homology group 𝐻ℐ𝑘 (𝑋;𝜌) isdefined as the 𝑘-th homology of the chain complex 𝐶ℐ* (𝑋;𝜌) where the differential 𝜕is given by 𝜕 = 𝑑⊗ 1.

Model structure. We now describe the model category structure, known as theprojective model structure, first introduced in [4], on a category sSetℐ . Analogously tothe situation in simplicial sets, we will do so by listing the classes of weak equivalences,fibrations and cofibrations:

We say that the map of diagrams 𝑓 : 𝑋 → 𝑌 is a weak equivalence if it is a weakequivalence pointwise, i.e. we assume 𝑓(𝑖) : 𝑋(𝑖) → 𝑌 (𝑖) induces isomorphism on thehomotopy groups for all 𝑖. If 𝑓 is a weak equivalence, then it induces an isomorphismof diagrams 𝜋*(𝑋),𝜋*(𝑌 ).

Similarly, 𝑓 : 𝑋 → 𝑌 is a fibration if it is a (Kan) fibration pointwise. A diagram𝑌 is called fibrant if the unique map to Δ0 = * is a fibration. Here, we interpret Δ0 astrivial diagram (all maps are identity). As in simplicial sets, for any diagram 𝑌 , thereexists a fibrant diagram 𝑌 fib ∈ sSetℐ and weak equivalence 𝑌 → 𝑌 fib. Such 𝑌 fib iscalled a fibrant replacement of 𝑌 .

The only remaining type of morphism in the model category is the cofibration,which we define as for the simplicial sets. It is a map of diagrams which has the leftlifting property with respect to all maps that are at the same time weak equivalencesand fibrations.

For any diagram 𝑋, there exists a unique map ∅ → 𝑋 from the trivial diagram∅. In the case of simplicial sets, this map is always a cofibration. However, for general

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sSetℐ this does not hold. We thus call a diagram 𝑋 cofibrant, if the unique map ∅ →𝑋is a cofibration. We remark that for any 𝑋, there exists a cofibrant diagram 𝑋cof and aweak equivalence 𝑋cof →𝑋. This follows from the model category structure on sSetℐ .

To define the notion of homotopy for maps between diagrams, we use the standardapproach as in [11], section 4. We will introduce cylinder objects and left homotopyand we omit similar definitions of path objects and right homotopy.

Definition 2.5. A (good) cylinder object for 𝑋 ∈ sSetℐ is an object Cyl𝑋 togetherwith a diagram

𝑋 ⨿𝑋 // 𝜄 // Cyl𝑋 ∼𝑝//𝑋

which is a factorization of the folding map id𝑋 + id𝑋 : 𝑋 ⨿ 𝑋 → 𝑋. Here 𝜄 is acofibration and 𝑝 is a weak equivalence. We further denote the two maps 𝑋 →𝑋 ⨿𝑋by 𝜄0, 𝜄1.

For an object 𝑋, there might be multiple cylinders. To give an example of onesuch an object, we first define the cartesian product 𝑋 × 𝑌 : ℐ → sSet of diagrams𝑋,𝑌 : ℐ → sSet as (𝑋 × 𝑌 )(𝑖) = 𝑋(𝑖) × 𝑌 (𝑖). For any 𝑓 : 𝑖 → 𝑗 we define (𝑋 ×𝑌 )(𝑓) = 𝑋(𝑓)× 𝑌 (𝑓).

We can now state (without proof) that in the case 𝑋 is cofibrant, the diagramΔ1 ×𝑋 ∈ sSetℐ , where Δ1 is seen as a constant diagram, is a cylinder object for 𝑋.We will, however, only use the fact that one such cylinder exists as in [11].

If we assume that 𝑋 is cofibrant, then according to [11], Lemma 4.4, the maps 𝜄0, 𝜄1are weak equivalences and cofibrations (such maps are usually called trivial cofibrationsor acyclic cofibrations).

Definition 2.6. We say that 𝑓, 𝑔 : 𝑋 → 𝑌 are left homotopic and denote this by𝑓 ∼ 𝑔 if there exists 𝐻 : Cyl𝑋 → 𝑌 such that the following diagram commutes:

𝑋𝜄0 //

𝑓##

Cyl𝑋

𝐻��

𝑋𝜄1oo

𝑔{{

𝑌

We finally define the set of homotopy classes of maps [𝑋,𝑌 ]ℐ assSetℐ(𝑋cof ,𝑌 fib)/∼. This is a standard definition from homotopy theory, see [11], andit can be shown that the definition is independent of the choice of the replacements.

Homotopy left Kan extension. In this section, we will describe the Bousfield–Kanmodel of a homotopy left Kan extension and the models for cofibrant replacement andhomotopy colimit will be obtained as special cases of this model. We choose this specificmodel because it has advantageous properties with regards to the effective homologywhich will be discussed later in Propositions 2.49 and 2.48.

The homotopy left Kan extension ([35, chapter 8], [22]) hoLan𝑝𝑋 for functors𝑋 : ℐ → sSet and 𝑝 : ℐ → 𝒥 is a functor hoLan𝑝𝑋 : 𝒥 → sSet that fits in the fol-lowing commutative diagram:

ℐ 𝑋 //

𝑝��

sSet

𝒥hoLan𝑝𝑋

==

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If we choose 𝒥 = * and 𝑝 the unique functor to the terminal category, then hoLan𝑝𝑋is the homotopy colimit of the diagram 𝑋 and setting 𝒥 = ℐ and 𝑝 = id results in thecofibrant replacement 𝑋cof , see [22], [28, chapter 5].

Further by the symbol hoLan𝑝𝑋, we will mean the following Bousfield–Kan formula(model) (see [4, chapter 11], [22, 9]):

(hoLan𝑝𝑋)(𝑗) =⨆𝑛

⨆𝑖0,··· ,𝑖𝑛

Δ𝑛 ×𝑋(𝑖0)× ℐ(𝑖0, 𝑖1)× · · · × ℐ(𝑖𝑛−1, 𝑖𝑛)× 𝒥 (𝑝(𝑖𝑛), 𝑗)/∼ (2.1)

where the relation ∼ is given as

(𝑑𝑘𝑡, 𝑥, 𝑓1, 𝑓2, . . . 𝑓𝑛, 𝑔) ∼ (𝑡, 𝑥, 𝑓1, 𝑓2, . . . , 𝑓𝑘+1𝑓𝑘, . . . , 𝑓𝑛−1, 𝑓𝑛, 𝑔), 0 < 𝑘 < 𝑛;

(𝑑𝑘𝑡, 𝑥, 𝑓1, 𝑓2, . . . 𝑓𝑛, 𝑔) ∼ (𝑡, 𝑥, 𝑓1, 𝑓2, . . . , 𝑓𝑛−2, 𝑓𝑛−1, 𝑔𝑝(𝑓𝑛)), 𝑘 = 𝑛;

(𝑑𝑘𝑡, 𝑥, 𝑓1, 𝑓2, . . . 𝑓𝑛, 𝑔) ∼ (𝑡, 𝑓1(𝑥), 𝑓2, . . . , 𝑓𝑛−1, 𝑓𝑛, 𝑔), 𝑘 = 0;

(𝑠𝑘𝑡, 𝑥, 𝑓1, 𝑓2, . . . 𝑓𝑛, 𝑔) ∼ (𝑡, 𝑥, 𝑓1, . . . , 𝑓𝑘, id, 𝑓𝑘+1, . . . , 𝑓𝑛−1, 𝑓𝑛, 𝑔), 0 ≤ 𝑘 ≤ 𝑛.

where 𝑑𝑘 : Δ𝑛−1 → Δ𝑛 is the inclusion into the 𝑘-th face and 𝑠𝑘 : Δ𝑛 → Δ𝑛−1 is the𝑘-th degeneracy.

An 𝑟-simplex 𝑦 ∈ (hoLan𝑝𝑋)(𝑗)𝑟 corresponds to an 𝑟-simplex 𝑧 ∈ (Δ𝑛 ×𝑋(𝑖0))𝑟and a collection of maps 𝑓𝑘 : 𝑖𝑘−1 → 𝑖𝑘 and 𝑔 : 𝑝(𝑖𝑛)→ 𝑗.

The morphisms (hoLan𝑝𝑋)(𝑗0) → (hoLan𝑝𝑋)(𝑗1) are induced by maps𝒥 (𝑝(𝑖𝑛), 𝑗0) → 𝒥 (𝑝(𝑖𝑛), 𝑗1).

The particular choices for 𝒥 and 𝑝 described above lead to models of homotopycolimit and cofibrant replacement of diagram 𝑋:

hocolim𝑋 =⨆𝑛

⨆𝑖0,··· ,𝑖𝑛

Δ𝑛 ×𝑋(𝑖0)× ℐ(𝑖0, 𝑖1)× · · · × ℐ(𝑖𝑛−1, 𝑖𝑛)/∼ ∈ sSet

and

𝑋cof(−) =⨆𝑛

⨆𝑖0,··· ,𝑖𝑛

Δ𝑛 ×𝑋(𝑖0)× ℐ(𝑖0, 𝑖1)× · · · × ℐ(𝑖𝑛−1, 𝑖𝑛)× ℐ(𝑖𝑛,−)/∼ ∈ sSetℐ .

Quillen equivalence between sSet𝐺 and sSet𝒪op𝐺 . Given a finite group 𝐺, one can

form the category of orbits 𝒪𝐺, where the objects are orbits 𝐺/𝐻 for any 𝐻 ≤ 𝐺and morphisms are 𝐺-equivariant maps of these sets. Then 𝒪𝐺 and its dual category𝒪op

𝐺 with the same objects and reversed arrows are both finite categories. Finally, thecategory sSet𝒪

op𝐺 is a diagram category and we will work with the projective model

structure on it.A group 𝐺 can be described as a category with one object and morphisms corres-

ponding to the elements of 𝐺 and any simplicial set 𝑋 with an action of 𝐺 can be seenas a functor 𝑋 : 𝐺→ sSet.

Simplicial sets with an action of 𝐺 and equivariant maps between them form acategory sSet𝐺. For a simplicial set 𝑋 ∈ sSet𝐺 and any subgroup 𝐻 ≤ 𝐺, there is asimplicial subset called a fixed–point (𝐻-fixed–point) set

𝑋𝐻 = {𝑥 ∈ 𝑋 | ℎ𝑥 = 𝑥 for allℎ ∈ 𝐻}.

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We define a model structure on sSet𝐺 by describing the weak equivalences, fibrationsand cofibrations: 𝑓 : 𝑋 → 𝑌 is a weak equivalence if 𝑓𝐻 : 𝑋𝐻 → 𝑌 𝐻 is a weak equival-ence for all 𝐻 ≤ 𝐺. Similarly 𝑓 is a fibration if 𝑓𝐻 is a fibration for all 𝐻 ≤ 𝐺.Finally, cofibrations can be identified with monomorphisms in the category sSet𝐺.Given this model structure, we can define a set of homotopy classes of 𝐺–equivariantmaps [𝑋, 𝑌 ]𝐺. In the same way as in the category of the simplicial sets, we obtain[𝑋, 𝑌 ]𝐺 ∼= [|𝑋|, |𝑌 |]𝐺.

We define a fixed–point functor Φ: sSet𝐺 → sSet𝒪op𝐺 by Φ(𝑋)(𝐺/𝐻) = 𝑋𝐻 . One

can notice that this functor further transports weak equivalences to weak equivalencesand fibrations to fibrations.

In the opposite direction, there is a functor Θ: sSet𝒪op𝐺 → sSet𝐺 described as Θ(𝑇 ) =

𝑇 (𝐺/𝑒). We claim, without proof which can be found in [28], Chapter V, that Φ is rightadjoint to Θ and that ΘΦ = id. Informally speaking, we thus see that the categorysSet𝐺 can be included via Φ into sSet𝒪

op𝐺 and that this inclusion respects some of the

model category data and the functor Θ can be improved in such a way it also respectsthe model category structure. Formally, we describe this by

Theorem 2.7 (Elmendorf, [14]). Define the functor Ψ: sSet𝒪op𝐺 → sSet𝐺 by Ψ(𝑇 ) =

Θ(𝑇 cof). Then there is a natural bijection

[𝑋,Ψ(𝑇 )]𝐺 ∼= [Φ(𝑋),𝑇 ]𝒪op𝐺

The functors Φ and Θ form a Quillen equivalence, [11]. For our purposes there isone consequence to this fact, namely

[𝑋, 𝑌 ]𝐺 ∼= [Φ(𝑋),Φ(𝑌 )]𝒪op𝐺

This means that computing [𝑋, 𝑌 ]𝐺 corresponds to computing [Φ(𝑋),Φ(𝑌 )]𝒪op𝐺.

2.3 Effective homology of chain complexes

We give a definition of simplicial sets with effective homology and our generalizationsto diagrams of simplicial sets.

We remark that there is a collection of algorithms which for simplicial sets witheffective homology as inputs give again simplicial sets with effective homology as out-puts.

These algorithms usually describe commonly used constructions from algebraic to-pology such as Cartesian product, loop space, bar construction, mapping cylinder, totalspace of a fibration i.e. the twisted Cartesian product (see [39, 5, 7, 16]) and homotopypushout [8].

Given a finite diagram of simplicial sets 𝑋 ∈ sSetℐ such that 𝑋 has pointwiseeffective homology and a (computable) morphism 𝑝 : ℐ → 𝒥 , where ℐ,𝒥 are finite, wewill present an algorithm that outputs the Bousfield-Kan model of hoLan𝑝𝑋 ∈ sSet𝒥

as diagram that has effective homology.In special cases, our algorithm computes the homotopy colimit hocolim𝑋 as a

simplicial set with effective homology and cofibrant replacement 𝑋cof of 𝑋 as a diagramof simplicial sets that has effective homology.

There are, however, constructions such as colimits over finite diagrams of simplicialsets with effective homology that do not result in a simplicial set with effective homologyand we will demonstrate this later in Example 2.37.

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Effective chain complexes, reductions and strong equivalences. In this section,we give a complete definition of a simplicial set with effective homology by definingthe strong equivalences. We first define the reduction, which is in literature sometimesreferred to as a strong deformation retraction or a strong contraction.

Definition 2.8. Let 𝐶*, 𝐶 ′* be chain complexes. A reduction 𝐶* ⇒⇒ 𝐶 ′* is a triple(𝛼, 𝛽, 𝜂) called reduction data and pictured as below, where 𝛼, 𝛽 are chain maps and𝜂 : 𝐶* → 𝐶*+1 is a morphism of graded groups

(𝛼, 𝛽, 𝜂) : 𝐶* ⇒⇒ 𝐶 ′* ≡ 𝐶*𝜂55

𝛼**𝐶 ′*

𝛽

jj

satisfying𝜂𝛽 = 0 𝛼𝜂 = 0 𝜂𝜂 = 0𝛼𝛽 = id 𝜕𝜂 + 𝜂𝜕 = id−𝛽𝛼 (2.2)

One of the most important and well known examples of a reduction is the followingexample, first given in [12, 13]:

Proposition 2.9 (Eilenberg–Zilber reduction). Let 𝑋, 𝑌 be simplicial sets. Then thereis a reduction

(AW,EML, SH): 𝐶*(𝑋 × 𝑌 )⇒⇒ 𝐶*(𝑋)⊗ 𝐶*(𝑌 )

The operators in the reduction data are called Alexander–Whitney, Eilenberg–MacLane and Shih respectively. They can be computed using the acyclic models the-orem as e.g. in [27], § 28 and they are not unique. However, in this thesis, we usethe reduction data presented in Theorem 2.1a, [12]. An important observation is thatthe operators of the reduction data are based on the face and degeneracy maps whichmeans that the reduction is functorial (in simplicial sets).

Definition 2.10. A strong equivalence between chain complexes 𝐶* ⇐⇐⇒⇒ 𝐶 ′* consistsof a span of reductions 𝐶* ⇐⇐ 𝐶* ⇒⇒ 𝐶 ′*.

Strong equivalences can be composed, as seen in [39], Proposition 125, producinganother strong equivalence.

We now give a proper definition of chain complex (and simplicial set) with effectivehomology, that were discussed in chapter 1.

Definition 2.11 (Effective chain complex). Let 𝐶* be a chain complex and let 𝐴 be aset such that for every 𝛼 ∈ 𝐴 we are given 𝑐𝛼 ∈ 𝐶*. We call 𝐶* free with basis {𝑐𝛼} ifevery chain 𝑐 ∈ 𝐶* can be expressed uniquely as a combination

𝑐 =∑

𝑘𝛼𝑐𝛼 (2.3)

with integer coefficients 𝑘𝛼 in Z.We call a chain complex 𝐶* locally effective if the elements 𝑐 ∈ 𝐶* have finite

(agreed upon) encoding and there are algorithms computing the addition, zero, inverseand differential for the elements of 𝐶*.

Finally, a locally effective chain complex 𝐶* is called effective if there is an algorithmthat for given 𝑛 ∈ N generates a finite basis 𝑐𝛼 ∈ 𝐶𝑛 and an algorithm that for every𝑐 ∈ 𝐶* outputs the unique description (2.3).

The chain complex 𝐶* has effective homology (𝐶* is a chain complex with effectivehomology) if there is a strong equivalence 𝐶* ⇐⇐⇒⇒ 𝐶ef

* where 𝐶ef* is an effective chain

complex. We say that a (locally effective) simplicial set 𝑋 has effective homology (𝑋is a simplicial set with effective homology) if 𝐶*(𝑋) has.

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Objects with effective homology have nice properties:

Lemma 2.12. Let 𝐶*, 𝐶 ′* be chain complexes with effective homology and let 𝑋, 𝑌 besimplicial sets with effective homology. Then the following holds:

1. 𝐶* ⊕ 𝐶 ′*, 𝐶* ⊗ 𝐶 ′* have effective homology.

2. The space 𝑋 × 𝑌 has effective homology.

Proof. Let 𝜌𝐶 = (𝑓𝐶 , 𝑔𝐶 , ℎ𝐶) : 𝐶* ⇒⇒ 𝐶ef* and 𝜌𝐶′ = (𝑓𝐶′ , 𝑔𝑣, ℎ𝐶′) : 𝐶 ′* ⇒⇒ 𝐶 ′ef* be

reductions. The proof that 𝐶*⊕𝐶 ′* have effective homology is easy – we define the newreduction by

𝜌𝐶⊕𝐶′ = (𝑓𝐶 ⊕ 𝑓𝐶′ , 𝑔𝐶 ⊕ 𝑔𝐶′ , ℎ𝐶 ⊕ ℎ𝐶′) : 𝐶* ⊕ 𝐶 ′* ⇒⇒ 𝐶ef* ⊕ 𝐶 ′

ef* .

For the tensor product, there is a reduction

𝜌𝐶⊗𝐶′ = (𝑓𝐶⊗𝐶′ , 𝑔𝐶⊗𝐶′ , ℎ𝐶⊗𝐶′) : 𝐶* ⊗ 𝐶 ′* ⇒⇒ 𝐶ef* ⊗ 𝐶 ′

ef* .

The new reduction is defined by 𝑓𝐶⊗𝐶′ = 𝑓𝐶 ⊗ 𝑓𝐶′ , 𝑔𝐶⊗𝐶′ = 𝑔𝐶 ⊗ 𝑔𝐶′ , ℎ𝐶⊗𝐶′ = ℎ𝐶 ⊗𝑖𝑑𝐶′ + 𝑔𝐶𝑓𝐶 ⊗ ℎ𝐶′ , or ℎ𝐶⊗𝐶′ = ℎ𝐶 ⊗ 𝑔𝐶′𝑓𝐶′ + 𝑖𝑑𝐶 ⊗ ℎ𝐶′ .

For the second part, we use the Eilenberg–Zilber reduction from Proposition 2.9.The proof is then finished using the first part of the statement, because reductions (asstrong equivalences) are composable.

Perturbation Lemmas. Consider a reduction 𝐶* ⇒⇒ 𝐷*. Assume we change thedifferential of one of the complexes, i.e. we replace either 𝐶* with some 𝐶 ′* or 𝐷* with𝐷′*. Then the Perturbation Lemmas provide us with new reductions 𝐶 ′* ⇒⇒ 𝐷* and𝐶* ⇒⇒ 𝐷′* where the 𝐶*, 𝐷* are again the original chain complexes with changed(perturbed) differential.

Definition 2.13. Let (𝐶*, 𝜕) be a chain complex. We call a collection of maps 𝛿 : 𝐶* →𝐶*−1 perturbation if the sum 𝜕 + 𝛿 is also a differential on 𝐶*.

The following perturbation lemmas are well–known, they constitute one of the basictools in homological perturbation theory. Their genesis can be traced back to [12, 3, 40]and their full proofs can be found e.g. in [39].

Lemma 2.14 (Easy Perturbation Lemma). Let (𝛼, 𝛽, 𝜂) : (𝐶*, 𝜕) ⇒⇒ (𝐶 ′*, 𝜕′) be a

reduction. Let 𝛿′ be a perturbation of differential 𝜕′. Then there is a reduction (𝛼, 𝛽, 𝜂) :(𝐶, 𝜕 + 𝛽𝛿′𝛼)⇒⇒ (𝐶 ′, 𝜕′ + 𝛿).

Proof. Check the formulas for the new reduction given in the statement.

Lemma 2.15 (Basic Perturbation Lemma). Let (𝛼, 𝛽, 𝜂) : (𝐶*, 𝜕) ⇒⇒ (𝐶*, 𝜕′) be a

reduction. Let 𝛿 be a perturbation of differential 𝜕 such that for every 𝑐 ∈ 𝐶* there issome 𝑖 ∈ N satisfying (𝜂𝛿)𝑖(𝑐) = 0. Then there is a perturbation 𝛿′of the differential 𝜕′and a reduction (𝛼′, 𝛽′, 𝜂′) : (𝐶, 𝜕 + 𝛿)⇒⇒ (𝐶, 𝜕′ + 𝛿′).

Proof. We describe the formulas for maps in the new reduction and the rest is left forthe reader. We set

𝜙 =∞∑𝑖=0

(−1)𝑖(𝜂𝛿)𝑖; 𝜓 =∞∑𝑖=0

(−1)𝑖(𝛿𝜂)𝑖

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By the condition in the statement, both sums are finite. The maps in the reduction aregiven as follows:

𝛿′ = 𝛼𝜓𝛿𝛽 = 𝛼𝛿𝜙𝛽; 𝛼′ = 𝛼𝜓; 𝛽′ = 𝜙𝛽; 𝜂′ = 𝜙𝜂 = 𝜂𝜓. (2.4)

2.4 Effective homology of twisted products

In this Section we describe in detail an application of the Basic Perturbation Lemmaon the Eilenberg–Zilber reduction which will be used to obtain a reduction from a chaincomplex of a twisted cartesian product 𝐶*(𝐹 ×𝜏 𝐵) to a chain complex which we willdenote 𝐶*(𝐹 )⊗𝜏 𝐶*(𝐵).

We will then describe the differential on 𝐶*(𝐹 ) ⊗𝜏 𝐶*(𝐵) and we give conditionsunder which the twisted cartesian product has effective homology. The content of thisSection comes mostly from the paper [16].

We now introduce the following notation: If 𝑋 is a simplicial set and 𝑥 ∈ 𝑋𝑛 we put𝑑𝑛−𝑖𝑥 = 𝑑𝑖+1 · · · 𝑑𝑛𝑥 and 𝑑0𝑥 = 𝑥. Given (𝑥, 𝑦) ∈ (𝑋 × 𝑌 )𝑛 we define the Alexander-Whitney operator:

AW(𝑥, 𝑦) =𝑛∑

𝑖=0

𝑑𝑛−𝑖𝑥⊗ 𝑑0𝑖𝑦.

For a non–twisted product 𝐹 ×𝐵, we have the Eilenberg–Zilber reduction

(AW,EML, SH) : (𝐶(𝐹 ×𝐵), 𝜕)⇒⇒ (𝐶*(𝐹 )⊗ 𝐶*(𝐵), 𝜕𝐹⊗𝐵).

The full description of the reduction can be found in [12].The only difference between the chain complexes (𝐶*(𝐹 ×𝜏 𝐵), 𝜕𝜏 ) and

(𝐶*(𝐹 ×𝐵), 𝜕) is in their differentials (to be precise in the face map 𝑑0) and it iseasy to see that

𝜕𝜏 = 𝜕 + (𝑑0(𝑦) · 𝜏(𝑏), 𝑑0(𝑏))− (𝑑0(𝑦), 𝑑0(𝑏)).

So the differential 𝜕𝜏 of 𝐶*(𝐸) is just 𝜕 with the added perturbation

𝛿𝜏 = (𝑑0(𝑦) · 𝜏(𝑏), 𝑑0(𝑏))− (𝑑0(𝑦), 𝑑0(𝑏)).

Definition 2.16. Let 𝐵 and 𝐹 be simplicial sets and let 𝐸 = 𝐹 × 𝐵. Let (𝑦, 𝑏) ∈ 𝐸.We may assume 𝑏 = 𝑠*𝑏

′ ∈ 𝐵, where 𝑠* is a composition of degeneracy operators and𝑏′ is nondegenerate. The filtration degree of (𝑦, 𝑏) is the dimension of 𝑏′. The filtrationdegree of an nonzero element 𝑦 ⊗ 𝑏 ∈ 𝐶*(𝐹 )⊗ 𝐶*(𝐵) is the dimension of 𝑏.

Proposition 2.17 ([39], Theorem 131). Let 𝐹 ×𝜏 𝐵 be a twisted product of simplicialsets and let 𝐺 be the simplicial group acting on 𝐹 . Then the Basic Perturbation Lemmacan be applied to the reduction data (AW,EML, SH) : 𝐶*(𝐹 × 𝐵, 𝜕) ⇒⇒ (𝐶*(𝐹 ) ⊗𝐶*(𝐵), 𝜕𝐹⊗𝐵) to obtain the reduction

(𝑓, 𝑔, ℎ) : (𝐶*(𝐹 ×𝜏 𝐵), 𝜕𝜏 )⇒⇒ (𝐶*(𝐹 )⊗𝜏 𝐶*(𝐵), 𝜕𝐹⊗𝐵𝜏 ),

where 𝐶*(𝐹 )⊗𝜏 𝐶*(𝐵) is just 𝐶*(𝐹 )⊗ 𝐶*(𝐵) with a new differential 𝜕𝐹⊗𝐵𝜏 .

According to [40], the perturbation 𝜕𝐹⊗𝐵𝜏 −𝜕𝐹⊗𝐵 can be seen as a cap product withso called twisting cochain, which is induced by 𝜏 . We will now give definitions of thesenotions.

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Let 𝑡 : 𝐶*(𝐵) → 𝐶*−1(𝐺) be a sequence of abelian group homomorphisms 𝑡𝑛 :𝐶(𝐵)𝑛 → 𝐶(𝐺)𝑛−1. We define a few operators that will be used within the construction:

𝐷 = AW ∘ 𝐶*(Δ) : 𝐶*(𝐵)→ 𝐶*(𝐵)⊗ 𝐶*(𝐵),

where 𝐶*(Δ) is induced by the diagonal map Δ : 𝐵 → 𝐵 × 𝐵. Denoting the rightaction of the simplicial group 𝐺 on the fibre 𝐹 by ·, we obtain an operator

𝜎 = 𝐶(·) ∘ EML : 𝐶*(𝐹 )⊗ 𝐶*(𝐺)→ 𝐶*(𝐹 ).

Finally, we define the cap product (𝑡∩ ) : 𝐶*(𝐹 ) ⊗ 𝐶*(𝐵) → 𝐶*(𝐹 ) ⊗ 𝐶*(𝐵) as acomposition

(𝜎 ⊗ 1)(1⊗ 𝑡⊗ 1)(1⊗𝐷).

Observe, that the cap product is a homomorphism of graded abelian groups and notof chain complexes. We say that 𝑡 is a twisting cochain if the following holds:

(𝜕𝐹⊗𝐵 + (𝑡∩))2 = 𝜕𝐹⊗𝐵(𝑡∩) + (𝑡∩)𝜕𝐹⊗𝐵 + (𝑡∩)(𝑡∩) = 0.

Proposition 2.17 implies that the twisting operator 𝜏 induces a new differential𝜕𝐹⊗𝐵𝜏 on the chain complex 𝐶*(𝐹 )⊗ 𝐶*(𝐵) via the Basic Perturbation Lemma. Thenthe same twisting operator 𝜏 (this time seen as a part of the twisted cartesian product𝐺×𝜏 𝐵) also induces a differential 𝜕𝐺⊗𝐵𝜏 on the chain complex 𝐶*(𝐺)⊗ 𝐶*(𝐵).

According to [40], the twisting operator 𝜏 can further be used to define a specialtwisting cochain which we will again denote by 𝑡 as follows:

𝑡𝑛 : 𝐶𝑛(𝐵)𝑒0⊗1−−−→ 𝐶0(𝐺)⊗ 𝐶𝑛(𝐵)

𝜆0(𝜕𝐺⊗𝐵𝜏 −𝜕𝐺⊗𝐵)−−−−−−−−−−→ 𝐶𝑛−1(𝐺)⊗ 𝐶0(𝐵)

𝑝−→ 𝐶𝑛−1(𝐺),

here 𝑒0 is the unit element of 𝐺0, 𝜆0 is a projection on the summand 𝐶𝑛−1(𝐺)⊗𝐶0(𝐵)of the sum

(𝐶*(𝐺)⊗ 𝐶*(𝐵))𝑛−1 =𝑛−1∑𝑖=0

𝐶𝑛−1−𝑖(𝐺)⊗ 𝐶𝑖(𝐵)

and 𝑝(𝑥⊗ 𝑏) = (𝜀𝑏)𝑥 where the map 𝜀 : 𝐶0(𝐵)→ Z is the augmentation.The following proposition was formulated and proved by Shih in [40] and describes

the relation between the differerential 𝜕𝐹⊗𝐵𝜏 and twisting cochain 𝑡 derived from 𝜏 and𝜕𝐺⊗𝐵𝜏 as above:

Proposition 2.18 ([40], Theorem 2). Let 𝐹 ×𝜏 𝐵 be a twisted cartesian product andlet 𝑡 be the twisting cochain induced by the differential 𝜕𝐺⊗𝐵𝜏 of the chain complex𝐶*(𝐺)⊗𝜏 𝐶*(𝐵). Then 𝜕𝐹⊗𝐵𝜏 − 𝜕𝐹⊗𝐵 = 𝑡 ∩ .

Let 𝐸 = 𝐹 ×𝜏 𝐵 be a twisted product of simplicial sets, 𝑡 be a twisting cochaininduced by the differential 𝜕𝐺⊗𝐵𝜏 on the chain complex 𝐶*(𝐺)⊗𝜏 𝐶*(𝐵) and 𝑏 ∈ 𝐵𝑛, 𝑦 ∈𝐹𝑘. Then using the definition of AW and 𝑡∩ together with the fact that 𝑡(𝑑𝑛𝑏) = 0 weobtain the following formula:

𝑡 ∩ (𝑦 ⊗ 𝑏) = (−1)𝑘𝜎(𝑦 ⊗ 𝑡(𝑑𝑛−1𝑏))⊗ 𝑑0𝑏+𝑛∑

𝑖=2

(−1)𝑘𝜎(𝑦 ⊗ 𝑡(𝑑𝑛−𝑖𝑏))⊗ 𝑑0𝑖𝑏. (2.5)

Using this formula we can summarize some properties of 𝑡∩.

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Corollary 2.19 ([25], Lemma 3.4). Let 𝐸 = 𝐹 ×𝜏 𝐵 be a twisted product of simplicialsets and let 𝑡 be a twisting cochain induced by the differential 𝜕𝐺⊗𝐵𝜏 on the chain complex𝐶*(𝐺)⊗𝜏 𝐶*(𝐵). Then the following holds:

1. The perturbation (𝑡∩) : 𝐶*(𝐹 )⊗ 𝐶*(𝐵) → 𝐶*(𝐹 )⊗ 𝐶*(𝐵) lowers the filtrationdegree by at least one.

2. If for all 𝑏 ∈ 𝐵1, 𝑡(𝑏) = 0, then the perturbation (𝑡∩) lowers the filtration degreeby at least two.

Proof. The first part is clear by the formula (2.5). If 𝑡(𝑑𝑛−1𝑏) = 0 for all 𝑏 ∈ 𝐵𝑛, then

𝑡 ∩ (𝑦 ⊗ 𝑏) =𝑛∑

𝑖=2

(−1)𝑘𝜎(𝑦 ⊗ 𝑡(𝑑𝑛−𝑖𝑏))⊗ 𝑑0𝑖𝑏

which proves the second part.

Effective chain complex for twisted product. We would like to find an answer tothe following problem: Let 𝐵 and 𝐹 be simplicial sets, 𝐺 a simplicial group, 𝐸 = 𝐹×𝜏𝐵a twisted cartesian product, and 𝜌𝐵 : 𝐶*(𝐵) ⇒⇒ 𝐶ef

* (𝐵), 𝜌𝐹 : 𝐶*(𝐹 ) ⇒⇒ 𝐶ef* (𝐹 ) be

reductions to effective chain complexes. Is there a reduction of the chain complex𝐶*(𝐸) to an effective chain complex which can be obtained from 𝜌𝐵, 𝜌𝐹 and 𝜏 by theapplication of the Basic Perturbation Lemma?

Our aim is to find an answer using the composition of given reductions. Accordingto Lemma 2.12, having reductions 𝜌𝐵, 𝜌𝐹 we can construct the reduction

𝜌𝐹⊗𝐵 : 𝐶*(𝐹 )⊗ 𝐶*(𝐵)⇒⇒ 𝐶ef* (𝐹 )⊗ 𝐶ef

* (𝐵).

We know that the chain homotopy ℎ𝐹⊗𝐵 from the reduction 𝜌𝐹⊗𝐵 raises the filtrationdegree by at most 1. This follows from the fact that ℎ𝐵 raises the filtration degree byat most 1 and the proof of Lemma 2.12. We can use the basic perturbation lemma toconstruct a reduction

𝜌𝐸 = (𝑓, 𝑔, ℎ) : 𝐶*(𝐸)⇒⇒ 𝐶*(𝐹 )⊗𝜏 𝐶*(𝐵).

From Corollary 2.19, the perturbation operator 𝜕𝐹⊗𝐵𝜏 −𝜕𝐹⊗𝐵 = 𝑡∩ lowers the filtrationdegree by at least one. If the composition ℎ𝐹⊗𝐵∘(𝜕𝐹⊗𝐵𝜏 −𝜕𝐹⊗𝐵) decreased the filtration,it would be nilpotent and hence we could use the basic perturbation lemma on thereduction data 𝜌𝐹⊗𝐵 and the perturbation 𝜕𝐹⊗𝐵𝜏 − 𝜕𝐹⊗𝐵 to get a reduction

𝜌𝑡 : 𝐶*(𝐹 )⊗𝜏 𝐶*(𝐵)⇒⇒ 𝐶ef* (𝐹 )⊗𝜏 𝐶

ef* (𝐵)

to an effective chain complex 𝐶ef* (𝐹 )⊗𝜏 𝐶

ef* (𝐵) which is 𝐶ef

* (𝐹 )⊗ 𝐶ef* (𝐵) with a new

differential obtained from the Basic Perturbation Lemma. However, in full generalityℎ𝐹⊗𝐵 ∘ (𝜕𝐹⊗𝐵𝜏 − 𝜕𝐹⊗𝐵) = ℎ𝐹⊗𝐵 ∘ (𝑡∩) preserves the filtration degree.

From (2.5) we see that in the composition ℎ𝐹⊗𝐵 ∘ (𝑡∩)(𝑦 ⊗ 𝑏), where 𝑏 ∈ 𝐵𝑛, thereis only one element with the filtration degree 𝑛, namely

𝑔𝐹𝑓𝐹𝜎(𝑦 ⊗ 𝑡(𝑑𝑛−1𝑏))⊗ ℎ𝐵𝑑0𝑏 (2.6)

and the degree 𝑛 element in (ℎ𝐹⊗𝐵 ∘ (𝑡∩))𝑖(𝑦 ⊗ 𝑏) is 𝑦𝑖 ⊗ 𝑏𝑖 where

𝑏0 = 𝑏, 𝑏𝑖+1 = ℎ𝐵𝑑0𝑏𝑖 = (ℎ𝐵𝑑0)𝑖𝑏,

𝑦0 = 𝑦, 𝑦𝑖+1 = 𝑔𝐹𝑓𝐹𝜎(𝑦𝑖 ⊗ 𝑡(𝑑𝑛−1𝑏𝑖)).(2.7)

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Now we can establish conditions for (ℎ𝐹⊗𝐵 ∘ (𝑡∩))𝑖 to decrease the filtration and thuswe get a proof of the following theorem:

Theorem 2.20. Let 𝐵 and 𝐹 be simplicial sets, 𝐺 a simplicial group with an action on𝐹 , 𝐸 = 𝐹 ×𝜏 𝐵 a twisted cartesian product, and 𝜌𝐵 : 𝐶*(𝐵)⇒⇒ 𝐶ef

* (𝐵), 𝜌𝐹 : 𝐶*(𝐹 )⇒⇒ 𝐶ef

* (𝐹 ) be reductions to effective chain complexes.If for all 𝑛 ∈ N, 𝑏 ∈ 𝐵𝑛, 𝑦 ⊗ 𝑏 ∈ 𝐶*(𝐹 ) ⊗ 𝐶*(𝐵), there exists 𝑖 ∈ N such that

(ℎ𝐵𝑑0)𝑖𝑏 = 0 (thus ℎ𝐵𝑑0 is nilpotent) or 𝑦𝑖 defined by (2.7) is zero, then there is a

reduction from the chain complex 𝐶*(𝐸) to an effective chain complex 𝐶ef* (𝐹 )⊗𝜏𝐶

ef* (𝐵)

which can be obtained from 𝜌𝐵, 𝜌𝐹 and 𝜏 by the application of the Basic PerturbationLemma.

Corollary 2.21. If 𝐺 is 0–reduced or 𝜌𝐵 is trivial (i.e. 𝑓𝐵 = 𝑔𝐵 = 𝑖𝑑, ℎ𝐵 = 0), 𝐶*(𝐸)can be reduced to an effective chain complex using the basic perturbation lemma.

Proof. If the reduction 𝜌𝐵 is trivial, then the chain homotopy ℎ𝐵 is trivial, so ℎ𝐵 = 0and hence 𝑏1 = ℎ𝐵𝑑0 = 0. To prove the case when 𝐺 is 0–reduced we compute 𝑡(𝑏)where 𝑏 ∈ 𝐵1. According to the definition we get

𝑡(𝑏) = 𝑡1(𝑏) = 𝑝𝜆0(𝜕𝐺⊗𝐵𝜏 − 𝜕𝐺⊗𝐵)(𝑒0 ⊗ 𝑏).

From the Basic Perturbation Lemma we get

(𝜕𝐺⊗𝐵𝜏 − 𝜕𝐺⊗𝐵)(𝑒0 ⊗ 𝑏) = AW(1 + 𝛿𝜏SH + (𝛿𝜏SH)2 + (𝛿𝜏SH)

3 + . . .)𝛿𝜏EML(𝑒0 ⊗ 𝑏)= AW(1 + 𝛿𝜏SH + (𝛿𝜏SH)

2 + (𝛿𝜏SH)3 + . . .)𝛿𝜏 (𝑠0(𝑒0), 𝑏)

= AW(1 + 𝛿𝜏SH + (𝛿𝜏SH)2 + (𝛿𝜏SH)

3 + . . .)(𝑑0𝑠0(𝑒0) · 𝜏(𝑏), 𝑑0(𝑏))− (𝑑0𝑠0(𝑒0), 𝑑0(𝑏))= AW(1 + 𝛿𝜏SH + (𝛿𝜏SH)

2 + (𝛿𝜏SH)3 + . . .)(𝜏(𝑏), 𝑑0(𝑏))− (𝑒0, 𝑑0(𝑏)).

As the operator SH = 0 on (𝐹 ×𝐵)0 the only nonzero term of (𝜕𝐺⊗𝐵𝜏 − 𝜕𝐺⊗𝐵)(𝑒0 ⊗ 𝑏)is

AW(𝜏(𝑏), 𝑑0(𝑏))− (𝑒0, 𝑑0(𝑏)) = (𝜏(𝑏)⊗ 𝑑0(𝑏))− (𝑒0 ⊗ 𝑑0(𝑏)),

so we have

𝑡(𝑏) = 𝑡1(𝑏) = 𝑝𝜆0(𝜏(𝑏)⊗ 𝑑0(𝑏))− (𝑒0 ⊗ 𝑑0(𝑏)) = 𝜏(𝑏)− 𝑒0.

If the group 𝐺 is 0–reduced, 𝜏(𝑏) = 𝑒0 as 𝑒0 is the only element in 𝐺0 and we have𝑡(𝑏) = 0 for 𝑏 ∈ 𝐵1. That is why 𝑦1 = 𝑔𝐹𝑓𝐹𝜎(𝑦 ⊗ 𝑡(𝑑𝑛−1𝑏)) = 0 and we can apply theprevious theorem.

Now we turn to strong equivalences.

Corollary 2.22. Let 𝐵 and 𝐹 be simplicial sets, 𝐺 a simplicial group, 𝐸 = 𝐹 ×𝜏 𝐵 atwisted cartesian product, and 𝐶*(𝐵)⇐⇐⇒⇒ 𝐶ef

* (𝐵), 𝐶*(𝐹 )⇐⇐⇒⇒ 𝐶ef* (𝐹 ) strong equival-

ences with effective chain complexes. If 𝐺 is 0-reduced or 𝜌𝐵 is trivial (i.e. 𝐶ef* (𝐵) =

𝐶*(𝐵) and all reductions are trivial) then 𝐶*(𝐹 ×𝜏 𝐵) is strongly equivalent to aneffective chain complex 𝐶ef

* (𝐹 ) ⊗𝜏 𝐶ef* (𝐵) which can be obtained from the strong equi-

valences for 𝐶*(𝐵) and 𝐶*(𝐹 ) representing 𝐶*(𝐸) and an effective chain complex usingthe Basic and Easy Perturbation Lemmas.

Proof. By Proposition 2.17 we have a reduction 𝐶*(𝐹 ×𝜏 𝐵) ⇒⇒ 𝐶*(𝐹 ) ⊗𝜏 𝐶*(𝐵).Since strong equivalences are composable, it remains to show that there is a strongequivalence 𝐶*(𝐹 )⊗𝜏 𝐶*(𝐵)⇐⇐⇒⇒ 𝐶ef

* (𝐹 )⊗𝜏 𝐶ef* (𝐵).

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Having strong equivalences 𝐶*(𝐵) ⇐⇐ 𝐶*(𝐵) ⇒⇒ 𝐶ef* (𝐵) and 𝐶*(𝐹 ) ⇐⇐ 𝐶*(𝐹 ) ⇒⇒

𝐶ef* (𝐹 ) then by Lemma 2.12 there is a strong equivalence

𝐶*(𝐹 )⊗ 𝐶*(𝐵)⇐⇐ 𝐶*(𝐹 )⊗ 𝐶*(𝐵)⇒⇒ 𝐶ef* (𝐹 )⊗ 𝐶ef

* (𝐵)

consisting of two reductions:

𝜌1 = (𝑓1, 𝑔1, ℎ1) : 𝐶*(𝐹 )⊗ 𝐶*(𝐵)⇐⇐ 𝐶*(𝐹 )⊗ 𝐶*(𝐵),

𝜌2 = (𝑓2, 𝑔2, ℎ2) : 𝐶*(𝐹 )⊗ 𝐶*(𝐵)⇒⇒ 𝐶ef* (𝐹 )⊗ 𝐶ef

* (𝐵).

Given the perturbation (𝑡∩) on the chain complex 𝐶*(𝐹 ) ⊗ 𝐶*(𝐵), we can use theEasy Perturbation Lemma on the reduction 𝜌1 = (𝑓1, 𝑔1, ℎ1) : 𝐶*(𝐹 ) ⊗ 𝐶*(𝐵) ⇐⇐𝐶*(𝐹 )⊗ 𝐶*(𝐵) to get a new reduction

𝜌′1 = (𝑓1, 𝑔1, ℎ1) : 𝐶*(𝐹 )⊗𝜏 𝐶*(𝐵)⇐⇐ 𝐶*(𝐹 )⊗𝜏𝐶*(𝐵),

where we introduce a perturbation 𝑔1(𝑡∩)𝑓1 to the differential of the chain complex𝐶*(𝐹 )⊗ 𝐶*(𝐵) and the reduction data remains unchanged. If the nilpotency conditionof the composition (𝑔1(𝑡∩)𝑓1) ∘ℎ2 was satisfied, we could apply the Basic PerturbationLemma on the reduction data

𝜌2 = (𝑓2, 𝑔2, ℎ2) : 𝐶*(𝐹 )⊗ 𝐶*(𝐵)⇒⇒ 𝐶ef* (𝐹 )⊗ 𝐶ef

* (𝐵)

to obtain a reduction

𝜌′2 : 𝐶*(𝐹 )⊗𝜏𝐶*(𝐵)⇒⇒ 𝐶ef

* (𝐹 )⊗𝜏 𝐶ef* (𝐵).

If 𝐺 is 0–reduced, then the filtration degree of the perturbation 𝑔1(𝑡∩)𝑓1 is −2 byCorollaries 2.19 and 2.21 and as the the filtration degree of ℎ2 is +1, the nilpotencycondition is satisfied. For 𝜌𝐵 trivial, ℎ2 is 0 and the nilpotency condition is triviallysatisfied.

The reductions 𝜌′1, 𝜌′2 therefore establish a strong equivalence

𝐶*(𝐹 )⊗𝜏 𝐶*(𝐵)⇐⇐⇒⇒ 𝐶ef* (𝐹 )⊗𝜏 𝐶

ef* (𝐵)

and, as the strong equivalences are composable, we get 𝐶*(𝐹 ×𝜏 𝐵) ⇐⇐⇒⇒ 𝐶ef* (𝐹 ) ⊗𝜏

𝐶ef* (𝐵).

Vector fields. We will now deal with the case in which we have more informationabout the reduction 𝜌𝐵 : 𝐶*(𝐵)⇒⇒ 𝐶ef

* (𝐵). In particular, 𝜌𝐵 is obtained via a discretevector field, see [37] or [18]. A discrete vector field 𝑉 on a simplicial set 𝑋 is a set ofordered pairs (𝜎, 𝜏), where 𝜎, 𝜏 are nondegenerate simplices of 𝑋, 𝜎 = 𝑑𝑖𝜏 for exactlyone index 𝑖 and for every two distinct pairs (𝜎, 𝜏), (𝜎′, 𝜏 ′) we have 𝜎′ = 𝜎, 𝜏 ′ = 𝜏, 𝜎′ = 𝜏and 𝜏 ′ = 𝜎. By writing 𝑉 (𝜎) = 𝜏 , we mean (𝜎, 𝜏) ∈ 𝑉 . Given a discrete vector field 𝑉 ,the nondegenerate simplices of 𝑋 are divided into three subsets 𝒮, 𝒯 , 𝒞 as follows:

∙ 𝒮 is the set of source simplices i.e. the simplices 𝜎 such that (𝜎, 𝜏) ∈ 𝑉 ,

∙ 𝒯 is the set of target simplices i.e. the simplices 𝜏 such that (𝜎, 𝜏) ∈ 𝑉 ,

∙ 𝒞 is the set of critical simplices i.e the remaining ones, not occurring in any edgeof 𝑉 .

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Proposition 2.23. Let 𝑉 be a discrete vector field on a simplicial set 𝑋. Then thereexists an induced reduction

𝜌𝑋 = (ℎ𝑋 , 𝑓𝑋 , 𝑔𝑋) : 𝐶*(𝑋)⇒⇒ 𝐶*(𝒞)

We will not give the proof of the previous Proposition (it can be found e.g. in [37] or[24]) and the way the reduction data are obtained, but we remark that the induced chainhomotopy operator ℎ𝑋 has the following property: for any 𝜎 ∈ 𝑋, we get ℎ𝑋(𝜎) ∈ Z𝒯and more importantly ℎ𝑋(𝜎) = 0 whenever 𝜎 ∈ 𝒞 ∪ 𝒯 .

Definition 2.24. Let 𝑋 be a simplicial set. For any nondegenerate simplex 𝜎 ∈ 𝑋𝑛

we will consider the following condition:

𝑑0𝜎 ∈ 𝒮 implies 𝜎 ∈ 𝒮 (2.8)

We say that a discrete vector field 𝑉 on a simplicial set satisfies (2.8) if all nonde-generate simplices of 𝑋 satisfy (2.8).

Corollary 2.25. Let 𝐵 and 𝐹 be simplicial sets, 𝐺 a simplicial group, 𝐸 = 𝐹 ×𝜏 𝐵a twisted cartesian product and 𝜌𝐵 : 𝐶*(𝐵) ⇒⇒ 𝐶ef

* (𝐵), 𝜌𝐹 : 𝐶*(𝐹 ) ⇒⇒ 𝐶ef* (𝐹 ) be

reductions to effective chain complexes. If the reduction 𝜌𝐵 is induced by a vector fieldsatisfying (2.8), then there exists a reduction from the chain complex 𝐶*(𝐸) to aneffective chain complex which can be obtained from 𝜌𝐵, 𝜌𝐹 and 𝜏 .

Proof. We show that (ℎ𝐵𝑑0)2 = 0 and apply Theorem 2.20. Because 𝜌𝐵 is induced by

a discrete vector field, for any 𝑏 ∈ 𝐵𝑛 we have 𝑏1 = ℎ𝐵(𝑑0𝑏) ∈ Z𝒯 . If 𝑑0𝑏1 ∈ Z𝒮, thenthe condition (2.8) would give us 𝑏1 = ℎ𝐵(𝑑0𝑏) ∈ Z𝒮. Therefore 𝑑0𝑏1 ∈ Z(𝒞 ∪ 𝒯 ).

Finally, from the fact that the homotopy operator ℎ𝐵 : 𝐶*(𝐵) → 𝐶*+1(𝐵) wasinduced by the vector field, we see that 𝑏2 = ℎ𝐵𝑑0𝑏1 = 0.

Example 2.26. An example of a vector field satisfying (2.8) is so called Eilenberg–MacLane vector field on a space 𝑋 = 𝐾(Z, 1).

The space 𝐾(Z, 1) is an Eilenberg-MacLane space. These spaces and their standardmodels will be introduced and discussed later in Section 2.9.

Here, we will use the fact that a simplex 𝜎 ∈ 𝑋𝑛 can be represented as an 𝑛–tuple[𝑎1| . . . |𝑎𝑛], where 𝑎1, . . . , 𝑎𝑛 ∈ Z (see [24], page 5). The face operators are 𝑑0𝜎 =[𝑎2| . . . |𝑎𝑛], 𝑑𝑛𝜎 = [𝑎1| . . . |𝑎𝑛−1], 𝑑𝑖𝜎 = [𝑎1| . . . |𝑎𝑖−1|𝑎𝑖 + 𝑎𝑖+1|𝑎𝑖+2| . . . |𝑎𝑛], where 1 <𝑖 < 𝑛.

For any 𝜎 = [𝑎1| . . . |𝑎𝑛] ∈ 𝑋𝑛, where 𝑎𝑛 = 1, we define the Eilenberg-MacLanevector field 𝑉EML in the following way:

𝑉EML(𝜎) =

{[𝑎1| . . . |𝑎𝑛−1|𝑎𝑛 − 1|1] for 𝑎𝑛 > 1,[𝑎1| . . . |𝑎𝑛−1|1] for 𝑎𝑛 < 0.

Now we can classify the simplices:

∙ 𝜎 ∈ 𝒮 has the form [𝑎1| . . . |𝑎𝑛], where 𝑎𝑛 = 1 and 𝑛 > 0.

∙ 𝜎 ∈ 𝒯 has the form [𝑎1| . . . |𝑎𝑛−1|1], where 𝑛 > 1.

∙ 𝜎 ∈ 𝒞 is [ ] and [1].

It is easy to check that the vector field 𝑉EML satisfies (2.8) and that it gives us areduction 𝐶*(𝐾(Z, 1)) ⇒⇒ 𝐶*([ ], [1]). Note that Corollary 2.25 implies that for any𝐸 = 𝐹 ×𝜏 𝐾(Z, 1) there is a reduction 𝐶*(𝐸)⇒⇒ 𝐶ef

* (𝐸) to an effective chain complexif there is a reduction 𝐶*(𝐹 )⇒⇒ 𝐶ef

* (𝐹 ) to an effective chain complex.

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Twisted division. In this section we present a construction, which can be seen asan opposite construction to the twisted cartesian product: We would like to showthat if a twisted cartesian product 𝐺 ×𝜏 𝐵 and the simplicial group 𝐺 have effectivehomology and some specific conditions are satisfied then the base space 𝐵 has effectivehomology. To sum up, instead of doing a twisted multiplication, we are computing atwisted division.

The main result is summarized in the following Proposition that was described e.g.in [7]. We will not give any details about this construction, we only remark that it isbased on the so-called bar construction and applications of the perturbation lemmas.

Proposition 2.27 ([7], Proposition 3.13 ). Let 𝐺 be a 0-reduced abelian simplicialgroup, let 𝐵 be a simplicial set, and let 𝜏 be a computable twisting operator. If both𝐺 and 𝐺×𝜏 𝐵 are equipped with effective homology, then 𝐵 can also be equipped witheffective homology.

2.5 Effective homology for diagrams

In this section, we make a formal definition of a diagram of simplicial sets that haseffective homology and describe some constructions with such diagrams. We begin byintroducing reduction and strong equivalence of diagrams:

Definition 2.28. Let 𝐶,𝐶 ′ : ℐ → Ch+ be diagrams of chain complexes. A reduction𝐶 ⇒⇒ 𝐶 ′ is a triple of natural transformations (𝛼, 𝛽, 𝜂)

(𝛼, 𝛽, 𝜂) : 𝐶 ⇒⇒ 𝐶 ′ ≡ 𝐶𝜂77

𝛼**𝐶 ′

𝛽

jj

which again satisfy the conditions (2.2). The strong equivalence 𝐶 ⇐⇐⇒⇒ 𝐶 ′ of diagramsof chain complexes is defined as a span of reductions 𝐶 ⇐⇐ 𝐶 ⇒⇒ 𝐶 ′.

The strong equivalences can be composed as in the case of strong equivalences ofchain complexes. We now define effective diagrams and introduce diagrams that haveeffective homology.

Definition 2.29. We call a diagram 𝐶 : ℐ → Ch+ of nonnegatively graded chaincomplexes locally effective if 𝐶(𝑖) is locally effective for every 𝑖 ∈ ℐ and if 𝐶(𝑓) is acomputable morphism for every morphism 𝑓 in the category ℐ.

We call a diagram of simplicial sets 𝑋 : ℐ → sSet locally effective if 𝑋(𝑖) is locallyeffective simplicial set for every 𝑖 ∈ ℐ and if 𝑋(𝑓) is a computable morphism for everymorphism 𝑓 in the category ℐ.

We say that a locally effective diagram of chain complexes 𝐶 has pointwise effectivehomology (or 𝐶 is a diagram with pointwise effective homology) if for every 𝑖 ∈ ℐ thereexists an effective chain complex 𝐶ef(𝑖) and a strong equivalence of chain complexes𝐶(𝑖)⇐⇐⇒⇒ 𝐶ef(𝑖).

A locally effective diagram of simplicial sets 𝑋 : ℐ → sSet is pointwise effective (orhas pointwise effective homology) if 𝐶(𝑋) has pointwise effective homology.

Definition 2.30. Let 𝐶 : ℐ → Ch+. We say 𝐶 is cellular if there exists an indexingset 𝐴 and for every 𝛼 ∈ 𝐴 there is

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𝑖𝛼 ∈ ℐ and a chain 𝑐𝛼 ∈ 𝐶(𝑖𝛼) such that the set

{𝑓𝛼*𝑐𝛼 | 𝛼 ∈ 𝐴, 𝑓𝛼 ∈ ℐ(𝑖𝛼, 𝑖)}

forms a basis for each 𝐶(𝑖).

We can formulate the cellularity also in a different way: given an element 𝑐 ∈ 𝐶(𝑖)there is a unique description of 𝑐 as

𝑐 =∑

𝛼,𝑓𝛼 : 𝑖𝛼→𝑖

𝑘𝑓𝛼𝑓𝛼*(𝑐𝛼) (2.9)

where 𝑘𝑓𝛼 ∈ Z.

Example 2.31. Let 𝑋 be a simplicial set with an action of a finite group 𝐺. Then𝐶*(𝑋) is a chain complex with an action of 𝐺 induced by the action of 𝐺 on 𝑋.Further, 𝐶*(𝑋) can be interpreted as a diagram of 𝐶 : 𝒪op

𝐺 → Ch+, where 𝐶(𝐺/𝐻) =𝐶*(𝑋)𝐻 = 𝐶*(𝑋

𝐻). We will show, that 𝐶 is a cellular diagram. To do so, we willdescribe a process that obtains the set 𝐴 of generators of 𝐶. We remark that the ideais to choose the appropriate generators in every chain complex 𝐶*(𝑋)𝐻 .

Let 𝒮 = {𝐻 | 𝐻 ≤ 𝐺} and let ≺ be a total order on 𝒮 that satisfies the condition𝐻 ≺ 𝐾 if |𝐻| < |𝐾|. We use ≺ to label the elements of 𝒮 by nonnegative integers andwe obtain a sequence {𝑒} = 𝐻0 ≺ 𝐻1 ≺ · · · ≺ 𝐻𝑛−1 ≺ 𝐻𝑛 = 𝐺.

Let 𝐵𝑗 be the set of generators (the basis) of 𝐶(𝐺/𝐻𝑗). We now construct the set𝐴 using the following iterative process:

First set 𝐴 B 𝐵𝑛. Next, if an element 𝑐𝛽 in 𝐵𝑛−1 cannot be expressed as

𝑐𝛽 =∑

𝛼,𝑓𝛼 : 𝐺/𝐻𝑛→𝐺/𝐻𝑛−1

𝑘𝑓𝛼𝑓𝛼*(𝑐𝛼)

where 𝑐𝛼 ∈ 𝐴, 𝑗 ≥ 𝑛− 1 and 𝑘𝑓𝛼 ∈ Z, we add 𝑐𝛽 to 𝐴 by setting 𝐴 B 𝐴 ∪ {𝑐𝛽}. Werepeat the process until we exhaust all elements in 𝐵𝑛−1 and then we continue on withelements in 𝐵𝑛−2, 𝐵𝑛−3, . . . , 𝐵0.

It remains to give a definition of effective diagram of chain complexes:

Definition 2.32. We call a locally effective diagram 𝐶 effective if it is cellular and thereis an algorithm that generates for given 𝑛 a finite list of all basis elements 𝑐𝛼 ∈ 𝐶(𝑖𝛼)𝑛and an algorithm computing (2.9) for every input 𝑐 ∈ 𝐶(𝑖).

Example 2.33. Let 𝑋 be a finite simplicial set with an action of a finite group 𝐺.We can view 𝐶*(𝑋) as a diagram of 𝐶 : 𝒪op

𝐺 → Ch+ as in Example 2.31. Then thefiniteness of 𝑋 implies that 𝐶 is effective.

The following effective chain complex will be utilized later:

Example 2.34. Let ℐ be a finite category. Then for any 𝑖 ∈ ℐ there is a func-tor Zℐ(𝑖,−) : ℐ → Ab, the free abelian group on the set ℐ(𝑖,−). We think of thisabelian group as a chain complex concentrated in degree 0 and thus obtain a functorZℐ(𝑖,−) : ℐ → Ch+. This diagram of chain complexes is effective.

We first show that it is cellular: The basis consists of one element only, namely id𝑖.Given some 𝑗 ∈ ℐ, the elements 𝑥 ∈ ℐ(𝑖, 𝑗) form the basis Zℐ(𝑖,−) and we can describethem as 𝑥 = 𝑥*(id𝑖). The finiteness of ℐ now implies that Zℐ(𝑖,−) is effective.

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The definition of a diagram of simplicial sets that has effective homology is similarto the definition for simplicial sets:

Definition 2.35. We say that a locally effective diagram of simplicial sets 𝑋 : ℐ → sSethas effective homology if there is a strong equivalence 𝐶(𝑋)* ⇐⇐⇒⇒ 𝐶ef

* between thediagram of chain complexes for 𝑋 and some effective diagram of chain complexes 𝐶ef

* .

To illustrate the fact that in general a construction with diagrams that have effectivehomology does not result in a diagram that has effective homology, we present twoexamples of diagrams that have pointwise effective homology, but their finite colimitsdo not have effective homology.

The first example shows that given a diagram 𝐶 : ℐ → Ch+ with pointwise effectivehomology, the colimit colimℐ𝐶 ∈ Ch+ does not have effective homology in general. Wewill then use the idea of the first example to give a finite diagram of simplicial sets thathas pointwise effective homology with a colimit that does not have effective homology.

Example 2.36. Let ℐ = C2, where C2 = {1, 𝑡} is the two element group. As a category,it has one object and two arrows. Any diagram of chain complexes 𝐷 : ℐ → Ch+ canbe seen as a chain complex of ZC2 modules. We take the following diagram of chaincomplexes 𝐶 : ℐ → Ch+, which is a free augmented ZC2 resolution of the module Zwith trivial group action:

· · · 𝜕3 // ZC2𝜕2 // ZC2

𝜕1 // ZC2𝜕0 // Z // 0

The differentials 𝜕𝑖 : 𝐶𝑖 → 𝐶𝑖−1 are 𝜕0(1) = 𝜕0(𝑡) = 1 and for 𝑖 ≥ 1, we have

𝜕2𝑖−1(𝑐) = (1 + 𝑡) · 𝑐; 𝜕2𝑖(𝑐) = (1− 𝑡) · 𝑐.

This chain complex is clearly contractible over Z as there is a reduction (0, 0, ℎ) : 𝐶 ⇒⇒ 0 to a trivial chain complex 0: ℐ → Ch+. The only nontrivial part of the reductiondata is the homotopy ℎ𝑖 : 𝐶𝑖 → 𝐶𝑖+1 which is given by

ℎ2𝑖−1(𝑡) = 𝑡, ℎ2𝑖−1(1) = 0, ℎ2𝑖(𝑡) = 0, ℎ2𝑖(1) = 1, ℎ0(1) = 1

and it follows that 𝐶 has pointwise effective homology.The colimit colimℐ𝐶 is the following chain complex which we obtain by factoring

out the group action:

· · · 2 // Z 0 // Z 2 // Z 0 // Z 1 // Z // 0

where the differentials 0, 2 alternate. Notice that colimℐ𝐶 has nontrivial homologygroups.

Let us now take (⨁

N 𝐶) : ℐ → Ch+, the countable sum of copies of 𝐶. It is stillcontractible and has effective homology (we use the reduction above on all factors).

On the other hand, colimℐ (⨁

N 𝐶) =⨁

N colimℐ𝐶 has infinitely generated homo-logy groups and therefore cannot have effective homology.

Example 2.37. We describe the main steps of the construction and omit most of thedetails. The main idea is similar to the idea of the previous example. We take the space𝐸C2, which is a contractible space with a free action of the group C2. It can be seen asa diagram 𝐸C2 : C2 → sSet. Using a specified simplicial model of 𝐸C2 described e.g. in[27, p. 101] given by

(𝐸C2)𝑛 = 𝐶0(Δ𝑛,C2)

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one can deduce that 𝐸C2 has effective homology [24].The unreduced suspension 𝑆𝐸C2 of 𝐸C2 has two distinct points which we denote

+1,−1 where the action of C2 is fixed and the action is free everywhere else. Further𝑆𝐸C2 has effective homology and we can construct a contraction to one of the points,say +1. We take a countable wedge sum of copies of 𝑆𝐸C2 with basepoints +1. Theresulting space

⋁N 𝑆𝐸C2 again has effective homology.

On the other hand, the space colimC2𝑆𝐸C2 has different homology. By factoring theaction of C2, we get colimC2𝑆𝐸C2 = 𝑆𝐵C2 an unreduced suspension of the space 𝐵C2.As 𝐵C2 = RP∞ has nontrivial homology, the space 𝑆𝐵C2 has nontrivial homology,namely for any even 𝑛 ∈ N, we have

𝐻𝑛(colimC2𝑆𝐸C2) ∼= Z/2Z.

It follows that 𝐻𝑛(colimC2

⋁N 𝑆𝐸C2) =

⨁N𝐻𝑛(colimC2𝑆𝐸C2) is an infinitely generated

group, so it cannot have effective homology.

Constructions with effective homology. Here, we describe some basic construc-tions with diagrams that have effective homology.

Lemma 2.38.

(1) Let 𝑋,𝑌 : ℐ → sSet be diagrams that have pointwise effective homology. Thenthe diagram (𝑋 × 𝑌 ) : ℐ → sSet has pointwise effective homology.

(2) Let 𝑋 : ℐ → sSet, 𝑌 : 𝒥 → sSet be diagrams that have pointwise effective ho-mology. Then 𝑋 × 𝑌 : ℐ × 𝒥 → sSet,𝑋 × 𝑌 (𝑖, 𝑗) = 𝑋(𝑖)× 𝑌 (𝑗), has effectivehomology.

(3) Let 𝐶 : ℐ → Ch+, 𝐶 ′ : ℐ → Ch+ be diagrams of chain complexes that have effectivehomology. Then 𝐶 ⊕𝐶 ′ has effective homology.

Proof. (1) It is enough to show that for each 𝑖 ∈ ℐ, 𝐶((𝑋 × 𝑌 )(𝑖)) has effectivehomology and this is proven by the second part of Lemma 2.12.

(2) We use the functoriality of Eilenberg–Zilber reduction (see [13], Theorem 2.1a).The diagram 𝐶(𝑋 × 𝑌 ) is strongly equivalent to diagram 𝐶ef(𝑋) ⊗ 𝐶ef(𝑌 ) and itremains to show the latter to be effective.

Let 𝑥𝛼 be the finite basis for 𝐶ef(𝑋) and 𝑦𝛽 be the basis of 𝐶ef(𝑌 ). It is well-knownthat the basis of tensor product is formed by tensor products of basis elements, so thebasis of 𝐶ef(𝑋)⊗𝐶ef(𝑌 ) is generated by the set

{𝑓*𝑥𝛼 ⊗ 𝑔*𝑦𝛽 | 𝑓 ∈ ℐ(𝑖𝛼, 𝑖), 𝑔 ∈ 𝒥 (𝑗𝛽,𝑖)} == {(𝑓, 𝑔)*𝑥𝛼 ⊗ 𝑦𝛽 | (𝑓, 𝑔) ∈ ℐ × 𝒥 ((𝑖𝛼, 𝑗𝛽), (𝑖, 𝑗))}.

The last part is trivial.

Corollary 2.39. Let 𝐶 : ℐ → Ch+ be a diagram that has effective homology and let 𝐶 ′be a chain complex with effective homology. Then the diagram 𝐶 ′ ⊗𝐶 : ℐ → Ch+ haseffective homology.

Proof. We can see 𝐶 ′ as a diagram 𝐶 ′ : * → Ch+. Lemma 2.38 (2) then gives theresult.

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The first statement in Lemma 2.38 is an example of a fact that holds true moregenerally and which we informally explain as follows:

Suppose that 𝐴 is a functorial construction that takes as an input a collectionof simplicial sets with effective homology (𝑋1, . . . , 𝑋𝑘) and outputs a simplicial setwith effective homology 𝐴(𝑋1, . . . , 𝑋𝑘). Let 𝐴 be a construction that given diagramsthat have pointwise effective homology (𝑋1, . . . ,𝑋𝑘), 𝑋𝑗 ∈ sSetℐ , 1 ≤ 𝑗 ≤ 𝑘 onthe input produces a diagram 𝐴(𝑋1, . . . ,𝑋𝑘) ∈ sSetℐ such that 𝐴(𝑋1, . . . ,𝑋𝑘)(𝑖) =𝐴(𝑋1(𝑖), . . . ,𝑋𝑘(𝑖)). Then 𝐴(𝑋1, . . . ,𝑋𝑘) has pointwise effective homology.

To sum up, if a construction on simplicial sets with effective homology results in asimplicial set with effective homology then the pointwise version on a pointwise effectivediagram results in a pointwise effective diagram.

The following Proposition will be used in the proof of our main result to computethe homotopy groups of a diagram 𝑌 . Before the statement itself, we define the diagramof cycles 𝑍: Given an effective diagram of chain complexes 𝐶 : ℐ → Ch+, there is adiagram of cycles 𝑍𝑘 : ℐ → Ch+ such that 𝑍𝑘(𝑖) is the subgroup of cycles in 𝐶𝑘(𝑖).

Proposition 2.40. Let 𝐶 : ℐ → Ch+ be an effective diagram of chain complexes suchthat 𝐻𝑘(𝐶) = 0 for 𝑘 ≤ 𝑛, 𝑖 ∈ ℐ. Then there is a (computable) retraction 𝑟 : 𝐶𝑘+1 →𝑍𝑘+1 i.e. a homomorphism that restricts to the identity on 𝑍𝑘+1.

Proof. The proof is a straightforward generalization of the proof of Proposition 2.12in [8]. We define a contraction 𝜎 : 𝐶𝑘 → 𝐶𝑘+1 i.e. a map satisfying 𝜕𝜎 + 𝜎𝜕 = id andwe use it to split off the cycles. We proceed by induction with respect to the dimension.

Basic step of induction: For every chain 𝑥 ∈ 𝐶(𝑖)0, we have 𝜕𝑥 = 0 and because𝐻0(𝐶)(𝑖) = 0, we conclude that 𝑥 is a boundary and hence there exists 𝑦 ∈ 𝐶1 suchthat 𝜕𝑦 = 𝑥. We want to compute such 𝑦 and set 𝜎(𝑥) = 𝑦.

Diagram 𝐶 is effective, so we have a finite set of generators 𝑥𝛼 ∈ 𝐶(𝑖𝛼)𝛼∈𝐴 and itsuffices to compute 𝜎(𝑥𝛼) = 𝑦𝛼 as any 𝑥 ∈ 𝐶0 can be expressed using the formula

𝑥 =∑

𝛼,𝑓𝛼 : 𝑖𝛼→𝑖

𝑘𝑓𝛼𝑓𝛼*(𝑥𝛼)

and it follows that

𝜎(𝑥) =∑

𝛼,𝑓𝛼 : 𝑖𝛼→𝑖

𝑘𝑓𝛼𝑓𝛼*(𝜎(𝑥𝛼)) =∑

𝛼,𝑓𝛼 : 𝑖𝛼→𝑖

𝑘𝑓𝛼𝑓𝛼*(𝑦𝛼)

It remains to show how to compute 𝑦𝛼 for some 𝑥𝛼 ∈ 𝐶(𝑖𝛼)0. Because 𝐶 is effective,one can compute the finite basis {𝑦𝛽}𝛽∈𝐵 of 𝐶1(𝑖𝛼). We can interpret 𝜕 : 𝐶(𝑖𝛼)1 →𝐶(𝑖𝛼)0 as a Z-linear map and use Smith normal form algorithm to compute 𝑦𝛼 as alinear combination of basis elements 𝑦𝛽.

Inductive step is similar: Suppose we have successfully constructed 𝜎𝑘−1 and wewant to construct 𝜎𝑘: Let again 𝑥𝛼 be a basis element, 𝛼 ∈ 𝐴𝑘. Then 𝑥𝛼 − 𝜎𝑘−1𝜕(𝑥𝛼)is a cycle. Because 𝐻𝑘(𝐶) = 0, we can compute 𝑦𝛼 such that 𝜕(𝑦𝛼) = 𝑥𝛼 − 𝜎𝑘−1𝜕(𝑥𝛼)and we set 𝜎𝑘(𝑥𝛼) = 𝑦𝛼. Similarly as above we compute 𝜎𝑘(𝑥) for 𝑥 ∈ 𝑍𝑘(𝑖). Finally,we set 𝑟 = id−𝜎𝜕. This completes the proof.

Perturbation lemmas for diagrams. In this section, we define perturbation andgive perturbation lemmas for diagrams of chain complexes.

Further on, we will use the following notation: For any diagram of chain complexes,𝐶 : ℐ → Ch+, the diagram 𝐶[𝑘], 𝑘 ∈ N is diagram of chain complexes 𝐶 with all thechain complexes moved up by dimension 𝑘, i.e. 𝐶[𝑘]𝑛 = 𝐶𝑛−𝑘.

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Definition 2.41. Let 𝐶,𝐶 ′ : ℐ → Ch+. Notice that the differential 𝜕 on 𝐶 can beseen as a natural transformation 𝐶 → 𝐶[1] satisfying 𝜕𝜕 = 0. We call homomorphism𝛿 : 𝐶 → 𝐶[1] perturbation of 𝜕 if the sum 𝜕 + 𝛿 is also a differential.

We now formulate the lemmas.

Lemma 2.42 (Easy Perturbation Lemma). Let (𝛼, 𝛽, 𝜂) : (𝐶, 𝜕) ⇒⇒ (𝐶 ′, 𝜕′) be a re-duction of diagrams of chain complexes. Let 𝛿′ be a perturbation of the differential 𝜕′.Then there is a reduction (𝛼, 𝛽, 𝜂) : (𝐶, 𝜕 + 𝛽𝛿′𝛼)⇒⇒ (𝐶 ′, 𝜕′ + 𝛿).

Lemma 2.43 (Basic Perturbation Lemma). Let (𝛼, 𝛽, 𝜂) : (𝐶, 𝜕) ⇒⇒ (𝐶, 𝜕′) be a re-duction of diagrams of chain complexes. Let 𝛿 be a perturbation of the differential 𝜕 suchthat for every 𝑖 ∈ ℐ and every 𝑐 ∈ 𝐶(𝑖) there is some 𝑘 ∈ N satisfying (𝜂𝛿)𝑘(𝑐) = 0.Then there is a perturbation 𝛿′ of the differential 𝜕′ and a reduction of diagrams ofchain complexes (𝛼′, 𝛽′, 𝜂′) : (𝐶, 𝜕 + 𝛿)⇒⇒ (𝐶 ′, 𝜕′ + 𝛿′).

We omit the proofs as they can be seen as simple consequences of the proofs of theoriginal perturbation Lemmas: For the Easy Perturbation Lemma the reduction dataare given in the statement and for the Basic Perturbation Lemma they are described bythe formulas (2.4). The new reduction data are therefore given as sums of compositionsof the operators 𝛼, 𝛽, 𝜂, 𝜕, 𝛿. As all the operators are natural, the proofs of the originalPerturbation Lemmas imply the diagrammatic versions.

As a first application, we will utilize the Perturbation Lemmas to give an effectivehomology to the algebraic mapping cone. We first define the object itself:

Definition 2.44. Suppose that we have diagrams of chain complexes 𝐶, 𝐷 : ℐ → Ch+and a natural transformation 𝜙 : 𝐶 → 𝐷. As a graded abelian group, the algebraicmapping cone Cone𝜙 is identical to the direct sum 𝐶[1]⊕𝐷 and these two diagramsdiffer in the differential only:

𝜕𝐶[1]⊕𝐷 =

(−𝜕𝐶 00 𝜕𝐷

)𝜕Cone𝜙 =

(−𝜕𝐶 0𝜙 𝜕𝐷

)We remark that choosing ℐ = *, we get a standard algebraic mapping cone.

Lemma 2.45. Let 𝐶,𝐷 : ℐ → Ch+ be diagrams that have effective homology andlet 𝜙 : 𝐶 → 𝐷 be a homomorphism. Then the mapping cone Cone𝜙 : ℐ → Ch+ is adiagram that has effective homology.

Proof. This is obtained using perturbation Lemmas 2.42 and 2.43 on the strong equi-valence of diagrams

𝐶[1]⊕𝐷 ⇐⇐⇒⇒ 𝐶ef [1]⊕𝐷ef .

In more detail, there is a span of reductions 𝐶[1]⊕𝐷 ⇐⇐ 𝐶[1]⊕ 𝐷 ⇒⇒ 𝐶ef [1]⊕𝐷ef .From the definition of the mapping cone, we can see that Cone𝜙 differs from 𝐶[1]⊕

𝐷 by the perturbation of the differential 𝜕𝐶 ⊕ 𝜕𝐷 which can described as

𝛿 =

(0 0𝜙 0

).

We now use the Easy Perturbation Lemma 2.42 on the reduction 𝐶[1] ⊕ 𝐷 ⇐⇐𝐶[1] ⊕ 𝐷 : (𝑓, 𝑔, ℎ) and we end up with a new differential which can de described asthe differential of the direct product plus the perturbation

𝛿 = (0 0

𝑔 ∘ 𝜙 ∘ 𝑓 0

).

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It remains to use the Basic Perturbation Lemma for 𝛿 on the reduction (𝑓 ef , 𝑔ef , ℎef) :𝐶[1]⊕ 𝐷 ⇒⇒ 𝐶ef [1]⊕𝐷ef so we need to check the nilpotency condition, but this easilyholds as the homotopy operator ℎef is given by

ℎef =

(ℎef𝐶[1] 0

0 ℎef𝐷

)and we obtain 𝛿ℎef𝛿 = 0.

If we restrict the resulting strong equivalence Cone𝜙 ⇐⇐⇒⇒ Coneef𝜙 to the secondsummand, we get the original strong equivalence 𝐷 ⇐⇐⇒⇒𝐷ef . We will utilize this factand the following remark in the proof of Theorem A.

Remark 2.46 (Topological and algebraic mapping cylinders). Given a map 𝜙 : 𝑋 →𝑌 of diagrams, we can construct a diagram

Cyl𝜙 = (Δ1 ×𝑋 ∪ 𝑌 )/ ∼,

where ∼ is the equivalence identifying (1, 𝑥) with 𝜙(𝑥), 𝑥 ∈ 𝑋. One can see Cyl𝜙as a particular case of homotopy colimit. We call the resulting diagram a topologicalmapping cylinder. In this construction, we can identify 𝑋 with {0}×𝑋, the “top copy”of 𝑋 in Cyl𝜙 and if we contract this copy of 𝑋, we obtain the topological mappingcone Cone𝜙.

Using the Eilenberg–Zilber reduction, one can check that the diagram of chaincomplexes of the pair (Cyl𝜙,𝑋) has a reduction to Cone𝜙*. Details can be found in[45], p. 20–22.

Filtrations. We are now going to formulate a general lemma about filtered diagramsof chain complexes and effective homology. This result will then be used in the followingsection to produce further constructions with effective homology. First, we introducesome notation. Let 𝐶 : ℐ → 𝐶ℎ+. We consider a filtration 𝐹 on diagram 𝐶 of chaincomplexes:

0 = 𝐹−1𝐶 ⊆ 𝐹0𝐶 ⊆ 𝐹1𝐶 ⊆ · · ·such that 𝐶 =

⋃𝑘 𝐹𝑘𝐶. We further assume that each 𝐹𝑘𝐶 is a cellular subcomplex

i.e. it is generated by a subset of the given basis of 𝐶 and that the filtration is locallyfinite i.e. for each 𝑛 we have 𝐹𝑘𝐶𝑛 = 𝐶𝑛 for 𝑘 ≫ 0.

Lemma 2.47 ([8], Lemma 7.3). Let 𝐶 be a diagram of chain complexes with filtration𝐹 satisfying the properties as above. If each filtration quotient 𝐺𝑘𝐶 = 𝐹𝑘𝐶/𝐹𝑘−1𝐶 isa diagram that has effective homology then 𝐶 has effective homology.

Proof. We define 𝐺 =⨁𝑘≥0

𝐺𝑘𝐶. The sum is not finite, but it is locally finite: By the

properties of 𝐹 , we get that 𝐺𝑘𝐶𝑛 = 0 for 𝑘 ≫ 0. Thus for each 𝑛, we get a finitedirect sum of diagrams 𝐺𝑘𝐶𝑛 : ℐ → Ch+ that have effective homology and it followsthat 𝐺 has effective homology.

If we take the direct sum of the given strong equivalences 𝐺𝑘 ⇐⇐ 𝐺𝑘 ⇒⇒ 𝐺ef𝑘 , we

obtain a strong equivalence 𝐺⇐⇐ 𝐺⇒⇒ 𝐺ef . The chain complex 𝐺ef is equipped witha filtration degree (coming from 𝐹 ) and the filtration degree can be extended via thereductions to 𝐺 and 𝐺.

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The diagram 𝐺 differs from 𝐶 only by a perturbation of its differential. Sincethis perturbation on 𝐺 decreases the filtration degree, while the homotopy operatorpreserves it, we can apply the perturbation lemmas 2.42, 2.43 to obtain a strong equi-valence 𝐶 ⇐⇐ 𝐶 ⇒⇒ 𝐶ef .

2.6 Homotopy colimit and cofibrant replacement have effectivehomology

In this section, we present results regarding homotopy colimits and cofibrant replace-ments:

Proposition 2.48. Let ℐ be a finite category and let 𝑋 : ℐ → sSet have pointwiseeffective homology. Then the space hocolim𝑋 has effective homology.

Proposition 2.49. Let ℐ be a finite category and let 𝑋 : ℐ → sSet have pointwiseeffective homology. Then there is an algorithm which provides a diagram 𝑋cof which isa cofibrant replacement of 𝑋 and has effective homology as a diagram.

We now formulate a theorem regarding effective homology of the Bousfield–Kanmodel of hoLan𝑝𝑋 and we obtain both Proposition 2.48 and 2.49 as straightforwardcorollaries.

Theorem 2.50. Let 𝑋 : ℐ → sSet be diagram that has pointwise effective homology,𝑝 : ℐ → 𝒥 a functor between finite categories. Then hoLan𝑝𝑋 : 𝒥 → sSet is a diagramthat has effective homology.

We remind that by choosing 𝒥 = ℐ and 𝑝 = id we get hoLan𝑝𝑋 = 𝑋cof and ifwe set 𝒥 = * we get hoLan𝑝𝑋 = hocolim𝑋. The following proof of Theorem 2.50therefore establishes proofs of Proposition 2.48 and 2.49.

Proof. For any category ℐ there is a simplicial set 𝑁ℐ, the nerve of ℐ. The simplicialset 𝑁ℐ can be seen as a homotopy colimit of the diagram consisting of points. Thenthere is a projection 𝑞 : hoLan𝑝𝑋 → 𝑁ℐ given as a projection onto⨆

𝑛

⨆𝑖0,··· ,𝑖𝑛

Δ𝑛 × ℐ(𝑖0, 𝑖1)× · · · × ℐ(𝑖𝑛−1, 𝑖𝑛)/∼

and we define the skeleton of hoLan𝑝𝑋:

sk𝑘hoLan𝑝𝑋 = 𝑞−1(𝑠𝑘𝑘𝑁ℐ).

We want to use Lemma 2.47 to prove that the diagram 𝐶(hoLan𝑝𝑋) : ℐ → Ch+ haseffective homology. Therefore, we first have to introduce a filtration 𝐹 on the diagramof chain complexes 𝐶(hoLan𝑝𝑋). We define 𝐹 as follows:

𝐹𝑘𝐶(hoLan𝑝𝑋) = 𝐶(sk𝑘hoLan𝑝𝑋)

Denoting 𝐺𝑘 = 𝐹𝑘/𝐹𝑘−1, we get

𝐺𝑘(𝐶(hoLan𝑝𝑋)) =⨁𝑖0→···→𝑖𝑘

nondeg.

𝐶(Δ𝑘×𝑋(𝑖0)× 𝒥 (𝑝(𝑖𝑘),−), 𝜕Δ𝑘 ×𝑋(𝑖0)× 𝒥 (𝑝(𝑖𝑘),−)).

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The sum is taken over chains of morphisms in ℐ that do not contain identity as thoseare factored out when computing 𝐺𝑘 = 𝐹𝑘/𝐹𝑘−1. By the finiteness of ℐ, the numberof nondegenerate chains of morphisms of length 𝑘 is finite, so the sum is finite.

Using the Eilenberg–Zilber reduction we get that 𝐺𝑘(𝐶(hoLan𝑝𝑋)) is strongly equi-valent to⨁

𝑖0→···→𝑖𝑘nondeg.

𝐶(Δ𝑘, 𝜕Δ𝑘)⊗ 𝐶(𝑋(𝑖0))⊗Z𝒥 (𝑝(𝑖𝑘),−) ∼=

∼=⨁

𝑖0→···→𝑖𝑘nondeg.

𝐶(𝑋(𝑖0))[𝑘]⊗ Z𝒥 (𝑝(𝑖𝑘),−)(2.10)

To finish the proof, it remains to show that under the assumptions of Theorem 2.50the diagrams 𝐺𝑘 have effective homology.

The diagram Z𝒥 (𝑝(𝑖𝑘),−) is an effective diagram of chain complexes, which wasdemonstrated in Example 2.34. Therefore it has effective homology. Further𝐶(𝑋(𝑖0))[𝑘] is a chain complex with effective homology. Using Corollary 2.39 we getthat the diagram of chain complexes 𝐶(𝑋(𝑖0))[𝑘]⊗Z𝒥 (𝑝(𝑖𝑘),−) has effective homology.As 𝐺𝑘(𝐶(hoLan𝑝𝑋)) is strongly equivalent to a finite direct sum of chain complexesthat has effective homology, it has effective homology. Now we can apply Lemma 2.47to complete the proof.

Functorial cofibrant replacement. In this section, we present an algorithm thatcomputes a cofibrant replacement of a map between two diagrams. We formally describethis result as the following Lemma:

Lemma 2.51. Let 𝜙 : 𝑌 → 𝑃 be a computable map of diagrams that have pointwiseeffective homology. Then there is an algorithm that computes 𝑌 cof ,𝑃 cof as diagramsthat have effective homology together with maps repl(𝜙), repl𝑃 and repl𝑌 in the com-mutative diagram

𝑌 cof repl(𝜙)//

repl𝑌

��

𝑃 cof

repl𝑃

��

𝑌𝜙

// 𝑃

Proof. We remind, that the diagram 𝑌 cof can be computed by Proposition 2.49 as adiagram that has effective homology and we have the following model for it:

𝑌 cof(−) =⨆𝑛

⨆𝑖0,··· ,𝑖𝑛

Δ𝑛 × 𝑌 (𝑖0)× ℐ(𝑖0, 𝑖1)× · · · × ℐ(𝑖𝑛−1, 𝑖𝑛)× ℐ(𝑖𝑛,−)/∼.

The description of the model can be further simplified (see [9]), which gives us

𝑌 cof(𝑖) =⨆𝑛

⨆𝑓 : 𝑖0→···→𝑖𝑛→𝑖

Δ𝑛 × 𝑌 (𝑖0)/≈.

We will not specify the relation ≈ here. However, we remark that each 𝑌 cof(𝑖) is gluedfrom simplicial sets of the form Δ𝑛×𝑌 (𝑖0), for some 𝑖0 ∈ ℐ and 𝑛 ≥ 0. Therefore it isenough to describe the mapping repl𝑌 : 𝑌 cof → 𝑌 on such cells, which will further beindexed by (𝑛, 𝑓), where 𝑓 : 𝑖0 → · · · → 𝑖𝑛 → 𝑖. Let 𝑓 : 𝑖0 → 𝑖 denote the compositionof maps in the chain 𝑓 .

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For any standard 𝑛-simplex Δ𝑛, there is a unique simplicial map 0: Δ𝑛 → Δ0 givenby mapping the unique 𝑛-cell 𝑒𝑛 of Δ𝑛 to 𝑛-fold degeneracy (𝑠0)

𝑛𝑒0. Further, one cansee that simplicial sets 𝑌 (𝑖) and Δ0 × 𝑌 (𝑖) are isomorphic. We then define

repl𝑌 (𝑛, 𝑓)(𝑒𝑛, 𝑥) = (0(𝑒𝑛), 𝑓(𝑥)).It remains to give repl(𝜙) : 𝑌 cof → 𝑃 cof :

repl(𝜙)(𝑛, 𝑓)(𝑒𝑛, 𝑥) = (𝑛, 𝑓)(𝑒𝑛, 𝜙(𝑥)).

By comparing the formulas, one can see that the diagram commutes.

2.7 Effective abelian groups

We have previously defined simplicial sets with effective homology in an abstract wayessentially as a black box with which we can perform certain computations. We furtherstated that given a simpicial set that has effective homology 𝑋, one can algorithmicallycompute the homology groups 𝐻𝑘(𝑋).

We will now describe the abstract computational (effective) model for finitely gen-erated abelian groups first introduced in [5]. The model enables us to compute kernels,cokernels and extensions in an algorithmic way. The main object of interest in thissection is a notion of fully effective abelian group.

Definition 2.52. A semi-effective group 𝐴 consists of

∙ a set of representatives 𝒜. The element of 𝐴 represented by an 𝛼 ∈ 𝒜 is denoted[𝛼],

∙ algorithms that provide us with a representative for neutral element, sum of twoelements and inverse. In more detail we can compute 0 ∈ 𝒜 such that [0] = 𝑒,given any 𝛼, 𝛽 ∈ 𝒜 we compute 𝛾 ∈ 𝒜 such that [𝛾] = [𝛼]+[𝛽] and for any 𝛼 ∈ 𝒜we can compute 𝛽 ∈ 𝒜 such that [𝛽] = −[𝛼],

A semi-effective abelian group 𝐴 is fully effective if we are further given

∙ a (computable) list of generators 𝑎1, . . . , 𝑎𝑟 of 𝐴 (given by the representatives)and numbers 𝑞1, . . . , 𝑞𝑟 ∈ {2, 3, . . .} ∪ {0}, denoting the orders of the generators,i.e. 𝑎𝑖 is seen as a generator of Z/𝑞𝑖 where Z/0 ∼= Z,

∙ an algorithm that given 𝛼 ∈ 𝒜 computes integers 𝑧1, . . . , 𝑧𝑟 such that [𝛼] =∑𝑟𝑖=1 𝑧𝑖𝑎𝑖.

We call a homomorphism 𝑓 : 𝐴 → 𝐵 of fully effective abelian groups computablehomomorphism if there is a computable mapping of sets 𝜑 : 𝒜 → ℬ such that 𝑓([𝛼] =[𝜑(𝛼)]).

The structure of fully effective abelian group 𝐴 gives us a computable isomorphism(together with its inverse) 𝐴 ∼= Z/𝑞1 ⊕ · · · ⊕ Z/𝑞𝑟.

There are several algorithmic constructions for effective abelian groups. The proofof the following two lemmas can be found in [5].

Lemma 2.53. Let 𝑓 : 𝐴→ 𝐵 be a computable homomorphism of fully effective abeliangroups. Then both Ker(𝑓) and Coker(𝑓) can be represented as fully effective abeliangroups.

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Lemma 2.54 (Lemma 3.7 in [5]). Let there be a short exact sequence of semi-effectiveabelian groups

0 // 𝐴𝑓// 𝐵 𝑔

//

𝑟tt

𝐶 //

𝜎tt

0

where

(1) Groups 𝐴 and 𝐶 are fully effective, 𝑓, 𝑔 are computable homomorphisms;

(2) 𝑟 : Ker 𝑔 → 𝐴 is a computable map (in general not a homomorphism) satisfying𝑓(𝑟(𝑏)) = 𝑏 for every 𝑏 ∈ 𝐵 with 𝑔(𝑏) = 0;

(3) 𝜎 is defined on the level of representatives, i.e. 𝜎 : 𝒞 → ℬ and behaves as a sectionfor 𝑔, so we have 𝑔([𝜎(𝛾)]) = [𝛾] for all 𝛾 ∈ 𝒞.

Then the structure of fully effective abelian group can be obtained on 𝐵.

The next result will be utilized in further proofs:

Lemma 2.55. Let 𝐴,𝐵 be fully effective abelian groups. Then Hom(𝐴,𝐵) and 𝐴⊗𝐵are fully effective abelian groups.

Proof. Suppose that 𝑎1, . . . , 𝑎𝑘 are the generators of 𝐴 and 𝑏1, . . . , 𝑏𝑛 are the generatorsof 𝐵. Let 𝑞1, . . . , 𝑞𝑘, 𝑝1, . . . , 𝑝𝑛 be the numbers denoting the orders as in the Defini-tion 2.52. Then we can identify Hom(𝐴,𝐵) with an abelian group, with generators 𝑥𝑖,𝑗,1 ≤ 𝑖 ≤ 𝑘, 1 ≤ 𝑗 ≤ 𝑛 and numbers 𝑟𝑖,𝑗 ∈ N∪ {0} when we identify Z/1 with the trivialgroup such that

𝑟𝑖,𝑗 =

⎧⎨⎩gcd(𝑞𝑖, 𝑝𝑗) if 𝑞𝑖, 𝑝𝑗 > 0,𝑝𝑗 if 𝑞𝑖 = 01 if 𝑞𝑖 > 0, 𝑝𝑗 = 0.

By removing the trivial groups (generators 𝑥𝑖,𝑗 where 𝑟𝑖,𝑗 = 1) from our description ofHom(𝐴,𝐵), we obtain a structure of a fully effective abelian group on Hom(𝐴,𝐵).

For the tensor product, we remark that the generators are

{𝑎𝑖 ⊗ 𝑏𝑗|1 ≤ 𝑖 ≤ 𝑘, 1 ≤ 𝑗 ≤ 𝑛}

and order of the generator is gcd(𝑞𝑖, 𝑝𝑗).

Let ℐ be a finite category and let 𝜋 : ℐ → Ab be a diagram such that every 𝜋(𝑖) isfully effective abelian and every morphism is a computable homomorphism. We thencall 𝜋 a diagram of fully effective abelian groups. As a consequence of the previouslemma, we get

Lemma 2.56. Let ℐ be a finite category and let 𝜋,𝜌 : ℐ → Ab be diagrams of fullyeffective abelian groups. Then Hom(𝜋,𝜌) is a fully effective abelian group.

Proof. Notice that each Hom(𝜋(𝑖),𝜌(𝑖′)) is a fully effective abelian group byLemma 2.55 and that Hom(𝜋,𝜌) ≤

∏𝑖∈ℐ Hom(𝜋(𝑖),𝜌(𝑖)). We define a homomorphism

𝐹 :∏𝑖∈ℐ

Hom(𝜋(𝑖),𝜌(𝑖))→∏

𝑓 : 𝑖→𝑖′

Hom(𝜋(𝑖),𝜌(𝑖′))

for any 𝑔 ∈∏

𝑖∈ℐ Hom(𝜋(𝑖),𝜌(𝑖)) as follows:

𝐹 (𝑔) = (𝜌(𝑓)𝑔(𝑖)− 𝑔(𝑖′)𝜋(𝑓))𝑓 : 𝑖→𝑖′ .

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Then the desired Hom(𝜋,𝜌) is equal to Ker𝐹 . As ℐ is a finite category, both∏𝑖∈ℐ Hom(𝜋(𝑖),𝜌(𝑖)) and

∏𝑓 : 𝑖→𝑖′ Hom(𝜋(𝑖),𝜌(𝑖′)) are fully effective abelian groups.

Because 𝐹 is computable, Lemma 2.53 gives us that Ker𝐹 is fully effective.

From the perspective of category theory, the previous lemma can be seen as aspecial case of a computation of the end of functor 𝐺 : ℐop × ℐ → Ab, where 𝐺(𝑖, 𝑗) BHom(𝜋(𝑖),𝜌(𝑗)). Formally, one writes

∫𝑖∈ℐ 𝐺(𝑖, 𝑖) =

∫𝑖∈ℐ Hom(𝜋(𝑖),𝜌(𝑖)) = Hom(𝜋,𝜌),

see e.g. chapter V in [28].Dually, to the end (or Hom) there is a construction known as a coend (or tensor

product): This time one of the diagrams of groups, say 𝜌 is contravariant, so 𝜌 : ℐop →Ab. We define ∫ 𝑖∈ℐ

𝜋(𝑖)⊗ 𝜌(𝑖) = 𝜋 ⊗ℐ 𝜌 =∑

𝑖∈ℐ,𝑖′∈ℐ

𝜋(𝑖)⊗ 𝜌(𝑖′)/ ∼ (2.11)

where ∼ is given by (𝑓*𝑎, 𝑏) ∼ (𝑎, 𝑓 *𝑏), where 𝑓 : 𝑖→ 𝑖′ is any arrow in ℐ, 𝑎 ∈ 𝜋(𝑖), 𝑏 ∈𝜌(𝑖′). We deduce the following result:

Lemma 2.57. Let ℐ be a finite category and let 𝜋 : ℐ → Ab and 𝜌 : ℐop → Ab bediagrams of fully effective abelian groups. Then 𝜋⊗ℐ 𝜌 is a fully effective abelian group.

Proof. The proof is similar to the proof of Lemma 2.56. We first need to show thattensor product of two fully effective abelian group is fully effective, but that is due toLemma 2.55. We define morphism

𝐹 :∑

𝑓 : 𝑖→𝑖′

𝜋(𝑖)⊗ 𝜌(𝑖′)→∑𝑖∈ℐ

𝜋(𝑖)⊗ 𝜌(𝑖)

by 𝐹 ((𝑎⊗𝑏)𝑓 ) = 𝑓*𝑎⊗𝑏−𝑎⊗𝑓 *𝑏 for some 𝑎 ∈ 𝜋(𝑖), 𝑏 ∈ 𝜌(𝑖′). Because Coker𝐹 = 𝜋⊗ℐ𝜌we can use Lemma 2.53 to obtain the result.

We remark that given a diagram 𝑋 that has effective homology, these computationscan be utilized to compute the chain complexes 𝐶ℐ* (𝑋,𝜋), 𝐶*ℐ(𝑋,𝜋) and further theBredon homology and cohomology groups.

2.8 Polycyclic groups

In Chapter 4, we will encounter computations with nonabelian groups and we will thusneed to extend some of the machinery from abelian groups to a wider class of groups,called polycyclic. Results presented here appeared first in [15].

Definition 2.58. A group 𝐺 is called polycyclic, if it has a subnormal series withcyclic factors. In detail, there exists a sequence of subgroups

𝐺 = 𝐺𝑟 ≥ 𝐺𝑟−1 ≥ · · · ≥ 𝐺1 ≥ 𝐺0 = 0 (2.12)

such that:∙ 𝐺𝑖−1 is a normal subgroup of 𝐺𝑖 for 𝑖 = 1, . . . , 𝑟,∙ 𝐺𝑖/𝐺𝑖−1 is a cyclic group for 𝑖 = 1, . . . , 𝑟.

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Example 2.59. Every finitely generated abelian group is polycyclic: when 𝐺 ∼= Z/𝑞1⊕· · · ⊕ Z/𝑞𝑟 with the corresponding generators 𝑔1, . . . , 𝑔𝑟, the filtration is given by 𝐺𝑖 =[𝑔1, . . . , 𝑔𝑖], i.e. the subgroup generated by 𝑔1, . . . , 𝑔𝑖.

Suppose that elements 𝑔𝑖 ∈ 𝐺𝑖 have been chosen in such a way that their imagesin 𝐺𝑖/𝐺𝑖−1 are the generators of these cyclic groups (clearly, such a choice is possible).Denoting by 𝑞𝑖 the order of 𝐺𝑖/𝐺𝑖−1, the following map

Z/𝑞1 × · · · × Z/𝑞𝑟 −→ 𝐺

(𝑧1, . . . , 𝑧𝑟) ↦−→ 𝑧1𝑔1 + · · ·+ 𝑧𝑟𝑔𝑟 𝑧𝑖 ∈ {0, . . . 𝑞𝑖−1}

is easily seen to be bijective: given 𝑔 ∈ 𝐺, consider its image 𝑧𝑟 ∈ 𝐺𝑟/𝐺𝑟−1 ∼= Z/𝑞𝑟.Then 𝑔 − 𝑧𝑟𝑔𝑟 ∈ 𝐺𝑟−1 and we continue in the same manner to show that 𝑔 − 𝑧𝑟𝑔𝑟 −· · · − 𝑧1𝑔1 ∈ 𝐺0 = 0, i.e. 𝑔 = 𝑧1𝑔1 + · · · + 𝑧𝑟𝑔𝑟 in a unique way. In particular, 𝐺 isgenerated by 𝑔1, . . . , 𝑔𝑟. At the same time, the word problem in 𝐺, i.e. the problem ofdeciding whether two given words in the generators 𝑔𝑖 are equal, can be translated toZ/𝑞1 × · · · × Z/𝑞𝑟 and easily solved there. This leads to our notion of a fully effectivepolycyclic group.

Definition 2.60. We say that a semi-effective group 𝐺, represented by a set 𝒢, isfully effective polycyclic if it is polycyclic with subnormal series (2.12) and a bijectionZ/𝑞1 × · · · ×Z/𝑞𝑟 ∼= 𝐺 as above is computable together with its inverse. In detail, thisconsists of∙ a finite list of elements 𝑔1 ∈ 𝐺1, . . . , 𝑔𝑟 ∈ 𝐺𝑟 (given by representatives) and the

orders 𝑞1, . . . , 𝑞𝑟 ∈ {2, 3, . . .} ∪ {0} of 𝐺𝑘/𝐺𝑘−1 (where 𝑞𝑘 = 0 gives Z/𝑞𝑘 = Z),∙ an algorithm that, given 𝛾 ∈ 𝒢, computes integers 𝑧1, . . . , 𝑧𝑟 so that [𝛾] = 𝑧1𝑔1 +· · ·+ 𝑧𝑟𝑔𝑟; each coefficient 𝑧𝑖 is unique within Z/𝑞𝑖, i.e. 𝑧𝑖 ∈ {0, . . . 𝑞𝑖−1}.

As explained just prior to the definition, the algorithm in the second point is equi-valent to the computability of the projections 𝑝𝑖 : 𝐺𝑖 → 𝐺𝑖/𝐺𝑖−1 ∼= Z/𝑞𝑖.

Computations with fully effective polycyclic groups. Here, we show that similarto fully effective abelian groups, fully effective polycyclic groups are closed under kernelsand extensions.

Proposition 2.61. Let 𝐺 be a fully effective polycyclic group, 𝐻 a fully effective abeliangroup and 𝑓 : 𝐺 → 𝐻 a computable homomorphism. Then it is possible to compute𝐾 = Ker 𝑓 as a fully effective polycyclic group.

Proof. We will proceed by induction with respect to the length 𝑟 of the subnormal seriesfor 𝐺. For 𝑟 = 1 we can apply Lemma 2.53. For 𝑟 > 1, we denote 𝐾𝑖 = Ker 𝑓 |𝐺𝑖

=𝐺𝑖 ∩𝐾. Obviously, 𝐾 = 𝐾𝑟 ≥ 𝐾𝑟−1 ≥ · · · ≥ 𝐾1 is the subnormal series for the group𝐾. In the following diagram, every row is a short exact sequence and so are the solid

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columns.

0

��

0

��

0

��

0 // 𝐾𝑟−1� _

��

� � // 𝐾𝑟� _

��

// 𝐾𝑟/𝐾𝑟−1 //

� _

��

0

0 // 𝐺𝑟−1� � //

𝑓��

𝐺𝑟//

𝑓��

𝐺𝑟/𝐺𝑟−1 //

𝑓 ′��

0

0 // 𝑓(𝐺𝑟−1)� � //

��

𝑓(𝐺𝑟) //

��

𝑓(𝐺𝑟)/𝑓(𝐺𝑟−1) //

��

0

0 0 0

It is easy to see that the dashed column is then also exact. By induction, we supposethat𝐾𝑟−1 is fully effective polycyclic. By Lemma 2.53, it is possible to compute Ker 𝑓 ′ ∼=𝐾𝑟/𝐾𝑟−1; say that it is generated by 𝑡𝑟 ∈ 𝐺𝑟/𝐺𝑟−1 ∼= Z/𝑞𝑟. This means that 𝑓(𝑡𝑟𝑔𝑟) ∈𝑓(𝐺𝑟−1) and thus, from the knowledge of the generators of 𝐺𝑟−1, it is possible tocompute some ℎ ∈ 𝐺𝑟−1 with 𝑓(𝑡𝑟𝑔𝑟) = 𝑓(ℎ). Finally, −ℎ + 𝑡𝑟𝑔𝑟 ∈ 𝐾𝑟 is the requiredelement which maps to the generator 𝑡𝑟 ∈ 𝐾𝑟/𝐾𝑟−1. The projection 𝐾𝑟 → 𝐾𝑟/𝐾𝑟−1 ∼=Z/(𝑞𝑟𝑡−1𝑟 ) is the composition

𝐾𝑟� � // 𝐺𝑟

// 𝐺𝑟/𝐺𝑟−1 ∼= Z/𝑞𝑟𝑡−1𝑟 × // Z/(𝑞𝑟𝑡−1𝑟 )

(the multiplication by 𝑡−1𝑟 is defined on the image of 𝐾𝑟) and is thus computable.

The following corollary states that we can further compute kernels of computablemaps between fully effective polycyclic groups.

Corollary 2.62. Let 𝐺, 𝐻 be fully effective polycyclic groups and 𝑓 : 𝐺 → 𝐻 a com-putable homomorphism. Then it is possible to compute 𝐾 = Ker 𝑓 as a fully effectivepolycyclic group.

Proof. Suppose that 𝐻 has a subnormal series

𝐻 = 𝐻𝑠 ≥ 𝐻𝑠−1 ≥ · · · ≥ 𝐻0 = 0

We put 𝐾𝑗 = 𝑓−1(𝐻𝑗) and observe that 𝐾0 = 𝐾 and 𝐾𝑠 = 𝐺 so 𝐾𝑠 is a fullyeffective polycyclic group. Now we can carry out the proof by induction from 𝑠 to 0

using Proposition 2.61 and the fact that 𝐾𝑗−1 is the kernel of the composition 𝐾𝑗 𝑓−−→𝐻𝑗 −→ 𝐻𝑗/𝐻𝑗−1 with abelian codomain.

Remark 2.63. It is also possible to compute cokernels, see [42]. However, we do notsee a way of controlling the running time of such an algorithm.

Finally, we generalize lemma 2.54 and show that fully effective polycyclic groupsare closed under extensions in the following sense:

Proposition 2.64. Suppose that there is given a short exact sequence of semi-effectivegroups

0 // 𝐾𝑓// 𝐺 𝑔

//

𝑟tt

𝐻 //

𝜎tt

0

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with 𝐾, 𝐻 fully effective polycyclic, 𝑓 , 𝑔 computable homomorphisms, 𝑟 : im 𝑓 → 𝐾 acomputable inverse of 𝑓 and 𝜎 : ℋ → 𝒢 a computable mapping such that 𝑔[𝜎(𝜂)] = [𝜂].Then there is an algorithm that equips 𝐺 with a structure of a fully effective polycyclicgroup.

Proof. Let 𝐻𝑠 ≥ 𝐻𝑠−1 ≥ · · · ≥ 𝐻0 and 𝐾𝑡 ≥ 𝐾𝑡−1 ≥ · · · ≥ 𝐾0 = 0 be the subnormalseries for 𝐻 and 𝐾, respectively. Then we have the following filtration

𝐺 = 𝑔−1(𝐻𝑠) ≥ 𝑔−1(𝐻𝑠−1) ≥ · · · ≥ 𝑔−1(𝐻0) = 𝑓(𝐾𝑡) ≥ 𝑓(𝐾𝑡−1) ≥ · · · ≥ 𝑓(𝐾0) = 0

with the filtration quotients either 𝐻𝑖/𝐻𝑖−1 or 𝐾𝑗/𝐾𝑗−1, the corresponding projections

𝑔−1(𝐻𝑖)𝑔−−→ 𝐻𝑖 −→ 𝐻𝑖/𝐻𝑖−1,

𝑓(𝐾𝑗)𝑟−→ 𝐾𝑗 −→ 𝐾𝑗/𝐾𝑗−1,

and the generators either [𝜎(𝜂𝑖)] when 𝜂𝑖 represents the generator ℎ𝑖 ∈ 𝐻𝑖 or 𝑓(𝑘𝑗)when 𝑘𝑗 ∈ 𝐾𝑗 is the generator.

2.9 Eilenberg–MacLane spaces and diagrams

Given a group 𝜋 and an integer 𝑘 ≥ 0 an Eilenberg–MacLane space 𝐾(𝜋, 𝑘) is a spacesatisfying

𝜋𝑖(𝐾(𝜋, 𝑘)) =

{𝜋 for 𝑖 = 𝑘,0 else.

In the rest of the thesis, by 𝐾(𝜋, 𝑘) we will always mean the simplicial model which isdefined in [27, page 101]:

𝐾(𝜋, 𝑘)𝑞 = 𝑍𝑘(Δ𝑞; 𝜋),

where Δ𝑞 ∈ sSet is the standard 𝑞–simplex and 𝑍𝑘 denotes the cocycles. This meansthat each 𝑞–simplex is regarded as a labelling of the 𝑘–dimensional faces of Δ𝑞 byelements of 𝜋 such that they add up to 0 on the boundary of every (𝑘 + 1)-simplex inΔ𝑞. The boundary and degeneracy operators in 𝐾(𝜋, 𝑘) are given as follows: For any𝜎 ∈ 𝐾(𝜋, 𝑘)𝑞, 𝑑𝑖(𝜎) ∈ 𝐾(𝜋, 𝑘)𝑞−1 is given by a restriction of 𝜎 of 𝐾(𝜋, 𝑘) on the 𝑖-thface of Δ𝑞. To define the degeneracy we first introduce mapping 𝜂𝑖 : {0, 1, . . . , 𝑞+1} →{0, 1, . . . , 𝑞} given by

𝜂𝑖(𝑗) =

{𝑗 for 𝑗 ≤ 𝑖𝑗 − 1 for 𝑗 > 𝑖

Every mapping 𝜂𝑖 defines a map 𝐶*(𝜂𝑖) : 𝐶*(Δ𝑞+1) → 𝐶*(Δ

𝑞). The degeneracy 𝑠𝑖𝜎 isnow a labelling of the 𝑘–faces of Δ𝑞+1 induced by 𝐶*(𝜂𝑖) (see [27], p.101).

In the same way as the Eilenberg–MacLane space, we define the path space 𝐸(𝜋, 𝑘)by the formula

𝐸(𝜋, 𝑘)𝑞 = 𝐶𝑘(Δ𝑞; 𝜋).

Given a 𝑞–simplex 𝜎 ∈ 𝐸(𝜋, 𝑘)𝑞, the coboundary operator 𝛿 : 𝐶𝑘(Δ𝑞; 𝜋)→ 𝐶𝑘+1(Δ𝑞; 𝜋)produces a 𝑞–simplex 𝛿𝜎 of 𝐾(𝜋, 𝑘+1). Abusing the notation, the coboundary operatordescribes a simplicial map 𝛿 : 𝐸(𝜋, 𝑘) → 𝐾(𝜋, 𝑘 + 1). In fact, 𝛿 is a Kan fibration.Further, according to [27, Theorem 23.10],

𝛿 : 𝐸(𝜋, 𝑘)→ 𝐾(𝜋, 𝑘 + 1)

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is a fibration with fibre 𝐾(𝜋, 𝑘) and we have a simplicial description of 𝐸(𝜋, 𝑘) as atwisted product

𝐾(𝜋, 𝑘)×𝜏 𝐾(𝜋, 𝑘 + 1),

where we remark, that a 𝑘-simplex of Δ𝑞 can be represented by an increasing (𝑘 + 1)-tuple (𝑖0, 𝑖1, . . . , 𝑖𝑘) and the twisting operator 𝜏 : 𝐾(𝜋, 𝑘 + 1) → 𝐾(𝜋, 𝑘) is definedby

𝜏(𝑧)((𝑖0, 𝑖1, . . . , 𝑖𝑘)) = 𝑧(0, 𝑖0+1, 𝑖1+1, . . . , 𝑖𝑘+1)−𝑧(1, 𝑖0+1, 𝑖1+1, . . . , 𝑖𝑘+1). (2.13)

This description of 𝐸(𝜋, 𝑘) as a twisted product can be used to show that 𝐸(𝜋, 𝑘)has effective homology.

Using these models, we define diagrams of Eilenberg–MacLane spaces 𝐾(𝜋, 𝑘) : ℐ →sSet for 𝜋 : ℐ → Ab by setting 𝐾(𝜋, 𝑘)(𝑖) = 𝐾(𝜋(𝑖), 𝑘). The maps in the diagram𝐾(𝜋, 𝑘) are induced by the homomorphisms in 𝜋. Analogously, we define the diagram𝐸(𝜋, 𝑘) of spaces 𝐸(𝜋(𝑖), 𝑘).

The simplicial sets 𝐾(𝜋, 𝑘), 𝐸(𝜋, 𝑘) defined above have the property that simplicialmaps 𝑋 → 𝐾(𝜋, 𝑘) correspond to cocycles 𝑍𝑘(𝑋; 𝜋) and simplicial maps 𝑋 → 𝐸(𝜋, 𝑘)correspond to cochains 𝐶𝑘(𝑋; 𝜋). This enables us to represent simplicial maps by co-chains and cocycles. We further remark that given a simplicial set 𝑋 the set of homo-topy classes of maps [𝑋,𝐾(𝜋, 𝑘)] corresponds to the elements of the group 𝐻𝑘(𝑋; 𝜋).

For Bredon cohomology (see Definition 2.4), a similar thing holds as well: Morph-isms 𝑋 → 𝐸(𝜋, 𝑘) correspond to cochains 𝐶𝑘

ℐ(𝑋;𝜋), morphisms 𝑋 → 𝐾(𝜋, 𝑛) cor-respond to cocycles 𝑍𝑘

ℐ(𝑋;𝜋). If the diagram 𝑋 is further cofibrant (and 𝐾(𝜋, 𝑘) isfibrant but this is always true for our model), the set of homotopy classes of maps ofdiagrams [𝑋,𝐾(𝜋, 𝑘)]ℐ corresponds to the elements in 𝐻𝑘

ℐ (𝑋;𝜋).This correspondence will be described in more detail in further sections.

Evaluation maps. Assume that 𝑘 ≥ 1 and 𝜋 ∈ Ab. As 𝐸(𝜋, 𝑘)𝑘 = 𝐶𝑘(Δ𝑘; 𝜋), the 𝑘-simplices of 𝐸(𝜋, 𝑘) can be seen as labellings of the unique 𝑘–simplex Δ𝑘 by an elementof 𝜋. Using the definition of the model of 𝐸(𝜋, 𝑘), we have

𝐶𝑘(Δ𝑘; 𝜋) ∼= Hom(𝐶𝑘(Δ𝑘;Z), 𝜋) ∼= Hom(Z, 𝜋) ∼= 𝜋.

This gives us a bijection ev : 𝐸(𝜋, 𝑘)𝑘 = 𝐶𝑘(Δ𝑘; 𝜋) ∼= 𝜋. The map ev is defined byreading the value on the unique 𝑘–simplex. This is why we will call this map evalu-ation. We remark that in this situation we further have 𝐸(𝜋, 𝑘)𝑘 = 𝐾(𝜋, 𝑘)𝑘, because𝐶𝑘(Δ𝑘; 𝜋) = 𝑍𝑘(Δ𝑘; 𝜋).

Next, we define a map 𝐶𝑘(𝐸(𝜋, 𝑘)) → 𝜋 as the unique extension of ev to the freeabelian group generated by 𝐸(𝜋, 𝑘)𝑘.

Abusing the notation, we will call this homomorphism ev. Similarly, we get homo-morphism 𝐶𝑘(𝐾(𝜋, 𝑘))→ 𝜋 and we again denote this homomorphism by ev.

Homomorphism ev is functorial, hence we can extend its definition to diagrams. Toemphasise the difference, we denote the resulting homomorphisms of diagrams ev, i.e.ev : 𝐶𝑘(𝐸(𝜋, 𝑘))→ 𝜋 and ev : 𝐶𝑘(𝐾(𝜋, 𝑘))→ 𝜋.

In the proof of Theorem A, the following two lemmas will be used.

Lemma 2.65. The homomorphism ev : 𝐶𝑘(𝐾(𝜋, 𝑘)) → 𝜋 induces an isomorphismof diagrams

𝐻𝑘(𝐾(𝜋, 𝑘))→ 𝜋.

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Proof. According to [7], Lemma 4.3, the statement of the Lemma holds pointwise andhence the mapping of diagrams 𝐻𝑘(𝐾(𝜋, 𝑘)) → 𝜋 induced by ev is an isomorphismof diagrams.

Given ev : 𝐶𝑘(𝐸(𝜋, 𝑘))→ 𝜋 and ev : 𝐶𝑘+1(𝐾(𝜋, 𝑘 + 1))→ 𝜋, we define a homo-morphism

ℎ B ev+ ev : 𝐶𝑘(𝐸(𝜋, 𝑘))⊕𝐶𝑘+1(𝐾(𝜋, 𝑘 + 1))→ 𝜋 ⊕ 𝜋 → 𝜋

where the last arrow is the standard addition. Using the map 𝛿 : 𝐸(𝜋, 𝑘)→𝐾(𝜋, 𝑘+1),which is a fibration pointwise, we can construct the algebraic mapping cone

Cone𝑘+1(𝛿*) = 𝐶𝑘(𝐸(𝜋, 𝑘))⊕𝐶𝑘+1(𝐾(𝜋, 𝑘 + 1)),

and we in fact have a homomorphism ℎ : Cone𝑘+1(𝛿*)→ 𝜋. This homomorphism hasa nice property:

Lemma 2.66 (Lemma 4.4 in [7]). The homomorphism ℎ sending (𝜎, 𝜏) to ev 𝜎+ ev 𝜏induces an isomorphism

𝐻𝑘+1(Cone*(𝛿*))→ 𝜋.

Proof. We will repeat the proof for simplicial sets (i.e. the pointwise version of thestatement) presented in [7], Lemma 4.4 as it fits our case as well. For brevity, we write𝐸 = 𝐸(𝜋, 𝑘) and 𝐾 = 𝐾(𝜋, 𝑘 + 1) since there are no other Eilenberg–MacLane spacesin this proof.

In order to claim that ℎ induces a map in homology, we verify that it vanishes onall boundaries. Thus, let (𝜎′, 𝜏 ′) ∈ Cone𝑘+2(𝛿*) be a generator, 𝜎′ ∈ 𝐸𝑘+1, 𝜏 ′ ∈ 𝐾𝑘+2.According to the description of the perturbation given in the proof of Lemma 2.45, wehave 𝜕Cone*(𝜎

′, 𝜏 ′) = (−𝜕𝐸𝜎′, 𝛿*(𝜎′)+𝜕𝐾𝜏 ′). Since 𝜏 ′ is a cocycle, we have ev(𝜕𝐾𝜏′) = 0

and it is easily checked that ev(𝜕𝐸𝜎′) = ev(𝛿*(𝜎

′)). It follows that ℎ indeed vanisheson boundaries and induces a homomorphism

ℎ* : 𝐻𝑘+1(Cone*(𝛿*))→ 𝜋.

Now we consider the canonical inclusion 𝐶*(𝐾)→ Cone*(𝛿*), which is a chain map,and thus it induces a map in homology, as in the following diagram (here we use that𝐶𝑘+1(𝐾) = 𝑍𝑘+1(𝐾) and 𝐶𝑘(𝐸) = 𝑍𝑘(𝐸)):

𝐶𝑘+1(𝐾) 𝜄 //

��

𝐶𝑘(𝐸)⊕ 𝐶𝑘+1(𝐾)= Cone𝑘+1(𝛿*)

��

𝜋 𝐻𝑘+1(𝐾)∼=ev*oo

∼=𝜄* // 𝐻𝑘+1(Cone*(𝛿*))

Here ev* on the left in the bottom row is the isomorphism induced by ev as inLemma 2.65.

The map 𝑖* is an isomorphism by the long exact homology sequence of the pair(Cone(𝛿*), 𝐶*(𝐾)), because the quotient Cone(𝛿*)/𝐶*(𝐾) ∼= 𝐶*(𝐸)[1] is the shift ofthe chain complex of a contractible simplicial set 𝐸 (see e.g. [27, Proposition 21.5,Theorem 23.10]), and thus all homology groups of this quotient vanish except for theone in dimension 1.

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Finally, it suffices to verify that ℎ*𝜄* = ev*, but this is clear, since the compositionon the left maps [𝜏 ]

𝜄*↦−−→ [(0, 𝜏)]ℎ*↦−−→ ev 𝜏 .

This finishes the proof of the pointwise verison presented in [7]. To complete theversion for diagrams, it remains to show that all maps in the diagram

𝜋 𝐻𝑘+1(𝐾)ev*oo

𝜄* //𝐻𝑘+1(Cone*(𝛿*))

ℎ*

ff

are natural, which is easy.

Simplicial maps to 𝐸(𝜋, 𝑘) and 𝐾(𝜋, 𝑘). Now we can describe the correspondencebetween simplicial maps to 𝐸(𝜋, 𝑘) and 𝐾(𝜋, 𝑘) and chain maps:

Lemma 2.67 (Lemma 24.2 in [27]). The simplicial maps 𝑓 : 𝑋 → 𝐸(𝜋, 𝑘) are inbijection with cochains 𝜅 : 𝐶𝑘(𝑋)→ 𝜋.

Proof. We present only the main ideas of the proof as the reader can see the fullformal proof in [27]. The most important observation here is that any simplicial map𝑓 : 𝑋 → 𝐸(𝜋, 𝑘) is completely determined by the map 𝑓𝑘 : 𝑋𝑘 → 𝐸(𝜋, 𝑘)𝑘 (the maps𝑓𝑗 : 𝑋𝑗 → 𝐸(𝜋, 𝑘)𝑗, for 0 ≤ 𝑗 < 𝑘 are trivial).

This is based on the fact that for any 𝑛 > 𝑘, 𝐸(𝜋, 𝑘)𝑛 corresponds to labellingsof 𝑘-faces of a standard 𝑛-simplex Δ𝑛 by elements of the group 𝜋. It follows that for𝑥 ∈ 𝑋𝑛, where 𝑛 > 𝑘, the image 𝑓(𝑥) ∈ 𝐸(𝜋, 𝑘) is determined by the images of𝑑𝑖0𝑑𝑖1 . . . 𝑑𝑖(𝑛−𝑘−1)

𝑥 ∈ 𝑋𝑘, i.e. the 𝑘-faces of the 𝑛-simplex 𝑥. The induced map of chaincomplexes 𝑓* : 𝐶*(𝑋)→ 𝐶*(𝐸(𝜋, 𝑘)) is then given by

𝑓𝑘* : 𝐶𝑘(𝑋)→ 𝐶𝑘(𝐸(𝜋, 𝑘)).

The corresponding cochain 𝜅𝑓 is then obtained as the composition

ev ∘𝑓𝑘* : 𝐶𝑘(𝑋)→ 𝜋.

In the opposite direction, take 𝜅 : 𝐶𝑘(𝑋) → 𝜋. Using 𝜋 ∼= 𝐶𝑘(Δ𝑘; 𝜋) ∼= 𝐸(𝜋, 𝑘)𝑘,we have 𝜅 : 𝐶𝑘(𝑋) → 𝐸(𝜋, 𝑘)𝑘. For any 𝜎 ∈ 𝑋𝑘, we define 𝑓𝑘 : 𝑋𝑘 → 𝐸(𝜋, 𝑘)𝑘 by𝑓𝑘(𝜎) = 𝜅(𝜎). Our previous observation implies that 𝑓𝑘 describes 𝑓 .

For Eilenberg-MacLane spaces, the same reasoning gives us

Lemma 2.68 (Lemma 24.3 in [27]). The simplicial maps 𝑓 : 𝑋 → 𝐾(𝜋, 𝑘) are inbijection with cocycles 𝜅 ∈ 𝑍𝑘(𝑋; 𝜋).

We omit the proof (it can be found in [27]) and remark that the correspondencebetween simplicial maps 𝑓 : 𝑋 → 𝐾(𝜋, 𝑘) and cochains 𝜅 ∈ 𝑍𝑘(𝑋; 𝜋) is the sameas described in the proof of Lemma 2.67. Here, one has to prove that for a given𝑓 : 𝑋 → 𝐾(𝜋, 𝑘) the corresponding cochain 𝜅 is a cocycle and, in the opposite direction,to show that for a given cocycle 𝜅 the image of the map 𝑓 : 𝑋 → 𝐸(𝜋, 𝑘) lies in 𝐾(𝜋, 𝑘).

When we generalize the previous results to diagrams, we obtain.

Lemma 2.69. Let 𝑋 : ℐ → sSet and let 𝜋 : ℐ → Ab. The maps 𝑓 : 𝑋 → 𝐸(𝜋, 𝑘) ofdiagrams of simplicial sets are in bijection with cochains 𝜅 ∈ 𝐶𝑘

ℐ(𝑋;𝜋).

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Proof. By Lemma 2.67, the collection of maps

𝑓𝑘(𝑖) : (𝑋(𝑖))𝑘 → 𝐸(𝜋(𝑖), 𝑘)𝑘, 𝑖 ∈ ℐ

is in a one-to-one correspondence with cochains

𝜅(𝑖) : 𝐶𝑘(𝑋(𝑖))→ 𝜋(𝑖), 𝑖 ∈ ℐ.

Let 𝜄 : 𝑖→ 𝑗 for any 𝑖, 𝑗 ∈ ℐ. The following diagrams commute

𝑋(𝑖)𝑓(𝑖)

//

𝑋(𝜄)��

𝐸(𝜋(𝑖), 𝑘)

𝐸(𝜋(𝜄),𝑘)��

𝑋(𝑗)𝑓(𝑗)// 𝐸(𝜋(𝑗), 𝑘)

𝐶𝑘(𝑋(𝑖))𝜅(𝑖)//

𝐶𝑘(𝑋(𝜄))��

𝜋(𝑖)

𝜋(𝜄)��

𝐶𝑘(𝑋(𝑗))𝜅(𝑗)// 𝜋(𝑗)

which means𝐸(𝜋(𝜄), 𝑘) ∘ 𝑓𝑘(𝑖) = 𝑓𝑘(𝑗) ∘𝑋(𝜄).

Using Lemma 2.67, this gives us 𝜋(𝜄)∘𝜅(𝑖) = 𝜅(𝑗)∘𝐶*(𝑋)(𝜄). Therefore the collectionsof cochains 𝜅(𝑖)𝑖∈ℐ describes a cochain 𝜅 ∈ 𝐶𝑘

ℐ(𝑋;𝜋).The other direction is similar: We take a cochain 𝜅 ∈ 𝐶𝑘

ℐ(𝑋;𝜋), which is a collectionof individual cochains 𝜅(𝑖)𝑖∈ℐ : 𝐶𝑘(𝑋(𝑖)) → 𝜋(𝑖), each of these cochains correspondsto a simplicial map 𝑓(𝑖) : 𝑋(𝑖)→ 𝐸(𝜋(𝑖), 𝑘) as in Lemma 2.67.

As above, we have a version for Eilenberg–MacLane spaces.

Lemma 2.70. Let 𝑋 : ℐ → sSet and let 𝜋 : ℐ → Ab. The maps 𝑓 : 𝑋 → 𝐾(𝜋, 𝑘) ofdiagrams of simplicial sets are in bijection with cocycles 𝜅 ∈ 𝑍𝑘

ℐ(𝑋;𝜋).

The proof is omitted as it is just a variation of the proof of Lemma 2.68 usingLemma 2.69.

Finally, we have the following result that can be seen as a generalization of The-orem 24.4 in [27].

Proposition 2.71. Let 𝑋 : ℐ → sSet be cofibrant and let 𝜋 : ℐ → Ab. Then homotopyclasses of maps of diagrams [𝑋,𝐾(𝜋, 𝑘)]ℐ are in bijection with Bredon cohomologygroups 𝐻𝑘

ℐ (𝑋;𝜋).

Proof. Our proof utilizes some basic facts from model categories about the existence ofcertain maps and uses cylinder objects Cyl𝑋 introduced in Definition 2.5. The readercan find details in [11]. For a map 𝑓 from 𝑋 to 𝐸(𝜋, 𝑘) or 𝐾(𝜋, 𝑘), the correspondingcochain (cocycle) will be denoted by 𝜅𝑓 .

Suppose we have 𝑓, 𝑔 : 𝑋 →𝐾(𝜋, 𝑘) and that they are (left) homotopic, i.e. thereexists a homotopy 𝐻 : Cyl𝑋 →𝐾(𝜋, 𝑘).

𝑋 //𝜄0∼//

𝑓 ##

Cyl𝑋

𝐻��

𝑋oo𝜄1∼

oo

𝑔{{

𝐾(𝜋, 𝑘)

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We aim to find some 𝜅𝑎 ∈ 𝐶𝑘−1ℐ (𝑋;𝜋) such that 𝛿(𝜅𝑎) = 𝜅𝑔 − 𝜅𝑓 . From the

projection 𝑝 : Cyl𝑋 → 𝑋, we have a left homotopy 𝐹 = 𝑓 ∘ 𝑝 : Cyl𝑋 → 𝐾(𝜋, 𝑘)between 𝑓 and 𝑓 .

𝑋 //𝜄0∼//

id$$

𝑓

""

Cyl𝑋

𝑝∼��

𝑋oo𝜄1∼

oo

idzz

𝑓

||

𝑋

𝑓��

𝐾(𝜋, 𝑘)

“Subtracting” the first diagram from the second we get the diagram

𝑋 //𝜄0∼

//

𝑓−𝑓 --

Cyl𝑋

𝐻−𝐹=𝑁��

𝑋oo𝜄1∼

oo

𝑔−𝑓qq𝑋

which says that 𝑁 : Cyl𝑋 →𝐾(𝜋, 𝑘) is a homotopy between trivial map 0 and 𝑔−𝑓 .So the following diagram commutes:

𝑋��

𝜄0∼��

0 // 𝐸(𝜋, 𝑘 − 1)

𝛿����

Cyl𝑋 𝑁 //

𝑏66

𝐾(𝜋, 𝑘)

Because 𝑋 is assumed to be cofibrant, the definition of a cylider object tells us that 𝜄0is a cofibration and a weak equivalence. Further 𝛿 is a fibration and using the modelcategory structure on sSetℐ , a lift (dotted arrow) 𝑏 has to exist.

We now set 𝑎 = 𝑏 ∘ 𝜄1 : 𝑋 → 𝐸(𝜋, 𝑘− 1). It remains to show that 𝛿(𝜅𝑎) = 𝜅𝑔 − 𝜅𝑓 ,but one can see that

𝜅𝛿(𝑎) = 𝜅(𝛿𝑏𝜄1) = 𝜅(𝑁𝜄1) = 𝜅(𝐻−𝐹 )𝜄1 = 𝜅𝐻𝜄1 − 𝜅𝐹𝜄1 = 𝜅𝑔 − 𝜅𝑓 .

In the opposite direction, the proof essentially follows the lines of proof of The-orem 24.4 in [27]:

We assume that 𝑓, 𝑔 are cohomologous i.e. there exists 𝑎 : 𝑋 → 𝐸(𝜋, 𝑘 − 1) suchthat 𝛿(𝜅𝑎) = 𝜅𝑔 − 𝜅𝑓 .

We use that fact that 𝐸(𝜋, 𝑘 − 1) is both fibrant and weakly equivalent to a point- this follows from the fact that any 𝐸(𝜋(𝑖), 𝑘 − 1) has this property. Further, 𝜄 is acofibration, see Definition 2.5. Due to the model category structure, there exists a lift𝑏 : Cyl𝑋 → 𝐸(𝜋, 𝑘 − 1) in the diagram

𝑋 ⨿𝑋��

𝜄

��

0⨿𝑎 // 𝐸(𝜋, 𝑘 − 1)

∼����

Cyl𝑋 0 //

𝑏66

Δ0

𝑏 ∘ 𝜄0 = 0 and 𝑏 ∘ 𝜄1 = 𝑎.Again, we take the homotopy 𝐹 = 𝑓 ∘ 𝑝 : Cyl𝑋 →𝐾(𝜋, 𝑘) between 𝑓 and 𝑓 and

we set 𝐻 = 𝐹 + 𝛿(𝑏). Thus we have 𝜅𝐻𝜄0 = 𝜅𝑓 and

𝜅𝐻𝜄1 = 𝜅𝑓 + 𝜅𝛿(𝑎) = 𝜅𝑓 + 𝜅𝑔 − 𝜅𝑓 = 𝜅𝑔.

Therefore 𝐻 is the desired homotopy between 𝑓 and 𝑔.

47

2. Tools

In case 𝑋 is not cofibrant, we get the bijection between [𝑋,𝐾(𝜋, 𝑘)]ℐ and thecohomology group 𝐻𝑘

ℐ (𝑋cof ;𝜋).

Representing a map of diagrams by an effective cocycle. In the Postnikov sys-tem algorithm, we will encounter the following situation: We consider a finite diagram𝑋 : ℐ → sSet that has effective homology and an effective diagram 𝐶ef

* (𝑋) of chaincomplexes such that 𝐶*(𝑋)⇐⇐⇒⇒ 𝐶ef

* (𝑋). Let 𝑓 : 𝐶*(𝑋)→ 𝐶ef* (𝑋) be the composite

(natural) map from the strong equivalence.Let us also consider a (𝑘 + 1)-cocycle

𝜓ef ∈ 𝑍𝑘+1(Hom(𝐶ef* (𝑋),𝜋)) = 𝑍𝑘+1

ℐ,ef (𝑋;𝜋)

for some diagram of fully effective Abelian groups 𝜋. The superscript “ef” emphasisethat the cocycle belongs to the “effective” cochain complex 𝐶*ℐ,ef(𝑋;𝜋) obtained fromthe effective diagram 𝐶ef

* (𝑋) associated to 𝑋. Then 𝜓ef can be represented by asystem of finite matrices, since it can be seen as a collection of maps from chain groups𝐶ef

𝑘+1(𝑋) of finite rank into 𝜋(𝑖), 𝑖 ∈ ℐ, as was described in Lemma 2.69.

Lemma 2.72. The cocycle 𝜓ef defines a simplicial map 𝑋 →𝐾(𝜋, 𝑘 + 1).

Proof. Take the chain map 𝑓 : 𝐶(𝑋)→ 𝐶ef(𝑋). We define a cocycle 𝜓 ∈ 𝑍𝑘+1ℐ (𝑋;𝜋)

as 𝜓 = 𝑓𝜓ef . We have seen in Lemma 2.70 that such a 𝜓 canonically defines a simplicialmap 𝜓 : 𝑋 →𝐾(𝜋, 𝑘 + 1).

A pullback from a fibration of Eilenberg–MacLane diagrams. For our con-struction of Postnikov systems, we will need an operation that is essentially a twistedCartesian product, but in a somewhat different representation. We will have the fol-lowing situation. We are given a diagram of simplicial sets 𝑃 ∈ sSetℐ , plus a mapping𝑓 : 𝑃 →𝐾(𝜋, 𝑛+1), for some diagram of abelian groups 𝜋 ∈ sSetℐ and a fixed 𝑛 ≥ 1.

Now we define a diagram 𝑄 ∈ sSetℐ as the pullback according to the followingcommutative diagram:

𝑄 //

��

𝐸(𝜋, 𝑛)

𝛿��

𝑃𝑓//𝐾(𝜋, 𝑛+ 1)

This means that each 𝑄(𝑖) is the simplicial subset of the Cartesian product 𝑃 (𝑖) ×𝐸(𝜋(𝑖), 𝑛) consisting of the pairs (𝛼, 𝛽) of simplices 𝛼 ∈ 𝑃 (𝑖)ℓ, 𝛽 ∈ 𝐸(𝜋(𝑖), 𝑛)ℓ with𝑓(𝛼) = 𝛿(𝛽).

Because the twisted products can be given effective homology, we get

Corollary 2.73. Given 𝜋, 𝑛,𝑃 , 𝑓 as above, where 𝜋 is a fully effective diagram ofabelian groups, the diagram 𝑃 has pointwise effective homology, and 𝑓 is computable,the pullback diagram 𝑄 has pointwise effective homology.

Proof. Let 𝜏 : 𝐾(𝜋, 𝑛 + 1) → 𝐾(𝜋, 𝑛) be the twisting operator (2.13) in the twis-ted product 𝐸(𝜋, 𝑛) ∼= 𝐾(𝜋, 𝑛)(𝑖) ×𝜏 𝐾(𝜋, 𝑛 + 1)(𝑖) and let 𝜏 * : 𝐾(𝜋, 𝑛 + 1) → 𝑃 ,where 𝜏 *(𝛼) B 𝜏(𝑓(𝛼)). Then Corollary 2.22 yields effective homology for each twistedproduct 𝐾(𝜋, 𝑛)(𝑖) ×𝜏* 𝑃 (𝑖). According to [27, Prop. 18.7] (which is formulated in amore general setting), there is a simplicial isomorphism 𝜙 : 𝐾(𝜋, 𝑛)(𝑖)×𝜏*𝑃 (𝑖)→ 𝑄(𝑖),given by

𝜙(𝛼, 𝛽) B (𝜓(𝑓(𝛼)) + 𝛽, 𝛼),

48

2. Tools

where 𝜓 : 𝐾(𝜋, 𝑛+ 1)→ 𝐸(𝜋, 𝑛) is the pseudo-section given by

𝜓(𝑧)(𝑖0, . . . , 𝑖𝑛) B 𝑧(0, 𝑖0 + 1, . . . , 𝑖𝑛 + 1),

with the same notation as in the definition of 𝜏 . Since both 𝜙 and its inverse are com-putable maps, we obtain effective homology for every 𝑄(𝑖) as needed. The functorialityof 𝜏 * and maps in diagrams 𝑃 ,𝐾(𝜋, 𝑛+1) further gives us maps in the diagram 𝑄.

Effective homology for 𝐸(𝜋, 𝑛) and 𝐾(𝜋, 𝑛). In this subsection, we show that𝐸(𝜋, 𝑛) and 𝐾(𝜋, 𝑛) have effective homology when 𝑛 ≥ 0 and 𝜋 is a finitely generatedabelian group.

These results were mainly developed by Sergeraert, his students and coworkers, seee.g P. Real’s thesis [33]. A strenghtening of these results was provided in [7], where theauthors present constructions that are polynomial.

We will use the obvious implication of these results: given 𝜋 : ℐ → Ab a finitediagram of finitely generated abelian groups and 𝑛 ≥ 1, the Eilenberg–MacLane space𝐾(𝜋, 𝑛) and the classifying space 𝐸(𝜋, 𝑛) have pointwise effective homology.

We really only need to concern ourselves with the effective homology for Eilenberg–MacLane spaces as there exists an explicit reduction 𝐶*(𝐸(𝜋, 𝑛)) ⇒⇒ Z (this is notsurprising since 𝐸(𝜋, 𝑛) is weakly equivalent to a point). Otherwise, one can see 𝐸(𝜋, 𝑛)as the twisted product 𝐾(𝜋, 𝑛)×𝜏𝐾(𝜋, 𝑛+1) and it follows that the effective homologyof Eilenberg–MacLane spaces implies effective homology of 𝐸(𝜋, 𝑛)

There is a (polynomial-time) reduction 𝐶*(𝐾(Z, 1)) ⇒⇒ 𝐶ef* (𝐾(Z, 1)) described in

[24] and we remark that in the section regarding the discrete vector fields, we haveshown a vector field producing a reduction 𝐶*(𝐾(Z, 1)) ⇒⇒ 𝐶ef

* (𝐾(Z, 1)), so 𝐾(Z, 1)has effective homology, this reduction, however is not polynomial time.

What remains is to show that 𝐾(𝜋, 𝑛) has effective homology for other dimensionsand groups. The entire discussion why this is true was given in [7], so we will omit manydetails and describe just the main ideas of how this result is achieved as a successionof steps:

∙ One proves that 𝐾(Z/𝑚, 1) has effective homology for 𝑚 ≥ 1 using effectivehomology for 𝐾(Z, 1), because there exists a (specific) twisted cartesian product𝑃 = 𝐾(Z, 1) ×𝜏 𝐾(Z/𝑚, 1) and one can show that 𝑃 is simplicially isomorphicto 𝐾(Z, 1). We will not give a description of this as it is described in detail byLemma 3.17 in [7], but we remark that conceptually, the isomorphism 𝜙 : 𝑃 →𝐾(Z, 1) can be obtained from the short exact sequence of abelian groups

0 // Z ×𝑚// Z mod 𝑚 // Z/𝑚 // 0

by passing to classifying spaces. The effective homology 𝐾(Z/𝑚, 1) now followsfrom effective homology for 𝐾(Z, 1) using Proposition 2.27.

∙ For 𝜋 arbitrary finitely generated abelian, we have a decomposition 𝜋 = 𝜋1 ⊕· · · ⊕ 𝜋𝑠, where 𝜋𝑖 is cyclic, 1 ≤ 𝑖 ≤ 𝑠. We obtain effective homology for 𝐾(𝜋, 1)using

𝐾(𝜋1 ⊕ · · · ⊕ 𝜋𝑠, 1) ∼= 𝐾(𝜋1, 1)× · · · ×𝐾(𝜋𝑠, 1),

which is easy to see from the definition of 𝐾(𝜋, 1), and a repeated use ofLemma 2.12 (product).

49

2. Tools

∙ An effective homology for 𝐾(𝜋, 𝑛), 𝑛 ≥ 2 is constructed by induction using theprevious results and the so–called 𝑊 construction: Given an Eilenberg–MacLanespace 𝐾(𝜋, 𝑛), there is a simplicial isomorphism 𝐾(𝜋, 𝑛 + 1)→ 𝑊𝐾(𝜋, 𝑛). Fur-ther, one uses a reduction

(𝑓, 𝑔, ℎ) : 𝐶*(𝐾(𝜋, 𝑛)×𝜏 𝑊𝐾(𝜋, 𝑛)) = 𝐶*(𝐸(𝜋, 𝑛))⇒⇒ Z

where the twisting operator 𝜏 is computable (for full description of this operator,see [7], Lemma 3.15). Using Proposition 2.27 and the induction assumption, weobtain that 𝑊𝐾(𝜋, 𝑛) ∼= 𝐾(𝜋, 𝑛+ 1) has effective homology.

50

3 Postnikov tower for diagrams

3.1 Reformulation of Theorem A

In this first section, we present a precise statement of Theorem A on an algorithmicconstruction of the Postnikov tower for diagrams and describe the objects that arecomputed during the run of the algorithm. In further sections, the algorithm itself isexplained and its correctness is proved.

We begin by describing formally what the Postnikov tower for a diagram means forus: Let 𝑌 : ℐ → sSet be a pointwise effective diagram of 1-connected simplicial sets,where the category ℐ is assumed to be finite. We remark that the 1-connectedness of𝑌 is needed for the proof of correctness of the algorithm, as the algorithm itself doesnot make use of any certificate of this fact and in particular, we do not assume thatspaces 𝑌 (𝑖) are even 1-reduced.

For our purposes, we define a Postnikov system of 𝑌 as above as the followingcommutative diagram:

𝑌𝑛𝜙𝑛

//

��

𝑃𝑛

𝑝𝑛��

...

��

...

��

𝑌2𝜙2

//

��

𝑃2

𝑝2��

𝑌1𝜙1

//

��

𝑃1

𝑝1��

𝑌 = 𝑌0𝜙0

// 𝑃0 = {*}

where 𝑃0 is the trivial diagram of points and the following conditions hold:

(i) For each 𝑛 ≥ 0, the map 𝜙𝑛 : 𝑌𝑛 → 𝑃𝑛 induces isomorphisms 𝜙𝑛* : 𝜋𝑗(𝑌𝑛) →𝜋𝑗(𝑃𝑛) of homotopy groups for 0 ≤ 𝑗 ≤ 𝑛, while 𝜋𝑗(𝑃𝑛) = 0 for 𝑗 ≥ 𝑛+ 1.

(ii) Each 𝜙𝑛 : 𝑌𝑛 → 𝑃𝑛, 𝑛 ≥ 1, is the cofibrant replacement of 𝜙′𝑛 : 𝑌𝑛−1 → 𝑃 ′𝑛, where𝑃 ′1 = {*} and for 𝑛 ≥ 1, 𝑃 ′𝑛 is the pullback in the following diagram

𝑃 ′cof𝑛 = 𝑃𝑛

��

𝑌𝑛 = 𝑌 cof𝑛−1

𝜙𝑛

88

��

𝑃 ′𝑛

𝑝𝑛����

// 𝐸(𝜋𝑛(𝑌 ), 𝑛)

����

𝑌𝑛−1𝜙𝑛−1

//

𝜙′𝑛

88

𝑃𝑛−1k𝑛−1

//𝐾(𝜋𝑛(𝑌 ), 𝑛+ 1)

The diagrams 𝑃0,𝑃1,𝑃2 . . . are called stages of the Postnikov system, and maps k𝑖

are called Postnikov classes (the terms Postnikov factors or Postnikov invariants arealso used in the literature).

51

3. Postnikov tower for diagrams

We now describe in full detail which objects will be constructed during the run ofthe algorithm:

Theorem A (Precise formulation). Let 𝑛 ≥ 2 be fixed, let 𝑌 : ℐ → sSet be a finitediagram that has pointwise effective homology and let us suppose that every 𝑌 (𝑖) is1-connected. Then there is an algorithm that computes the 𝑛-stage Postnikov systemfor 𝑌 . In detail, we get the following objects:

∙ Diagram 𝜋𝑗(𝑌 ) of homotopy groups as a diagram of fully effective abelian groups,0 ≤ 𝑗 ≤ 𝑛.

∙ Diagrams 𝑃0,𝑃′1,𝑃

′2, . . . ,𝑃

′𝑛 that have pointwise effective homology.

∙ Diagrams 𝑃1,𝑃2, . . . ,𝑃𝑛 and : 𝑌1,𝑌2, . . . ,𝑌𝑛 that have effective homology.

∙ Computable maps (natural transformations) 𝜙′𝑗 : 𝑌𝑗−1 → 𝑃 ′𝑗 , 𝜙𝑗 : 𝑌𝑗 → 𝑃𝑗 andmaps from the cofibrant replacement repl𝑃

′𝑗 : 𝑃𝑗 → 𝑃 ′𝑗 , repl𝑌𝑗−1 : 𝑌𝑗 → 𝑌𝑗−1,

1 ≤ 𝑗 ≤ 𝑛.

∙ Computable maps representing Postnikov classes k𝑗−1 : 𝑃𝑗−1 →𝐾(𝜋𝑗(𝑌 ), 𝑗+1),1 < 𝑗 ≤ 𝑛.

In case ℐ = 𝒪op𝐺 , we can use Theorem A to derive the following result

Theorem 3.1. Let 𝑛 ≥ 2 be fixed, let 𝑌 be a finite simplicial set with an action ofa finite group 𝐺. Further, let 𝑌 𝐻 be a 1-connected for all 𝐻 ≤ 𝐺. Then there is analgorithm that computes the 𝑛-stage Postnikov system for the diagram of fixed pointsΦ(𝑌 ) : 𝒪op

𝐺 → sSet.

3.2 Description of the algorithm

The algorithm we present here is in fact a modification of an algorithm that constructsa Postnikov tower for 1–connected simplicial sets presented in [7]. The main differencecan be seen in the application of Proposition 2.40, which will be stressed later. Further,we have to make sure that the constructions work for diagrams as well.

The following is a pseudo-code for the algorithm in Theorem A:

(1) Set 𝑌0 = 𝑌 , set 𝑃 ′1 = 𝑃0 = {*} and construct the (obvious) map 𝜙′1 : 𝑌0 → 𝑃 ′1.

(2) Compute cofibrant replacement of 𝜙′1 via Lemma 2.51 and denote 𝑌1 = (𝑌0)cof ,

𝑃1 = (𝑃 ′1)cof and 𝜙1 = repl(𝜙′1) : 𝑌1 → 𝑃1.

(3) For 𝑘 = 2 to 𝑛 do:

(4) Take the chain map (𝜙𝑘−1)* : 𝐶*(𝑌𝑘−1) → 𝐶*(𝑃𝑘−1) and using Lemma 2.45construct the algebraic mapping cone 𝑀 B Cone𝜙𝑘−1* together with a strongequivalence 𝑀 ⇐⇐⇒⇒𝑀 ef , where 𝑀 ef is an effective diagram of chain complexes.

(5) Compute a retraction 𝑟 : 𝑀 ef𝑘+1 → 𝑍𝑘+1(𝑀

ef) using Proposition 2.40.

(6) Compute the homology group 𝐻𝑘+1(𝑀ef) and the composite morphism

𝜌 : 𝑀 ef𝑘+1

𝑟−→ 𝑍𝑘+1(𝑀ef)→𝐻𝑘+1(𝑀

ef).

52

3. Postnikov tower for diagrams

(7) Set 𝜋𝑘 B𝐻𝑘+1(𝑀ef).

(8) Take the composite chain homomorphism 𝑓 : 𝑀𝑘+1 → 𝑀 ef𝑘+1 and compute

𝜆𝑘 : 𝐶𝑘(𝑌𝑘−1)→ 𝜋𝑘 as the restriction of 𝜌𝑓 to the summand 𝐶𝑘(𝑌𝑘−1). Computethe simplicial map ℓ𝑘 : 𝑌𝑘−1 → 𝐸(𝜋𝑘, 𝑘) corresponding to 𝜆𝑘 using Lemma 2.69.

Similarly, we obtain 𝜅𝑘−1 : 𝐶𝑘+1(𝑃𝑘−1) → 𝜋𝑘 as the restriction of 𝜌𝑓 to thesecond summand and we get a computable map k𝑘−1 : 𝑃𝑘−1 →𝐾(𝜋𝑘, 𝑘+1) viaLemma 2.70.

(9) Apply Corollary 2.73 to obtain 𝑃 ′𝑘 as a pullback in the diagram

𝑃 ′𝑘

����

// 𝐸(𝜋𝑘, 𝑘)

𝛿����

𝑌𝑘−1𝜙𝑘−1

//

ℓ𝑘−1

''

𝜙′𝑘

99

𝑃𝑘−1k𝑘−1

//𝐾(𝜋𝑘, 𝑘 + 1)

(3.1)

and set 𝜙′𝑘 = (𝜙𝑘−1, ℓ𝑘) as the map to the pullback 𝑃 ′𝑘.

(10) Compute 𝜙𝑘 : 𝑌𝑘 → 𝑃𝑘 using Lemma 2.51 as the cofibrant replacement of 𝜙′𝑘.

3.3 Correctness of the algorithm

In this section, we prove the correctness of the algorithm presented above. The proofis by induction with respect to 𝑘.

The basic step of the induction is in the pseudocode covered by (1) and (2). Inthese steps we compute the map 𝜙1 : 𝑌1 → 𝑃1 between diagrams that have effectivehomology. Because 𝑌 is 1-connected, 𝜙1 and 𝑃1 satisfy the properties in the Postnikovtower.

In the induction step, we assume that we have a (computable) map of diagramsthat have effective homology 𝜙𝑘−1 : 𝑌𝑘−1 → 𝑃𝑘−1. To produce a map of diagrams thathave effective homology 𝜙𝑘 : 𝑌𝑘 → 𝑃𝑘 we need to verify the following claims:

∙ In step (5) we have to make sure that 𝐻ef𝑗 (𝑀) = 0 for 𝑗 ≤ 𝑘 in order to use

Proposition 2.40.

∙ There is an isomorphism 𝜋𝑘∼= 𝜋𝑘(𝑌𝑘−1) ∼= 𝜋𝑘(𝑌 ).

∙ The cochain 𝜅𝑘−1 is a cocycle, thus we can define the Postnikov classesk𝑘−1 : 𝑃𝑘−1 →𝐾(𝜋𝑘, 𝑘 + 1) using Lemma 2.70.

∙ The image of the induced map 𝜙′𝑘 = (𝜙𝑘−1, ℓ𝑘) : 𝑌𝑘−1 ×𝐸(𝜋𝑘, 𝑘) lies in 𝑃 ′𝑘.

∙ The diagram 𝑃𝑘 is a Postnikov stage, has effective homology and 𝜙𝑘 satisfies theproperties in the definition of the Postnikov system.

The proof of most of these claims follows from proof of Theorem 4.1 in [7], the maindifference is that we make sure that all statements are true for diagrams, because theyare clearly true pointwise.

We will further frequently use the simple observation already mentioned in thedefinition of a weak equivalence in the projective model structure: given two diagrams

53

3. Postnikov tower for diagrams

of groups 𝐴,𝐵 : ℐ → Grp and a morphism 𝑓 : 𝐴→ 𝐵, to verify that 𝑓 : 𝐴→ 𝐵 is anisomorphism, it is enough to show that it is an isomorphism pointwise.

For brevity we further write 𝐾 = 𝐾(𝜋𝑘, 𝑘 + 1) and 𝐸 = 𝐸(𝜋𝑘, 𝑘).

Homology of the mapping cone 𝑀 . Let Cyl𝜙𝑘−1 be the mapping cylinder of𝜙𝑘−1 : 𝑌𝑘−1 → 𝑃𝑘−1. Clearly 𝜋*(Cyl𝜙𝑘−1) = 𝜋*(𝑃𝑘−1) because these diagrams arepointwise homotopy equivalent. Further 𝑌𝑘−1 is a subdiagram of Cyl𝜙𝑘−1. The chaincomplex of the pair (Cyl𝜙𝑘−1,𝑌𝑘−1) has a reduction to 𝑀 = Cone𝜙𝑘−1* , see Re-mark 2.46. Hence

𝐻*(Cyl𝜙𝑘−1,𝑌𝑘−1) ∼= 𝐻*(𝑀 ) ∼= 𝐻*(𝑀ef).

For the pair (Cyl𝜙𝑘−1,𝑌𝑘−1), we consider the long exact sequence of homotopygroups. Using the fact that 𝜙𝑘−1 induces isomorphism 𝜋𝑗(𝑌𝑘−1) ∼= 𝜋𝑗(𝑃𝑘−1) for 𝑗 ≤𝑘 − 1 and that 𝜋𝑗(𝑃𝑘−1) = 0 for 𝑗 ≥ 𝑘, we get that the pair (Cyl𝜙𝑘−1,𝑌𝑘−1) is 𝑘-connected and it follows that 𝐻𝑗(Cyl𝜙𝑘−1,𝑌𝑘−1) = 0 for 𝑗 ≤ 𝑘, so one can applyProposition 2.40 on 𝑀 ef , where 𝑀 ef ⇐⇐⇒⇒𝑀 = Cone𝜙𝑘−1* .

Checking that 𝜋𝑘∼= 𝜋𝑘(𝑌 ). We recall that the algorithm sets 𝜋𝑘 to be equal to

𝐻𝑘+1(𝑀ef), so we need to verify that 𝐻𝑘+1(𝑀

ef) ∼= 𝜋𝑘(𝑌 ). In the previous sec-tion, we have seen that the long exact sequence of homotopy groups for the pair(Cyl𝜙𝑘−1,𝑌𝑘−1)(𝑖) yields that this pair is 𝑘-connected and thus

𝜋𝑘(𝑌𝑘−1) ∼= 𝜋𝑘+1(Cyl𝜙𝑘−1,𝑌𝑘−1).

Due to the 𝑘-connectedness of (Cyl𝜙𝑘−1,𝑌𝑘−1) and the simple connectivity of 𝑌 ,the Hurewicz isomorphism produces an isomorphism

𝜋𝑘+1(Cyl𝜙𝑘−1,𝑌𝑘−1) ∼= 𝐻𝑘+1(Cyl𝜙𝑘−1,𝑌𝑘−1).

To sum up, we have a sequence of isomorphisms

𝜋𝑘 = 𝐻*(𝑀ef) ∼= 𝐻*(𝑀 ) ∼= 𝐻*(Cyl𝜙𝑘−1,𝑌𝑘−1)∼= 𝜋𝑘+1(Cyl𝜙𝑘−1,𝑌𝑘−1) ∼= 𝜋𝑘(𝑌𝑘−1) ∼= 𝜋𝑘(𝑌 ).

Hence we obtain 𝜋𝑘(𝑌 ) ∼= 𝜋𝑘, as desired.

The cochain 𝜅𝑘−1 is a cocycle. We aim to show that the following composition

𝜅𝑘−1 : 𝐶𝑘+1(𝑃𝑘−1) →˓ 𝐶𝑘(𝑌𝑘−1)⊕𝐶𝑘+1(𝑃𝑘−1) = 𝑀𝑘+1𝑓−−→𝑀 ef

𝑘+1

𝜌−−→ 𝜋𝑘

is a cocycle. The inclusion and 𝑓 are chain maps, and preserve boundaries. By definition,𝜌 vanishes on them. Thus the composite 𝜅𝑘−1 also vanishes on boundaries and is indeeda cocycle.

The map 𝜙′𝑘 takes values in 𝑃 ′𝑘. Denote the inclusion 𝐶𝑘+1(𝑃𝑘−1) →˓ 𝑀𝑘+1 as 𝑖and the inclusion 𝐶𝑘(𝑌𝑘−1) →˓ 𝑀𝑘+1 as 𝑗. Note that 𝑗 is not a chain map. We canwrite 𝜅𝑘−1 = 𝜌𝑓𝑖 and 𝜆𝑘 = 𝜌𝑓𝑗.

Now, we will verify that the image of the map 𝜙′𝑘 = (𝜙𝑘−1, ℓ𝑘) : 𝑌𝑘−1 → 𝑃𝑘−1×𝐸 liesin the pullback 𝑃 ′𝑘. According to (3.1), this translates to showing that k𝑘−1𝜙𝑘−1 = 𝛿ℓ𝑘.

54

3. Postnikov tower for diagrams

Using Lemma 2.67 and 2.69, we find that

𝜅𝑘−1𝜙𝑘−1* = (ev(k𝑘−1)*)𝜙𝑘−1* = ev(k𝑘−1𝜙𝑘−1)*.

It is easy to verify from the definitions that ev(𝛿ℓ𝑘)* = 𝜆𝑘𝑑𝑌𝑘−1 , where 𝑑𝑌𝑘−1 is the

differential in 𝐶*(𝑌𝑘−1). Therefore the desired equality k𝑘−1𝜙𝑘−1 = 𝛿ℓ𝑘 of simplicialmaps can be rewritten in terms of cochains in 𝐶𝑘+1

ℐ (𝑌𝑘−1,𝜋𝑘) as

𝜅𝑘−1𝜙𝑘−1* = 𝜆𝑘𝑑𝑌𝑘−1 , (3.2)

By the definitions of 𝜅𝑘−1 and 𝜆𝑘, we have

𝜅𝑘−1𝜙𝑘−1* − 𝜆𝑘𝑑𝑌𝑘−1 = 𝜌𝑓(𝑖𝜙𝑘−1* − 𝑗𝑑𝑌𝑘−1).

The composition 𝜌𝑓 maps boundaries in 𝑀 to 0, because 𝜌 does, so it suffices toshow that the images of 𝑖𝜙𝑘−1*− 𝑗𝑑𝑌𝑘−1 are boundaries. This follows from the fact thatthe formula for the differential in the algebraic mapping cone says that we have

(𝑖𝜙𝑘−1* − 𝑗𝑑𝑌𝑘−1)(𝜎) = 𝑑𝑀 (𝜎, 0)

for every 𝜎 ∈ 𝐶𝑘+1(𝑌𝑘−1).

𝑃𝑘 and 𝜙𝑘 satisfy properties of the Postnikov system. Since 𝜙𝑘 is a cofibrantreplacement of 𝜙′𝑘 and 𝑃𝑘 is a cofibrant replacement of 𝑃 ′𝑘, we prove that 𝑃𝑘 and 𝜙𝑘

satisfy properties of the Postnikov system by showing that 𝜙′𝑘* : 𝜋𝑗(𝑌𝑘−1) → 𝜋𝑗(𝑃′𝑘)

induces isomorphism for 𝑗 ≤ 𝑘 and that 𝜋𝑗(𝑃′𝑘) = 0 for 𝑗 > 𝑘.

We remark that that 𝑃 ′𝑘 has pointwise effective homology because it is a pullback(twisted product) of diagrams that have pointwise effective homology as in Corol-lary 2.73.

Because 𝛿 : 𝐸 → 𝐾 is a pointwise (principal and minimal) fibration with fibre𝐾(𝜋𝑘, 𝑘), the definition of 𝑃 ′𝑘 as a pullback, gives us a pointwise fibration

𝐾(𝜋𝑘, 𝑘)→ 𝑃 ′𝑘 → 𝑃𝑘−1.

By the induction assumption, 𝜋𝑗(𝑃𝑘−1) = 0 for 𝑗 ≥ 𝑘, it is straightforward to checkthat 𝜋𝑗(𝑃

′𝑘) = 0 for 𝑗 ≥ 𝑘 + 1, and that the maps 𝜋𝑗(𝑌𝑘−1) → 𝜋𝑗(𝑃

′𝑘) induced by 𝜙′𝑘

are isomorphisms for 𝑗 ≤ 𝑘−1. To show that 𝑃 ′𝑘 is up to homotopy the 𝑘-th Postnikovstage, it remains to verify that (𝜙′𝑘)* : 𝜋𝑘(𝑌𝑘−1)→ 𝜋𝑘(𝑃

′𝑘) is an isomorphism as well.

We will do this using the diagram

𝑌𝑘−1

𝜙𝑘−1

��

𝜙′𝑘 // 𝑃 ′𝑘

//

𝑝𝑘

��

𝐸

𝛿��

𝑃𝑘−1 𝑃𝑘−1 //𝐾

where the right square is the pullback diagram defining 𝑃 ′𝑘.We will now make all the vertical maps into inclusions by replacing the spaces in the

bottom row with the mapping cylinder of the respective vertical map. This constructionalso induces the horizontal maps between cylinders. We obtain the diagram.

55

3. Postnikov tower for diagrams

𝑌𝑘−1

��

𝜙′𝑘 // 𝑃 ′𝑘

//

��

𝐸

��

Cyl𝜙𝑘−1 // Cyl 𝑝𝑘 // Cyl 𝛿

(3.3)

Now we consider the long exact sequences for pairs (Cyl𝜙𝑘−1,𝑌𝑘−1) and(Cyl 𝑝𝑘,𝑃

′𝑘):

0 = 𝜋𝑘+1(Cyl𝜙𝑘−1) //

∼=��

𝜋𝑘+1(Cyl𝜙𝑘−1,𝑌𝑘−1)

∼=��

// 𝜋𝑘(𝑌𝑘−1)

𝜙′𝑘*��

// 𝜋𝑘(Cyl𝜙𝑘−1) = 0

∼=��

0 = 𝜋𝑘+1(Cyl 𝑝𝑘) // 𝜋𝑘+1(Cyl 𝑝𝑘,𝑃′𝑘)

// 𝜋𝑘(𝑃′𝑘)

// 𝜋𝑘(Cyl 𝑝𝑘) = 0

The first and the last vertical arrows are isomorphisms due to the fact that we have𝜋𝑗(𝑃𝑘−1) = 0 for 𝑗 ≥ 𝑘 and that both of the mapping cylinders deform onto the basespace 𝑃𝑘−1, i.e.

𝜋𝑗(Cyl𝜙𝑘−1) ∼= 𝜋𝑗(𝑃𝑘−1) ∼= 𝜋𝑗(Cyl 𝑝𝑘).

The exactness of the rows further implies isomorphisms 𝜋𝑘+1(Cyl𝜙𝑘−1,𝑌𝑘−1) ∼=𝜋𝑘(𝑌𝑘−1) and 𝜋𝑘+1(Cyl 𝑝𝑘,𝑃

′𝑘)∼= 𝜋𝑘(𝑃

′𝑘). If we prove that the second vertical homo-

morphism is an isomorphism, then 𝜙′𝑘* has to be an isomorphism on 𝜋𝑘. We formulatethe required claim as

Lemma 3.2 (Lemma 4.5 in [7]). The map

𝜋𝑘+1(Cyl𝜙𝑘−1,𝑌𝑘−1)→ 𝜋𝑘+1(Cyl 𝑝𝑘,𝑃′𝑘)

induced by the left square of diagram (3.3) is an isomorphism.

Proof. The proof of the pointwise version of the Lemma is technical and can be foundin [7]. We thus have a homomorphism of diagrams of groups

𝜋𝑘+1(Cyl𝜙𝑘−1,𝑌𝑘−1)→ 𝜋𝑘+1(Cyl 𝑝𝑘,𝑃′𝑘)

which is an isomorphism pointwise. It follows that this is an isomorphism of diagrams.This finishes the proof of the correctness of the algorithm and thus of Theorem A.

3.4 Computing equivariant cohomology operations

In the nonequivariant setting, one standardly computes cohomology operations as

[𝐾(𝜋, 𝑛), 𝐾(𝜌, 𝑘)].

Here, we will focus on the computation of 𝐺-invariant cohomology operations. Beforedoing that, we prove the following general statement:

Proposition 3.3. Let ℐ be a finite category and 𝑋 : ℐ → sSet a diagram with pointwiseeffective homology. Let 𝜋 : ℐ → Ab be a diagram of fully effective abelian groups. Thenthere is an algorithm which computes [𝑋,𝐾(𝜋, 𝑛)]ℐ.

56

3. Postnikov tower for diagrams

Proof. We will utilize the following sequence of isomorphisms

[𝑋,𝐾(𝜋, 𝑛)]ℐ ∼= [𝑋cof ,𝐾(𝜋, 𝑛)]ℐ ∼= 𝐻𝑛ℐ (𝑋

cof ;𝜋),

where the first (left) isomorphism is a standard result of homotopy theory and thesecond isomorphism follows from Proposition 2.71. Further 𝐶*(𝑋

cof) has effective ho-mology and this in particular implies that there is a chain complex 𝐶ef

* (𝑋cof) such that

𝐶ef𝑛 (𝑋

cof) is a diagram of fully effective abelian groups.By Lemma 2.56 each Hom(𝐶ef

𝑘 (𝑋cof),𝜋) is a fully effective abelian group, so we con-

struct an effective chain complex 𝐶*ℐ(𝑋cof ;𝜋). Using Lemma 2.53, we finally compute𝐻𝑛ℐ (𝑋

cof ;𝜋), which is an effective abelian group.

Let 𝜋 : 𝒪op𝐺 → Ab. We call a 𝐺-simplicial set 𝑋 Eilenberg–MacLane 𝐺-simplicial

set for the diagram 𝜋 in the dimension 𝑛 if

𝜋𝑗(𝑋𝐻) =

{𝜋(𝐺/𝐻) for 𝑗 = 𝑛,0 otherwise.

Similarly, we call a diagram 𝑋 : 𝒪op𝐺 → sSet Eilenberg-MacLane diagram for the

diagram 𝜋 in the dimension 𝑛 if

𝜋𝑗(𝑋) =

{𝜋 for 𝑗 = 𝑛,0 otherwise.

As an example, we remark that the diagram 𝐾(𝜋, 𝑛) introduced in Section 2.9 is anEilenberg-MacLane diagram for diagram 𝜋 in the dimension 𝑛.

We now formulate the following corollary of Proposition 3.3 that describes thecomputation of 𝐺-invariant cohomology operations.

Corollary 3.4. Let 𝐺 be a finite group and let 𝑋, 𝑌 be Eilenberg-MacLane 𝐺-simplicialsets for the diagrams of groups 𝜋 and 𝜌 in dimensions 𝑛 and 𝑘, respectively. Then thereexists an algorithm computing [𝑋, 𝑌 ]𝐺.

Proof. We use Elmendorf’s theorem [14, 43, 28] to transform our computation to com-putation with diagrams of simplicial sets:

[𝑋, 𝑌 ]𝐺 ∼= [Φ𝑋,Φ𝑌 ]𝒪op𝐺.

If we suppose that there exists an isomorphism

[Φ𝑋,Φ𝑌 ]𝒪op𝐺

∼= [𝐾(𝜋, 𝑛),𝐾(𝜌, 𝑘)]𝒪op𝐺, (3.4)

the proof can be finished using Proposition 3.3.It thus remains to prove the isomorphism (3.4): The model category structure on

sSet𝒪op𝐺 induces a homotopy category Ho(sSet𝒪

op𝐺 ), where maps Ho(sSet𝒪

op𝐺 )(𝐴,𝐵) cor-

respond to homotopy classes of maps [𝐴,𝐵]𝒪op𝐺

(see [11]). Further, any weak equivalencein the category sSet𝒪

op𝐺 turns into an isomorphism in Ho(sSet𝒪

op𝐺 ). It is thus enough to

construct a zigzag of weak equivalences

Φ𝑋 →←←→ · · · ←→→𝐾(𝜋, 𝑛).

We take (Φ𝑋)cof and compute

[(Φ𝑋)cof ,𝐾(𝜋, 𝑛)]𝒪op𝐺

∼= 𝐻𝑛𝒪op

𝐺((Φ𝑋)cof ;𝜋).

57

3. Postnikov tower for diagrams

As (Φ𝑋)cof is an Eilenberg-MacLane diagram, we get that 𝐻𝑖(Φ𝑋) = 0 for 𝑖 < 𝑛and 𝐻𝑛(Φ𝑋) = 𝜋 and thus 𝐻𝑛

𝒪op𝐺(Φ𝑋;𝜋) = Hom(𝜋,𝜋). Using Proposition 2.71, there

exists a map id* : (Φ𝑋)cof →𝐾(𝜋, 𝑛) corresponding to id ∈ Hom(𝜋,𝜋). Clearly, id* isa weak equivalence. This, together with weak equivalence (Φ𝑋)cof → Φ𝑋 gives us therequired zigzag. Similarly, we get a zigzag connecting Φ𝑌 and 𝐾(𝜌, 𝑘). This tells usthat

Ho(sSet𝒪op𝐺 )(Φ(𝑋,Φ𝑌 ) ∼= Ho(sSet𝒪

op𝐺 )(𝐾(𝜋, 𝑛),𝐾(𝜌, 𝑘))

and we have obtained (3.4).

58

4 How to decide if a map is homotopically trivial

In this chapter, we present a simplified version of the result achieved by L. Vokrinekand myself in the article [15].

The original article presents decision algorithm for the existence of a homotopybetween given maps 𝑓, 𝑔 : 𝑋 → 𝑌 where 𝑋 and 𝑌 are finite simplicial complexes ormore generally finite simplicial sets, 𝑓 and 𝑔 are simplicial maps and 𝑌 is assumed tobe 1-connected.

Here, we will present a simplified version which decides whether given 𝑓 : 𝑋 → 𝑌is homotopic to a constant (trivial) map to a chosen basepoint in 𝑌 . If a homotopyexists, we will call 𝑓nullhomotopic. The original, more general version relies heavily onthe fiberwise approach of the paper [8]. We feel that introducing further notions to thiswork would bring many unnecessary complications.

It is well known that no homotopy decision algorithm may exist if 𝑌 is allowedto be non-simply connected; this follows at once from Novikov’s result [32] on theunsolvability of the word problem in groups. Here, we will thus restrict our attentionto the case of a simply connected 𝑌 .

Theorem B. Let 𝑋, 𝑌 be finite simplicial sets where 𝑌 is simply connected, and let𝑓 : 𝑋 → 𝑌 be a simplicial map. Then there is an algorithm that decides whether 𝑓 ishomotopically trivial.

In the paper [5], the authors gave an algorithmic solution to the following problem:given two simplicial sets 𝑋, 𝑌 , compute [𝑋, 𝑌 ], i.e. the set of homotopy classes ofcontinuous maps from 𝑋 to 𝑌 . Their algorithm works under a certain connectivityrestriction on 𝑌 . This restriction can be removed when the domain is replaced by asuspension – this is our next result which, at the same time, generalizes the computationof homotopy groups of spaces described by Brown in [2].

We remark that in the case 𝑌 is simply connected, there is a bijection between setsof unpointed homotopy classes of maps [𝑋, 𝑌 ] and pointed homotopy classes of maps[(𝑋, *𝑋), (𝑌, *𝑌 )]. Further on, by [𝑋, 𝑌 ] we will always mean pointed classes.

Theorem C. There is an algorithm that computes the group [Σ𝑋, 𝑌 ] of pointed ho-motopy classes of maps from a suspension Σ𝑋 to a simply connected simplicial set𝑌 .

The group is presented on the output as a so-called fully-effective polycyclic group– this structure is introduced in Section 2.8 and allows one to compute a finite set ofgenerators and relations and solve the word problem.

The results described above are connected in the following way: In both cases,we replace the space 𝑌 by a Postnikov tower and the proof is by induction. Further,the induction assumption obtained used in Theorem B is utilized to give a proof ofTheorem C and vice versa.

Relative statement. In order to prove Theorem B and Theorem C we will work inthe comma category 𝐴/sSet i.e. the category of pointed simplicial sets under 𝐴, where𝐴 contains the basepoint. Here the objects are simplicial sets 𝑋 equipped with a maps𝐴 → 𝑋. Morphisms in this category are maps 𝑓 : 𝑋 → 𝑌 for which the following

59

4. How to decide if a map is homotopically trivial

diagram𝐴 𝛼 //

𝜄

��

𝑌

𝑋

>>

𝑓 (4.1)

commutes. There is also an obvious notion of homotopy (preserving the basepoint andrelative to 𝐴). In the case that 𝜄 is an inclusion, 𝛼 is fixed and 𝑌 is a Kan complex, theresulting set of homotopy classes of maps that are constant (equal to 𝛼) on 𝐴 will bedenoted by [𝑋, 𝑌 ]𝐴. For general 𝑋, 𝑌 ∈ 𝐴/sSet, we define [𝑋, 𝑌 ]𝐴 first by replacing 𝜄up to weak homotopy equivalence by an inclusion 𝐴 // //𝑋cof and 𝑌 by 𝑌 fib and then bysetting [𝑋, 𝑌 ]𝐴 = [𝑋cof , 𝑌 fib]𝐴, where the map 𝐴 → 𝑌 fib comes from the composition𝐴→ 𝑌 → 𝑌 fib.

The relative version of Theorems B and C reads as

Theorem D. Let there be a diagram

𝐴𝛼 //

��

𝜄

��

𝑌

��

𝑋 // {*}

given on the input, where all spaces are finite simplicial sets where 𝑌 is simply connec-ted. Then there are the following algorithms:D.1. Given a map 𝑓 : 𝑋 → 𝑌 in 𝐴/sSet, the algorithm decides whether 𝑓 is nullho-

motopic i.e. whether it represents the zero element in [𝑋, 𝑌 ]𝐴.D.2. The algorithm computes the group structure on the set of (pointed) homotopy

classes of maps [Σ𝑋, 𝑌 ]Σ𝐴 such that Σ𝐴 ⊆ Σ𝑋 is mapped to the basepoint in𝑌 .

Theorems B and C are obtained from Theorem D by setting 𝐴 = *, where weremark that Σ* = *.

4.1 Computations with Postnikov towers

The proof of both Theorems relies on computations in the Postnikov tower of 𝑌 . Theconstruction of the tower was described in details in previous chapters. Here, we willuse a standard property of Postnikov tower

Proposition 4.1 ([8, Theorem 3.3]). The map 𝜙𝑛 : 𝑌 → 𝑃𝑛 induces a bijection𝜙𝑛* : [𝑋, 𝑌 ]𝐴 → [𝑋,𝑃𝑛]

𝐴 for every 𝑛-dimensional simplicial set 𝑋 and 𝐴 ⊆ 𝑋.

This Proposition allows us to replace the diagram (4.1) by

𝐴𝛼𝑛 //

��

𝜄

��

𝑃𝑛

𝑋

>>

𝑓𝑛

in which 𝛼𝑛 = 𝜙𝑛𝛼 and 𝑓𝑛 = 𝜙𝑛𝑓 , where 𝜙𝑛 : 𝑌 → 𝑃𝑛. Since 𝑃𝑛 is a Kan simplicial set,the homotopy classes in [𝑋,𝑃𝑛]

𝐴 are represented by simplicial maps 𝑋 → 𝑃𝑛 under 𝐴(no replacements needed).

60

4. How to decide if a map is homotopically trivial

For our algorithm, it will be essential to lift homotopies. Moreover, homotopy con-catenation will serve as the main tool in the computations with maps defined on sus-pensions. The proofs of the results in this subsection can be found in [8]. We start witha general algorithm for lifting maps by one stage.

Proposition 4.2 ([8, Proposition 3.5]). There is an algorithm that, given a diagram

𝐴 //��

��

𝑃𝑛

𝑝𝑛����

𝑋 //

==

𝑃𝑛−1

decides whether a diagonal exists. If it does, it computes one.

The following two special cases apply even to lifting through multiple stages.

Proposition 4.3 (homotopy lifting, [8, Proposition 3.6]). Given a diagram

(𝑘 ×𝑋) ∪ (Δ1 × 𝐴) //

��

∼��

𝑃𝑛

����

Δ1 ×𝑋 //

77

𝑃𝑚

where 𝑘 ∈ {0, 1}, it is possible to compute a diagonal. In other words, one may lifthomotopies in Moore–Postnikov towers algorithmically.

The second special case will be used later to concatenate homotopies.

Proposition 4.4 (homotopy concatenation, [8, Proposition 3.7]). Given a diagram

( 2𝑘 ×𝑋) ∪ (Δ2 × 𝐴) //

��

∼��

𝑃𝑛

����

Δ2 ×𝑋 //

66

𝑃𝑚

where 𝑘 ∈ {0, 1, 2}, it is possible to compute a diagonal.

4.2 Maps out of suspensions

As the simplicial set 𝑌 is pointed, every Postnikov stage 𝑃𝑛 is equippped with a uniquebasepoint 𝑜 which we will call zero. Further, we will assume that 𝛼𝑛 = 𝑜, i.e. [𝑋,𝑃𝑛]

𝐴

will now denote the set of homotopy classes of maps 𝑓 : 𝑋 → 𝑃𝑛 that are zero on 𝐴.From now on, we use a shorthand notation 𝐼 = Δ1. Hence 𝐼𝑞 is the 𝑞-cube i.e. the

𝑞-fold product 𝐼𝑞 = 𝐼 × · · · × 𝐼 and 𝜕𝐼𝑞 is its boundary.

Homotopy concatenation. We will now use Proposition 4.4 to make a group struc-ture on [Σ𝑋,𝑃𝑛]

Σ𝐴. It is simple to see that this set is isomorphic to[𝐼 × 𝑋,𝑃𝑛]

(𝜕𝐼×𝑋)∪(𝐼×𝐴). We will work with the second description and represent theelements of [Σ𝑋,𝑃𝑛]

Σ𝐴 by homotopies 𝐼 ×𝑋 → 𝑃𝑛, starting and finishing at the zeromap and zero on 𝐼 × 𝐴.

Let ℎ2, ℎ0 : 𝐼 × 𝑋 → 𝑃𝑛 be two such homotopies. Viewing each ℎ𝑖 as defined on𝑑𝑖Δ

2 × 𝑋, we obtain a single map 21 × 𝑋 → 𝑃𝑛 which, together with the zero map

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4. How to decide if a map is homotopically trivial

𝑜 : Δ2 × 𝐴 → 𝑃𝑛, prescribes the top map in Proposition 4.4. The bottom map is thecomposition Δ2×𝑋 pr−→ 𝑋

𝛽−→ {*}, i.e. we take 𝑚 = 0. Let Δ2×𝑋 → 𝑃𝑛 be the diagonalmap computed by Proposition 4.4. Then we will call its restriction to 𝑑1Δ2 × 𝑋 theconcatenation of ℎ2 and ℎ0 and denote it by ℎ0 + ℎ2.

The inverse of a homotopy is computed similarly: For inverse of some ℎ : 𝐼×𝑋 → 𝑃𝑛,we see ℎ as a map ℎ : 𝑑2Δ

2 × 𝑋, we assume a zero map 𝑜 : 𝑑1Δ2 × 𝑋 and we use

Proposition 4.4 to compute −ℎ : 𝑑0Δ2×𝑋. The situation is summarized in the followingProposition:

Proposition 4.5. The set [Σ𝑋,𝑃𝑛]Σ𝐴 ∼= [𝐼 × 𝑋,𝑃𝑛]

(𝜕𝐼×𝑋)∪(𝐼×𝐴) is a semi-effectivegroup represented by the set of all simplicial maps 𝐼 × 𝑋 → 𝑃𝑛 that are zero on(𝜕𝐼 ×𝑋) ∪ (𝐼 × 𝐴).

4.3 Deciding the existence of a homotopy

An exact sequence associated with a fibration. We start with the followingnotation:𝜋𝑛 = 𝜋𝑛(𝑌 ), 𝐾𝑛+1 = 𝐾(𝜋𝑛, 𝑛+ 1) and 𝐿𝑛 = 𝐾(𝜋𝑛, 𝑛). There are maps

𝐿𝑛𝑗−→ 𝑃𝑛

𝑝𝑛−−→ 𝑃𝑛−1𝑘𝑛−−→ 𝐾𝑛+1,

where 𝑘𝑛 is the Postnikov invariant 𝑃𝑛−1 −→ 𝐾(𝜋𝑛, 𝑛+ 1) and 𝑗 is the inclusion suchthat the image of 𝑗 consists precisely of those simplices of 𝑃𝑛 that map to zero in 𝑃𝑛−1.The following sequence of pointed sets is exact by [8, Theorem 4.8] (the relevant partsof the proof do not use the stability assumption 𝑛 ≤ 2𝑑):

[Σ𝑋,𝑃𝑛−1]Σ𝐴 𝜕−−→ [𝑋,𝐿𝑛]

𝐴 𝑗*−−→ [𝑋,𝑃𝑛]𝐴 𝑝𝑛*−−−→ [𝑋,𝑃𝑛−1]

𝐴 𝑘𝑛*−−−→ [𝑋,𝐾𝑛+1]𝐴. (4.2)

The isomorphisms [𝑋,𝐿𝑛]𝐴 ∼= 𝐻𝑛(𝑋,𝐴; 𝜋𝑛) and [𝑋,𝐾𝑛+1]

𝐴 ∼= 𝐻𝑛+1(𝑋,𝐴; 𝜋𝑛) showthat these sets are abelian groups that can be computed easily. The group homomorph-ism 𝜕 is defined in the following way. Given a homotopy ℎ : 𝐼 ×𝑋 → 𝑃𝑛−1, lift it to ahomotopy ℎ : 𝐼×𝑋 → 𝑃𝑛 in such a way that (0×𝑋)∪(𝐼×𝐴) maps to the zero section,using Proposition 4.3. Since the restriction of ℎ to 1×𝑋 takes values in the image of𝑗, it could be interpreted as a map 𝑋 → 𝐿𝑛. This map is then a representative of 𝜕[ℎ].

Proof of Theorem D. The proof is by induction. First, we list a series of claims:

(gen)𝑛 It is possible to compute a finite set of generators of the group[𝐼 ×𝑋,𝑃𝑛]

(𝜕𝐼×𝑋)∪(𝐼×𝐴).(null)𝑛 It is possible to decide whether a given map 𝑓 : 𝑋 → 𝑃𝑛 under 𝐴 is nullho-

motopic; when this is the case, it is possible to compute a nullhomotopy, i.e.a homotopy from the zero map to 𝑓 .

We remark that if we perceive the claims above as algorithms, then the pair (𝑋,𝐴) ofsimplicial sets is seen as the input of these algorithms.

Proof of Theorem D.1 from (null)𝑛 We use the fact that [𝑋, 𝑌 ]𝐴 ∼= [𝑋,𝑃𝑛]𝐴,

therefore Theorem D.1 follows from (null)𝑛.

Proofs of (null)0 and (gen)0 The basic steps are trivial, because 𝑌 is 1-connectedand 𝑃0 = {*}. Thus the group is trivial and any given map is always nullhomotopic.

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4. How to decide if a map is homotopically trivial

The algorithm (null)𝑛 is given by induction using (gen)𝑛−1. This is essentially

contained in [8, Section 4.9]; we reproduce the algorithm here for reader’s conveniencebut omit the proof of correctness.

(null)𝑛−1 + (gen)𝑛−1 → (null)𝑛 on (𝑋,𝐴)

∙ We take 𝑓 : 𝑋 → 𝑃𝑛. By (null)𝑛−1, we decide whether the composition 𝑝𝑛𝑓 : 𝑋 →𝑃𝑛−1 is nullhomotopic and compute the nullhomotopy ℎ′ : 𝐼 × 𝑋 → 𝑃𝑛−1. If 𝑝𝑛𝑓fails to be nullhomotopic, then then 𝑓 cannot be nullhomotopic and the algorithmstops.∙ We lift nullhomotopy ℎ′ using Proposition 4.3 to a homotopy ℎ′ : 𝐼 ×𝑋 → 𝑃𝑛 such

that ℎ′ : 𝑓 ′ ∼ 𝑓 . Since 𝑝𝑛𝑓 ′ = 𝑜 (as 𝑝𝑛ℎ′ = ℎ′), we interpret 𝑓 ′ : 𝑋 → 𝑃𝑛 as a map𝑓 ′ : 𝑋 → 𝐿𝑛.∙ We use (gen)𝑛−1 to decide whether [𝑓 ′] ∈ im 𝜕 and to further compute ℎ′′ : 𝐼 ×𝑋 → 𝑃𝑛−1 with 𝜕[ℎ′′] = [𝑓 ′]. Using Proposition 4.2, it is possible to compute a liftℎ′′ : 𝐼 ×𝑋 → 𝑃𝑛 that starts at the zero map and finishes at 𝑓 ′

∙ Finally, the concatenation ℎ = ℎ′ + ℎ′′, computed by Proposition 4.4, is a homo-topy from the zero map to 𝑓 . If either of ℎ′, ℎ′′ fails to exist, the map 𝑓 is notnullhomotopic.

Thus, it remains to prove (gen)𝑛. To make the induction possible, we will have tostrengthen the claim and compute more than just generators, namely the structure ofa fully effective polycyclic group as the group [𝐼 ×𝑋,𝑃𝑛]

(𝜕𝐼×𝑋)∪(𝐼×𝐴) is polycyclic.From now on, we denote

G𝑛,𝑞 = [𝐼𝑞 ×𝑋,𝑃𝑛](𝜕𝐼𝑞×𝑋)∪(𝐼𝑞×𝐴)

and we formulate the new claim as(poly)𝑛 There is an algorithm which equips G𝑛,𝑞 with a structure of a fully effective

polycyclic group.As fully effective polycyclic group enables us to compute the generators, we get(poly)𝑛 → (gen)𝑛.

Since G𝑛,𝑞 for (𝑋,𝐴) is a G𝑛,1 for (𝐼𝑞−1, (𝜕𝐼𝑞−1 × 𝑋) ∪ (𝐼𝑞−1 × 𝐴)), it would beenough to restrict to the case 𝑞 = 1. This special case is just Theorem D.2 as for𝑛 = dim(𝐼 ×𝑋) = 1 + dim𝑋 we have

[Σ𝑋, 𝑌 ]Σ𝐴 ∼= [𝐼 ×𝑋, 𝑌 ](𝜕𝐼×𝑋)∪(𝐼×𝐴) ∼= [𝐼 ×𝑋,𝑃𝑛](𝜕𝐼×𝑋)∪(𝐼×𝐴) = G𝑛,1

and the last term is computable by (poly)𝑛. Hence (poly)𝑛 implies Theorem D.2.

(poly)𝑛−1 + (null)𝑛−1 → (poly)𝑛. We remind that

[𝐼𝑞 ×𝑋,𝐿𝑛](𝜕𝐼𝑞×𝑋)∪(𝐼𝑞×𝐴) ∼= 𝐻𝑛(𝐼𝑞 ×𝑋, (𝜕𝐼𝑞 ×𝑋) ∪ (𝐼𝑞 × 𝐴); 𝜋𝑛) ∼= 𝐻𝑛−𝑞(𝑋,𝐴; 𝜋𝑛)

and similarly [𝐼𝑞×𝑋,𝐾𝑛+1](𝜕𝐼𝑞×𝑋)∪(𝐼𝑞×𝐴) ∼= 𝐻𝑛+1−𝑞(𝑋,𝐴; 𝜋𝑛). The shifts in the degree

of cohomology groups follows from the properties of suspension.

Proof. The group G𝑛,𝑞 is semi-effective by Proposition 4.5. The exact sequence (4.2)applied to (𝐼𝑞 ×𝑋, (𝜕𝐼𝑞 ×𝑋) ∪ (𝐼𝑞 × 𝐴)) instead of (𝑋,𝐴) reads

G𝑛−1,𝑞+1𝜕−−→ 𝐻𝑛−𝑞(𝑋,𝐴; 𝜋𝑛)

𝑗*−−→ G𝑛,𝑞𝑝𝑛*−−−→ G𝑛−1,𝑞

𝑘𝑛*−−−→ 𝐻𝑛+1−𝑞(𝑋,𝐴; 𝜋𝑛)

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4. How to decide if a map is homotopically trivial

and induces a short exact sequence

0 // Coker 𝜕𝑗*// G𝑛,𝑞 𝑝𝑛*

//

𝑟tt

Ker 𝑘𝑛* //

𝜎tt

0.

Group 𝐻𝑛−𝑞(𝑋,𝐴; 𝜋𝑛) is a fully effective abelian group and by (poly)𝑛−1, groupsG𝑛−1,𝑞+1 and G𝑛−1,𝑞 are fully effective polycyclic. The images of generators of G𝑛−1,𝑞+1

under 𝜕 generate a subgroup in 𝐻𝑛−𝑞(𝑋,𝐴; 𝜋𝑛), so we can use Lemma 2.53 and findthat the group Coker 𝜕 is fully effective abelian. Proposition 2.61 then gives us Ker 𝑘𝑛*as a fully effective polycyclic group.

For the application of Proposition 2.64, we need to provide algorithms for the twoindicated sections 𝑟, 𝜎. The section 𝜎 is defined on the level of representatives (on whichit depends) by mapping a diagonal 𝑓 : 𝐼𝑞×𝑋 → 𝑃𝑛−1 to an arbitrary lift 𝑓 : 𝐼𝑞×𝑋 → 𝑃𝑛

of 𝑓 that is zero on (𝜕𝐼𝑞 ×𝑋) ∪ (𝐼𝑞 ×𝐴). The computation of 𝑓 can be performed byProposition 4.2.

For the construction of the partial inverse 𝑟 on im 𝑗* = Ker 𝑝𝑛*, let 𝑓 : 𝐼𝑞 × 𝑋 →𝑃𝑛 be a diagonal such that its composition with 𝑝𝑛 : 𝑃𝑛 → 𝑃𝑛−1 is homotopic tozero. Then we can compute such a nullhomotopy ℎ : 𝐼𝑞 ×𝑋 → 𝑃𝑛−1 by (null)𝑛−1 on(𝐼𝑞×𝑋, (𝜕𝐼𝑞×𝑋)∪ (𝐼𝑞×𝐴)). Using Proposition 4.3, we lift ℎ along 𝑝𝑛 to a homotopyfrom some 𝑓 ′ : 𝐼𝑞 × 𝑋 → 𝑃𝑛 to 𝑓 . Since 𝑝𝑛𝑓 ′ = 𝑜, the image of 𝑓 ′ lies in 𝐿𝑛 and wemay set 𝑟([𝑓 ]) = [𝑓 ′].

Remark 4.6. The computation of Coker 𝜕 in the above proof requires generators ofG𝑛−1,𝑞+1 – these are provided by an application of (poly)𝑛−1 which in turn requiresgenerators of G𝑛−2,𝑞+2, etc. Thus, in principle, we need to compute G𝑚,𝑝 for all 𝑚 < 𝑛and 𝑝 ≤ 𝑛+ 𝑞.

However, this is not necessary: for 𝑞 ≥ 2, G𝑛,𝑞 is an abelian group and one maycompute its generators from those of Ker 𝑘𝑛* and Coker 𝜕 via Lemma 2.54. Now, Coker 𝜕is generated by the images of generators of 𝐻𝑛−𝑞(𝑋,𝐴; 𝜋𝑛). Further, Ker 𝑘𝑛* can becomputed from a set of generators of G𝑛−1,𝑞.

Thus, for 𝑞 ≥ 2, the computation of generators of G𝑛,𝑞 can be executed by inductionon 𝑛 while 𝑞 is fixed. To summarize, it is thus possible to organize the computation ofG𝑛,1 in Theorem D.2 in such a way that in its course, we only need a fully effectivepolycyclic structure on G𝑚,1, 𝑚 ≤ 𝑛, and generators of G𝑚,2, 𝑚 < 𝑛.

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