nerve complexes

8/10/2019 Nerve Complexes

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NERVE COMPLEXES OF CIRCULAR ARCS

MICHAL ADAMASZEK, HENRY ADAMS, FLORIAN FRICK, CHRIS PETERSON,

AND CORRINE PREVITEJOHNSON

Abstract. We show that the nerve complex ofn arcs in the circle is homotopy equivalentto either a point, an odd-dimensional sphere, or a wedge sum of spheres of the same evendimension. Moreover this homotopy type can be computed in time O(n log n). For theparticular case of the nerve complex of evenly-spaced arcs of the same length, we determinethe dihedral group action on homology, and we relate the complex to a cyclic polytope withn vertices. We give three applications of our knowledge of the homotopy types of nervecomplexes of circular arcs. First, we use the connection to cyclic polytopes to give a noveltopological proof of a known upper bound on the distance between successive roots of ahomogeneous trigonometric polynomial. Second, we show that the Lovasz bound on thechromatic number of a circular complete graph is either sharp or off by one. Third, we showthat the VietorisRips simplicial complex ofn points in the circle is homotopy equivalentto either a point, an odd-dimensional sphere, or a wedge sum of spheres of the same evendimension, and furthermore this homotopy type can be computed in time O(n log n).

Contents

1. Introduction 12. Preliminaries 23. Nerve complexes of evenly-spaced arcs 34. Application to the Lovasz bound 65. The odd-dimensional spheres: cyclic polytopes and trigonometric polynomials 76. The even-dimensional spheres: minimal generators 9

7. Induced representation in homology 118. Clique complexes of evenly-spaced arcs 129. Nerve and clique complexes of arbitrary circular arcs 14References 16

1. Introduction

For U a collection of subsets of some topological space, the nerve simplicial complex N(U) contains ak-simplex for every subcollection of k + 1 sets with nonempty intersection. The Nerve Theorem, whichholds in a variety of contexts, states that if the intersection of each subcollection ofU is either empty orcontractible, then the nerve complex is homotopy equivalent to the union of the subsets [11, 10]. A coarser

representation of the incidences between sets in U is given by the clique complex N(U), which contains ak-simplex for every collection ofk+ 1 sets with pairwise nonempty intersections. In this paper we studynerve complexes and clique complexes of finite collections of arcs in the circle, which are known, respectively,asambient Cech complexesandVietoris-Rips complexeswhen all arcs have the same length. We completelyclassify their homotopy types.

2010 Mathematics Subject Classification. 05E45, 52B15, 68R05.

Key words and phrases. Nerve complex, Cech complex, VietorisRips complex, Circular arc, Cyclic polytope.Research of HA was supported by the Institute for Mathematics and its Applications. FF is supported by the German Science

Foundation DFG via the Berlin Mathematical School.

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Main result (Theorem 9.4). The nerve complex and the clique complex of any finite collection of arcs inthe circle are homotopy equivalent to either a point, an odd-dimensional sphere, or a wedge sum of spheresof the same even dimension.

The higher-dimensional spheres occur when the arcs are large enough so that the intersection of two arcsneed not be contractible.

We begin by studying the homotopy types and the combinatorics of the nerve complexes of evenly-spacedcircular arcs. For 0 k < n, letN(n, k) denote the nerve complex ofnevenly-spaced arcs each occupying a knfraction of the circumference of the circle. These are of fundamental interest since, as we shall see, the nerveof any finite configuration of arcs deformation retracts to a complex isomorphic to some N(n, k). We provea recursive relation from which we derive the homotopy types of the N(n, k). We further provide explicitgenerators of homology and cohomology ofN(n, k) and describe the induced action of their automorphismgroups on homology. In the generic case, when N(n, k) is homotopy equivalent to an odd-dimensional sphereS2l+1, we show that it contains the boundary complex of the n-vertex cyclic polytopeC2l+2(n) as a homotopyequivalent subcomplex.

Applications. We give two immediate applications of these calculations.1. We show that the Lovasz bound on the chromatic number of a circular complete graph is either sharp or

off by one (see Corollary 4.2).2. We use the relation with cyclic polytopes to give a novel topological proof of a known upper bound on

the distance between successive roots of a homogeneous trigonometric polynomial (see Theorem 5.12).

In the last section we study arbitrary collectionsUofn arcs inS1. We show that each such nerve complexN(U) has an explicit homotopy-preserving combinatorial reduction to one of the form N(n, k) withn n.We can compute the reduction in time O(n log n), and as a result we obtain an efficient algorithm fordetermining the homotopy type ofN(U), even though the worst-case size ofN(U) is exponential in n. Thereductions are independent of any knowledge of theN(n, k), and they also carry over to the clique complexesN(U).

WhenUis a collection of balls of fixed radius r in a Riemannian manifoldM, the clique (or flag) complexN(U) is called a VietorisRips complex[31]. Such complexes arise in manifold reconstruction [14, 4] and intopological data analysis [16, 12]. If the radius ris sufficiently small and if the balls are sufficiently dense, thenHausmann and Latschev prove the VietorisRips complex is homotopy equivalent to manifold M [21, 27].However, VietorisRips complexes with larger radii parameters r are not well understood, even for simple

spaces such as spheres. We show that the VietorisRips complex of an arbitrary subset ofn points in thecircle with arbitrary radius parameterr is homotopy equivalent to either a point, an odd-dimensional sphere,or a wedge sum of spheres of the same even dimension. Moreover this homotopy type can be computed intimeO(n log n). We also prove a surprising relationship, different from the usual inclusion, between the Cechand VietorisRips complexes of evenly-spaced points on the circle.

2. Preliminaries

We assume the reader is familiar with basic concepts in topology and combinatorial topology, and referto Hatcher [20] and Kozlov [23].

Simplicial complexes. Let Kbe a simplicial complex, let V(K) be its vertex set, and let K(i) be itsi-skeleton. We will identify an abstract complex with its geometric realization and use the symbol todenote homotopy equivalence and = to denote isomorphism of simplicial complexes. For V V(K), let

K[V] be the induced subcomplex of K containing only those simplices with all vertices in V. We letK\ {v}= K[V(K) \ {v}] be the simplicial complex obtained from K by removing all simplices containingv. The link of vertex v is lkK(v) = { K | v / and {v} K}.

Domination. We say vertexv is dominatedby vertex v if each Kcontainingv satisfies {v} K,i.e. if lkK(v) is a cone with apex v . If vertexv Kis dominated, thenKK\{v}because we are removinga vertex v whose link is contractible. In fact there is a deformation retractionK K\ {v} which sends vto v and also simplicially collapses K to K\ {v}. These removals go by various names: folds, elementarystrong collapses, and LC reductions [6, 9, 29]. An analogous operation for graphs is known as dismantling.

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We say that simplicial complex K is minimal if it contains no dominated vertices.

Nerves and cliques. Let Ybe a topological space, and let U ={Ui}iI with =Ui Y be a collectionof subsets. We say U is a covering ofY if

iIUi= Y.

Definition 2.1. Given a collection of subsets U = {Ui}iI in a topological space, the nerve simplicial

complexN(U) has vertex setIand containsk-simplex[i0, . . . , ik] ifkj=0 Uij =.

The Nerve Theorem is generally attributed to Borsuk [11], and the version we use is due to Bj orner [10,Theorem 10.6].

Theorem 2.2 (Nerve Theorem). LetKbe a simplicial complex and letUbe a covering by subcomplexes. Ifevery nonempty finite intersection of complexes inU is contractible, thenN(U) K.

In Sections 8 and 9 we will consider clique complexes of nerves. For G a simple, loopless, undirectedgraph, the clique simplicial complex Cl(G) has V(G) as its vertex set and a face for each clique (completesubgraph) ofG.

Definition 2.3. Given a collection of subsets U = {Ui}iI in a topological space, the clique simplicialcomplexN(U) has vertex setIand containsk-simplex[i0, . . . , ik] ifUij Uij = for all0 j, j

k.

We noteN(U) = Cl(N(U)(1)).

Cech and VietorisRips complexes. In the particular case whenYis a metric space andUis a collectionof balls, the nerve complex is also known as a Cech complex, and the clique complex is also known as aVietorisRips complex. ForY a metric space, we denote the closed ball of radiusr 0 centered at y Y byB(y, r) = {y Y |d(y, y) r}. Fix someXY and letU(X, r) = {B(x, r)| x X}. ThenN(U(X, r)) isisomorphic to the ambient Cech complex Cech(X, Y; r) with landmark set Xand witness set Y , as definedby Chazal, de Silva, & Oudot [13, Section 4.2.3] 1. TheVietorisRips complex VR(X, r) is defined to be thesimplicial complex on vertex set X containing finite Xas a simplex if the distance between any twopoints in is at most r . IfY is a geodesic space then VR(X, r) is isomorphic to N(U(X,r/2)).

Conventions regarding S1. In this paper we study the setting where U is a finite collection of arcs inthe circle S1. We identifyS1 with R/Z, where the positive orientation on R corresponds to the clockwiseorientation on S1. For x, y R with x y we denote by [x, y]S1 the closed circular arcobtained as the

image of the interval [x, y] under the quotient map R R/Z. Similarly, for a, b S1 we denote by [a, b]S1the closed circular arc obtained by moving from a to b in a clockwise fashion. Open and half-open intervalsin S1 are obtained by removing endpoints from closed intervals. The intersection ofk such arcs is eitherempty, contractible, or homotopy equivalent to a disjoint union of at most k points;Uis known as an acyclicfamily [15].

We also equip the circle S1 of circumference 1 with the natural arc-length distance. Under this metricthe diameter ofS1 is 12 . The choice of this particular metric does not influence the generality of our results.

Other conventions. We denote the topological space consisting of a single point by . For a topologicalspaceY we let

i Ydenote the wedge sum ofi copies ofY, where by convention0 Y =. The symbol denotes unreduced suspension.

All homology and cohomology is taken with integer coefficients.If is an oriented d-simplex in K(an element of the standard basis of the chain group Cd(K)) then

denotes the dual d-cochain which assigns 1 to ,1 to the reverse oriented , and 0 to other d-simplices.

3. Nerve complexes of evenly-spaced arcs

We begin by giving a combinatorial model for nerve complexes of evenly-spaced circular arcs.

Definition 3.1. Forn 1 andi, j Z withi j, let thediscrete circular arc [i, j]n be the image of the set{i, i + 1, . . . , j} under the quotient map Z Z/n,zz mod n.

1Attali & Lieutier [3] refer to the ambient Cech complex as a restricted Cech complex.

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For most of this paper we will be studying the topology and combinatorics of the following family ofabstract simplicial complexes.

Definition 3.2. Forn 1 andk 0, thenerve complex N(n, k) has vertex set{0, . . . , n 1}, and its setof maximal simplices is

[i, i + k]n | i = 0, . . . , n 1

.

Ifk n 2 then N(n, k) has n maximal simplices given by the n rotations of [0, k]n, and ifk n 1

thenN(n, k) is the (n 1)-simplex.To see the connection with evenly-spaced circular arcs, for 0 k < n consider the collection

(1) Un,k= i

n,i + k

n

S1

i = 0, . . . , n 1ofn evenly-spaced arcs of length kn . Also, letXn S

1 be a set ofn evenly-spaced points. It is an easyexercise to verify the isomorphisms of simplicial complexes

(2) Cech

Xn, S1; k2n

=N(Un,k)=N(n, k).

For k even, the complex N(n, k) can also be described as a distance-neighborhood complex of the cyclegraphCn, as studied by the last author [30].

The following regimes are simple.

Disconnected: N(n, 0) is the disjoint union ofn points, i.e.n1 S0.

Circle: For 1 k < n /2 we have N(n, k) S1 by the Nerve Theorem. Indeed, consider thetriangulation ofS1 with vertices in and edges [

in ,

i+1n ]S1 fori = 0, . . . , n 1. Since 1 k < n/2, the

covering Un,k ofS1 has all nonempty intersections contractible. We haveN(n, k) = N(Un,k), andTheorem 2.2 gives N(Un,k) S1.

Top-dimensional sphere: N(n, n 2) is the boundary of the (n 1)-simplex. Contractible: Fork n 1 the complex N(n, k) is the full (n 1)-simplex.

Example 3.3. The nerve complex N(6, 3) is the nerve of the 6 equally-spaced closed arcs of length 36 = 12 ;

see Figure 1. The Nerve Theorem does not apply since [ i6 , i+3

6 ]S1 [i+3

6 , i6 ]S1 S

0 is not contractible. The

complexN(6, 3) has six maximal 3-simplices, and as we shall seeN(6, 3)2 S2.

Figure 1. (Top) The six arcs ofU6,3. (Bottom) The six maximal 3-simplices in N(6, 3)

We will now determine the homotopy types of the complexes N(n, k). For this we repeatedly use thefollowing lemma, which is a simple version of [10, Lemma 10.4.(ii)].

Lemma 3.4. If the simplicial complex K is the union of two contractible subcomplexesK1 and K2, thenK (K1 K2).

Proposition 3.5. Forn/2 k < n we haveN(n, k) 2N(k, 2k n).

Proof. Denote the maximal simplices ofN(n, k) byi = [i, i + k]n fori = 0, . . . , n 1. Then we can write

N(n, k) = nk2

i=0

i

n1j=nk1

j

,

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where by a slight abuse of notation we write

tTt for the subcomplex ofN(n, k) with maximal simplices{t | t T}. Eachj containsn 1 and eachi containsk sincen k 2 k, hence both unions are cones.Moreover, the simplices i do not contain n1. By Lemma 3.4 we haveN(n, k) K, where K is thecomplex with vertex set {0, . . . , n 2}whose maximal simplices are the inclusion-wise maximal elements inthe family

{i j | i = 0, . . . , n k 2 and j = n k 1, . . . , n 1}.

The intersections i j fall into three categories, see Figure 2.

a) If 0 i j + k n i+k < j n 1 then i j = {i , . . . , j+ k n}. We have i j 0 n1 = {0, . . . , k 1}.

b) If 0 j + k n < i j i+ k n 2 then i j = {j, . . . , i+ k}. We have i j nk2 nk1 = {n k 1, . . . , n 2}.

c) If i j + k n and j i+ k then i j = {i , . . . , j+ k n} {j, . . . , i+ k}. These are notcontained in any other set of the form i j .

We conclude that the maximal simplices ofKare

={0, . . . , k 1}, ={n k 1, . . . , n 2}, and i,j ={i , . . . , j+ k n} {j, . . . , i + k},

subject to the conditions

0 i j + k n and j i + k n 2.

0i

j + k n

i+ k

j

0i + k

j

i

j + k n

0

i + k

j

i

j + k n

a) b) c)

Figure 2. The intersections i j inN(n, k).

We claim that the subcomplex T = (i,ji,j) of K is contractible. Let Tl = T[{l , . . . , n 2}] forl = 0, . . . , n k 1. For l = n k 1 the maximal simplices ofTl containing l are of the form l,j, sincea maximal simplex ofT containing l is of the form i,j for some i l j + k n and i,j V(Tl) l,j .Since eachl,j containsl + k, vertexl is dominated by l + k in Tl, givingTl Tl \ {l}= Tl+1. It follows thatT =T0 is homotopy equivalent to Tnk1 =

, which is is contractible.We writeK= Tas the union of two contractible subcomplexes, and by Lemma 3.4 there is a homotopy

equivalence K ( T). Note the vertex set of T is {0, . . . , k 1}, and its maximal simplices are theinclusion-wise maximal elements in the family consisting of and all i,j. These maximal elementsare

={n k 1, . . . , k 1}

0,j ={j, . . . , k 1} {0, . . . , j+ k n}, n k j k 1

i,i+k = {i , . . . , i + 2k n}, 0 i n k 2.

These are precisely all the cyclic intervals of the form [i, i+ (2k n)]k in {0, . . . , k 1}, hence T =N(k, 2k n). By combining the two suspension steps we obtain

N(n, k) K 2 ( T) = 2N(k, 2k n).

The homotopy types of the nerve complexes N(n, k) follow.5


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Theorem 3.6. Let0 k n 2. Then

N(n, k)

nk1 S2l if kn = ll+1S2l+1 if ll+1 n k 2,

where the second inequality is equivalent to ll+1 < kn . Theng n k 1, and so there is a segment of

n k 1 consecutive elements in Z/n disjoint from , meaning N(n, k).

For example, if l = 0 then the boundary C2(n) N(n, k) is just the n-cycle passing through all thevertices ofN(n, k) in their natural ordering. If 1 k < n/2 then it is not difficult to describe a deformationretraction fromN(n, k) to this embedded circle. Guided by this intuition we will now prove that in generalthe embedded C2l+2(n) generates the homotopy type ofN(n, k). Note that the proof relies on the priorknowledge ofN(n, k) S2l+1, i.e. it cannot be used as a replacement for the proof of Theorem 3.6.

Theorem 5.9. Suppose l

l+1 < k

n < l+1

l+2 . Then the inclusionC2l+2(n) N(n, k) is a homotopy equiva-lence.

Proof. The condition ll+1 < kn

dd+1 >

d1d , Lemma 5.8 ensures that C2d(n) N(n, k)

with the natural ordering of the vertices. It follows that the inclusion : C2d(n) C2d(n)\ {0} factors

through N(n, k). This is a contradiction, since is a homotopy equivalence between spaces homotopyequivalent to S2d1, butH2d1(N(n, k)) is trivial for

dd+1

nerve complexes

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