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ALGEBRAIC CYCLES ON REAL VARIETIES ANDZ/2-EQUIVARIANT HOMOTOPY THEORY

PEDRO F. DOS SANTOS

Abstract. In this paper the spaces of algebraic cycles on a real projective

variety X are studied as Z/2-spaces under the action of the Galois group

Gal(C/R). In particular, the equivariant homotopy type of the group of al-

gebraic p-cycles Zp(PnC) is computed. A version of Lawson homology for real

varieties is proposed. The real Lawson homology groups are computed for a

class of real varieties.

1. Introduction

In the past ten years there has been renewed interest in the study of the topo-

logical groups of algebraic cycles on complex projective varieties, using homotopy

theoretic techniques. The idea behind these results is that, for a complex projective

variety X, the homotopy invariants of the groups of p-dimensional algebraic cycles

(denoted here by Zp(X), with p ≤ dim X) carry information about the algebraic

structure of X. In this context, the case X = PnC was particularly important. The

computation of the homotopy type of Zp(PnC) in [16] was the starting point for the

definition, due to Friedlander, of a homology theory for projective varieties called

Lawson homology [7]. The Lawson homology groups of a projective variety X are

a set of bigraded invariants, LpHk(X), defined by

LpHk(X) = πk−2pZp(X),

for all k ≥ 2p. This theory was later extended to arbitrary complex varieties by

Lima-Filho [22].

The importance of Lawson homology comes from the fact that it has a rather ac-

cessible definition and still carries a lot of information about the algebraic structure

of the variety X. For example,

the groups of algebraic equivalence classes of p-cycles on X are the Lawson

homology groups LpH2p(X) (see [7]). Another reason for the interest in Lawson

homology comes from the fact that it interpolates between motivic homology and

2000 Mathematics Subject Classification. 55P91; Secondary 14C05, 19L47, 55N91.

1

2 PEDRO F. DOS SANTOS

singular homology (see [8]). This can be seen as a generalization of the fact that

algebraic equivalence interpolates between rational and homological equivalence of

algebraic cycles.

Lawson homology groups also have independent interest as they are as homotopy

groups of colimits of Chow varieties. Therefore they provide information about

the topological properties of these varieties which are classical objects in algebraic

geometry.

The definition of Lawson homology for complex varieties relies heavily on the

fact that, when X is a complex variety, the analytic topology can be used to endow

Zp(X) with a Hausdorff topology. For varieties over more general fields no such

topology is available and the definition of similar invariants requires the much more

sophisticated machinery of etale homotopy theory [7].

In this paper we address the of problem defining a version of Lawson homology

for varieties defined over R, namely, one that reflects the real structure. For this, we

will adopt the point of view that a real variety X is (by Galois descent) a complex

variety with the extra structure provided by the action of the Galois group of

Gal(C/R) on its set of complex points. This Z/2-action naturally extends to the

groups of cycles Zp(X) and is continuous w.r.t. the analytic topology. Thus,

it is natural to define a version Lawson homology for real varieties in terms of

equivariant homotopy invariants of its groups of algebraic cycles, endowed with the

analytic topology.

To introduce such a theory we must at least understand the equivariant homo-

topy type of Zp(PnC). The first steps in this direction were taken by Lam. In his

thesis [15] he proved that Lawson’s suspension Theorem holds equivariantly. More

recently, the homotopy type of the spaces of real cycles (i.e. invariant under the

action of the Galois group) was computed in [18].

Non-equivariantly, Zp(PnC) is a product of classifying spaces for singular coho-

mology with integer coefficients in the even dimensions 0, 2, . . . , 2(n − p). Our

first result is an equivariant version of this. It parallels the non-equivariant result

very closely with the appropriate changes. Singular cohomology is replaced with

RO(Z/2)-graded equivariant cohomology; Z is replaced with the constant Mackey

functor Z; the dimensions 2, . . . , 2(n− p) above are replaced with the non-integral

dimensions R1,1, . . . , Rn−p,n−p — we use Rr,s to denote Rr+s with the Z/2-action

of multiplication by −1 in the last s coordinates. Equivariant cohomology with Z

coefficients in dimension Rr,s is denoted by Hr,s(−; Z).

ALGEBRAIC CYCLES ON REAL VARIETIES 3

Theorem 3.7. The space Zp(PnC) is Z/2-homotopy equivalent to the following

product of equivariant Eilenberg-Mac Lane spaces

(3.1) Zp(PnC) '−→

n−p∏k=0

K(Z, Rk,k).

Furthermore, under these equivalences, Zp(Pn−1C ) includes in Zp(Pn

C) as a factor

in the product (3.1) with constant last coordinate. Hence, the space of stabilized

cycles, Z = colimn,p→∞Zp(PnC), is Z/2-homotopy equivalent to a weak product of

equivariant Eilenberg-Mac Lane spaces

(3.2) Z '−→∞∏

k=0

K(Z, Rk,k).

Before considering cycles on other real varieties we look more closely at the

space of stabilized cycles Z of (3.2). In [2] it was shown that Z has an infinite loop

space structure induced by the operation of joining algebraic cycles. The inclusion

of linear spaces into all cycles induces an infinite loop space map P : BU → Z

that classifies the total Chern class. The existence an infinite loop structure with

these properties proved a longstanding conjecture of Segal in homotopy theory.

This result has an equivariant counterpart. Indeed, the pairing on the space of

stabilized cycles, µ : Z × Z → Z induced by the join of algebraic cycles, and the

map P : BU → Z are both equivariant (here BU is given the Z/2-action induced

by complex conjugation). In Proposition 3.10 we show that µ classifies the cup

product in H∗,∗(−; Z).

The equivariant map P : BU → Z leads to an interesting connection with

Atiyah’s KR-theory. Recall that a real bundle in the sense of Atiyah [1] is a complex

bundle over a Z/2-space with an anti-linear involution which covers the involution

on the base. Since BU with the Z/2-action mentioned above is the classifying space

for real bundles, it is natural to expect that P should be closely related to charac-

teristic classes for real bundles. A simple computation shows that H∗,∗(BU; Z) is a

polynomial ring on certain classes c1, . . . , cn, . . . in the dimensions R1,1, . . . , Rn,n, . . .

which appear is the decomposition (3.2). In Proposition 3.12 we show that, under

the isomorphism (3.2), P classifies the total Chern class 1 + c1 + c2 · · · .

Having obtained a complete description of the Z/2-homotopy type of Zp(PnC)

and of the effect of the join map we propose a definition of real Lawson homology

for real quasi-projective varieties. For a projective real variety X, we define the


real Lawson homology groups by (Definition 4.4)

LpRn,m(X) def= [Sn−p,m−p,Zp(X)]Z/2 for n, m ≥ p and p ≤ dim X;

where Sn−p,m−p is the one-point compactification of the representation Rn−p,m−p

and [−,−]Z/2 is the set of equivariant homotopy classes.

Given that LpH2p(−) is the group of p-cycles modulo algebraic equivalence, a

relation between LpRp,p(−) and algebraic equivalence is to be expected. In Propo-

sition 4.6, we show that, for a real variety X, LpRp,p(X) is the quotient of the

group of real p-cycles (i.e. defined over R) on X modulo an equivalence relation

closely related to algebraic equivalence. This equivalence relation is a version in

the level of algebraic cycles of Friedlander and Walker’s real topological equivalence

for real algebraic bundles [11, Prop.1.6].

Using the techniques developed by Friedlander, Gabber [8] and Lima-Filho [23],

we establish the existence of exact sequences, excision and a cycle map which takes

values in the Z/2-equivariant homology of X with Z coefficients.

In Proposition 4.20 we relate real Lawson homology to Lawson homology for

complex varieties by showing that, considering a complex variety U as a real variety

UR (see Example 4.19), we recover the Lawson homology of the complex variety

U , i.e. we have LpRr,s(UR) ∼= LpHr+s(U). Furthermore, we prove the following

result.

Proposition 4.21. Let U be a real quasi-projective variety. Then there is a transfer

map π∗ : LpRr,s(U) → LpHr+s(U) and a restriction map π∗ : LpHr+s(U) →

LpRr,s(U) such that the composition π∗ π∗ is multiplication by 2. In particular,

for any finitely generated module M over Z[ 12 ], LpRr,s(U)⊗M is isomorphic to a

submodule of LpHr+s(U)⊗ M .

In Section 5 we compute compute real Lawson homology in several examples. In

particular, the real Lawson homology groups of affine space An = PnC−Pn−1

C , with its

standard real structure, and the effect of the cycle map are computed. Moreover,

excision and the cycle map are used to establish a general result for a class of

examples: real varieties with a real cell decomposition. A real variety X has a real

cell decomposition if there is a filtration X = Xn ⊃ Xn−1 ⊃ · · ·X0 ⊃ X−1 = ∅ by

real subvarieties, such that Xi−Xi−1 is a union of affine spaces Anij (Definition 5.3).


Theorem 5.4. Let X be a real quasi-projective variety with a real cell decom-

position, then the map

sp : Zp(X) −→ Ωp,pZ0(X)

is an equivariant homotopy equivalence. In particular, the cycle map induces an

isomorphism

LpRn,m(X) ∼= Hn,m(X; Z),

so that LpRn,m(X) is independent of p in this case.

This class includes, for example, the Grassmannians Gp(Cn) (with the real struc-

ture induced by sending a complex plane to its complex conjugate plane) and

PnC × Pm

C with the product of the standard real structures on PnC and Pm

C .

By considering different real structures one can produce many of different ex-

amples of real varieties for which the non-equivariant Lawson homology groups are

known. Examples of this are PnC×Pn

C with the action τ ·(x, y) = (y, x), real quadrics

and the real Severi-Brauer varieties, PC(Hn) (whose real structure is induced by

multiplication of the homogeneous coordinates by the imaginary quaternion j).

The real Lawson homology groups of some of these examples are computed in Sec-

tion 5 and we expect that they should all be possible to compute. In the case of the

Severi-Brauer varieties PC(Hn) the homotopy groups of the real cycles have been

computed in [17]. The complete equivariant homotopy type of their spaces of cycles

is computed in [5] where a beautiful connection between the spaces Zp(PC(Hn)),

Atiyah’s KR-theory and Dupont’s symplectic K-theory [6] is established.

The paper is organized as follows. We summarize in §2 the results, definitions

and notation from equivariant homotopy theory we will need. In §3 we compute

the equivariant homotopy type of Zp(PnC) and Z and study the natural map BU →

Z. In §4 we propose a definition of real Lawson homology and prove its basic

properties: exactness, homotopy invariance and existence of cycle map with values

in equivariant homology with Z coefficients. In §5 we compute examples and show

that the cycle map is an isomorphism for the class of real varieties with a real cell

decomposition. The details of the proof of proposition 4.9 are given in §6.

Acknowledgement. The author wishes to thank H. Blaine Lawson, Jr., for in-

troducing him to this subject, suggesting the problem and for his guidance and

support during the elaboration of this work; Daniel Dugger and Gustavo Granja

for many fruitful discussions; Paulo Lima-Filho for suggestions and corrections.


2. Definitions and results from equivariant homotopy theory

In this section we summarize the definitions, results and notation from equivari-

ant homotopy theory needed to state the results concerning the Z/2-equivariant

homotopy type of spaces of algebraic cycles on real varieties. Our general reference

for equivariant homotopy theory is [27].

2.1. G-CW-complexes and G-homotopies. Let Dn, Sn−1 be the unit disk and

unit sphere of Rn, respectively, considered as a G-spaces with the trivial action.

The unit interval I = [0, 1] is also equipped with the trivial action.

Definition 2.1. A G-CW complex X is the union of G-spaces

X0 ⊂ X1 ⊂ · · ·Xn ⊂ · · ·

such that X0 is a disjoint union of orbits G/H and Xn is obtained from Xn−1 by

attaching G-cells Dn×G/H along maps Sn−1×G/H → Xn−1. We say that Xn is

the n-skeleton of X.

Given G-maps f, g : X → Y , a G-homotopy from f to g is a homotopy from f to

g which is equivariant for the diagonal action of G on X × I. A G-map f : X → Y

is a G-homotopy equivalence (or equivariant homotopy equivalence) if there exist

a G-map h : Y → X such that f h and h f are G-homotopic to the identity. We

will use the symbol ' to denote G-homotopy equivalence.

2.2. Coefficient systems. Let G be a finite group and let FG be category of

finite G-sets and G-maps. The coefficients for ordinary equivariant (co)homology

are (contravariant) covariant functors from FG to the category Ab of abelian groups

which sends disjoint unions to direct sums.

Notation 2.2. If V is a representation of a finite group G, SV denotes the one

point compactification of V and, for a based G-space X, ΩV X denotes the space

of based maps F (SV , X). The space F (SV , X) is equipped with the its standard

G-space structure. The set of equivariant homotopy classes [SV , X]G is denoted by

πV (X).

Example 2.3. For each G-space X and G-representation V , there is a contravariant

coefficient system πV (X) whose value on G/H is [SV ∧G/H+, X]G. The value of

πV (X) on a G-map f : G/H → G/K is induced by id∧f . Actually, without

extra assumptions on X, we must assume V contains two copies of the trivial


representation to ensure that πV (X) takes values in Ab. If X is, for example, a

topological abelian G-group this extra hypothesis is not necessary.

If V is the trivial representation Rn, the coefficient system πV is just denoted

by πn. The functors πn are also called Bredon homotopy groups.

The following result will be used throughout: a G-map f : X → Y between G-

CW complexes is a G-homotopy equivalence if and only if it induces an isomorphism

on the Bredon homotopy groups.

A Mackey functor M is a pair (M∗,M∗) of additive functors M∗ : FG → Ab and

M∗ : FopG → Ab with the same value on objects and which transform each pull-back

diagram

Af−−−−→ B

g

y yh

Ck−−−−→ D

in FG into a commutative diagram in Ab

M(A)M∗(f)−−−−→ M(B)

M∗(g)

x xM∗(h)

M(C)M∗(k)−−−−→ M(D)

In this paper we will be interested in the case where M = Z is the Mackey functor

constant at Z. This Mackey functor is uniquely determined by the conditions:

(i) Z(G/H) = Z, for H ≤ G;

(ii) If K ≤ H, the value of the contravariant functor Z∗ on the projection ρ :

G/K → G/H is the identity.

2.3. RO(G)-graded homology and cohomology. Given a Mackey functor M

there exists an RO(G)-graded cohomology theory H∗G(−;M) such that H0

G(pt;M) =

M . For each real orthogonal representation V there is a classifying space K(M,V )

which classifies H∗G(−;M) in dimension V . It is important to note that the spaces

K(M,V ) can have homotopy in several integral dimensions. This is precisely the

case of the K(M,V )-spaces considered below (see 3.6). The spaces K(M,V ) fit

together to give an equivariant Eilenberg-Mac Lane spectrum HM. In particular,

this means that, given G-representations V , W , there is a G-homotopy equivalence

K(M,V ) ' ΩW K(M,V + W ).

In this paper we will be interested in the case where G = Z/2 and M = Z

is the Mackey functor constant at Z. Recall that RO(Z/2) ∼= Z ⊕ Z and let 1

and ρ denote the one-dimensional trivial representation, and the one-dimensional


non-trivial representation, respectively. Then, for p, q ∈ N, Rp,q ∼= p · 1 + q · ρ

where Rp,q denotes Rp+q with the Z/2-action of multiplication by −1 in the last q

coordinates. For simplicity, we will use the notation Hp,q(−; Z) for Z/2-equivari-

ant cohomology with Z coefficients in dimension Rp,q. In order to avoid confusion,

singular cohomology groups with Z coefficients will be denoted by H∗sing(−; Z).

Similar conventions will be used for homology groups.

Notation 2.4. In the case G = Z/2, we will use the notation Sp,q instead of SRp,q

,

πp,q(−) instead of πRp,q , Ωp,q instead of ΩRp,q

and πp,q instead of πRp,q .

Given a Mackey functor M the computation of H∗G(pt;M) is a non-trivial prob-

lem. In the case G = Z/p and M = Z these groups were originally computed by

Stong and appear in [3]. In the case p = 2 they are as follows

(2.1) Hn,m(pt; Z) =

Z/2 n even, 0 ≤ −n < m

Z n even, −n = m

Z/2 n odd, 1 < n ≤ −m

0 otherwise

There is also a cup product ∪ : Hn,m(−; Z)⊗Hn′,m′(−; Z) → Hn+n′,m+m′

(−; Z)

which, in particular, gives a ring structure to H∗(pt; Z). We will denote this ring

be R. The product on R appears also in [3] but we will not give here as it is not

so easy to describe and we will not need it. It is important to note that for any

Z/2-space X, H∗(X; Z) is a module over R.

2.4. G-fibrations. Since restricting to the case of Z/2 is not more elucidating we

will again give the definitions for a finite group G.

A G-fibration is an equivariant map π : E → B with the G-homotopy lifting

property with respect to for G-CW-complexes. A G-fibration π : E → B gives rise

to exact sequences in homotopy groups. In the particular case of G = Z/2, for each

q, there is an exact sequence

· · · → πp,qE → πp,qB → πp−1,qF → · · ·

ending at

· · · → π0,qF → π0,qE → π0,qB.


2.5. Dold-Thom theorem. In the non-equivariant case, the computation of the

homotopy type of the spaces of algebraic cycles Zp(PnC) is a direct consequence of the

suspension theorem and classical Dold-Thom Theorem. To compute the equivariant

homotopy type of Zp(PnC) we will need a generalization of the this classical theorem.

Definition 2.5. Let X be a G-space. The topological group of zero cycles on X is

denoted by Z0(X). Its elements are formal sums∑

i nixi, with ni ∈ Z and xi ∈ X.

There is an augmentation homomorphism deg : Z0(X) → Z. Its kernel is denoted

by Z0 (X). Note that, denoting by X+ the union of X with a disjoint point fixed

by the G-action, Z0(X) is isomorphic to Z0 (X+).

Theorem 2.6. [4] Let G be a finite group, let X be a based G-CW-complex and let

V be a finite dimensional G-representation, then there is a natural equivalence

πV Z0 (X) ∼= HGV (X; Z).

In particular, Z0

(SV)

is a K(Z, V ) space.

Given a G-space X there is a G-spectrum HZ ∧X+ such that π?(HZ ∧X+) =

H?(X; Z) (where ? refers to RO(G)-grading). The theorem above shows that Z0(X)

is G-homotopy equivalent to the zero-th space of the G-spectrum HZ ∧X+.

3. The equivariant homotopy type of Z and real vector bundles

In this section we study the space of algebraic p-cycles in PnC as a Z/2-space under

the action of Gal(C/R). We begin by reviewing some basic definitions concerning

algebraic cycles.

An effective algebraic p-cycle in PnC is a finite formal c =

∑i niVi where each ni

is a positive integer and the Vi’s are irreducible subvarieties of dimension p in PnC.

The degree of c is defined as deg(c) =∑

i ni deg(Vi), where deg(Vi) is the degree of

as an irreducible subvariety of PnC. The support of c is the algebraic set

⋃i Vi and

is denoted by |c|.

The set of effective algebraic p-cycles of degree d in PnC can be given the structure

of an algebraic set, denoted by Cp,d(PnC) (see [28]). If X is an algebraic subset of Pn

C,

the subset Cp,d(X) ⊂ Cp,d(PnC) consisting of those cycles whose support is contained

in X, is an algebraic subset of Cp,d(PnC). The algebraic structure of Cp,d(Pn

C) depends

on the embedding of X in PnC.


Definition 3.1. Let X ⊂ PnC be a projective subvariety. The Chow monoid of X

is the set

Cp(X) def=∐d≥0

Cp,d(X) = 0∐∐

d>0

Cp,d(X)

.

Here we consider Cp,d(X) with its analytic topology. The Grothendieck group (or

naive group completion) of Cp(X) is denoted Zp(X) and will be called the group of

p-cycles on X. The group Zp(X) is endowed with the quotient topology induced

by the map π : Cp(X)× Cp(X) → Zp(X) such that π(c, c′) = c− c′.

Remark 3.2. It is important to note that although the algebraic structure of

Cp,d(X) depends on the embedding, the homeomorphism type of Cp(X) does not,

cf. [7].

Definition 3.3. Let X be a subvariety of PnC. The algebraic suspension of X is

the subset of Pn+1C consisting of all points of the lines joining points of X to the

point (0 : · · · : 0 : 1). It is a subvariety of Pn+1C defined by the same equations as

X but now considered as equations in n + 2 variables. The operation Σ/ increases

dimension by one and keeps the codimension fixed.

Note that, as a topological space, Σ/ X is the Thom space of the line bundle

O(1)|X.

More generally, given varieties X ⊂ PnC and Y ⊂ Pm

C the set of points of the

lines joining points of X to points of Y — which we denote by X#Y — is a

subvariety of Pn+m+1C = Pn

C#PnC. Observe that dim X#Y = dim X +dim Y +1 and

Σ/ X = X#P0C.

The importance of the operation Σ/ for the computation of the Z/2-homotopy

type of Zp(PnC) is given by the following result of Lam. This result is an equivariant

version of Lawson’s suspension theorem [16].

Theorem 3.4. [15] The suspension map

Σ/ : Zp(PnC) −→ Zp+1(Pn+1

C )

is a Z/2-homotopy equivalence.

Proof. This is proved in [18, Prop.9.1].

Definition 3.5. The space Z of stabilized cycles is

Z = limn,p→∞

Zp(PnC)


where the limit is defined w.r.t. the suspension map and the natural inclusions

Zp(PnC) ⊂ Zp(Pn+1

C ).

Observe that, by Theorem 3.4, Z is Z/2-homotopy equivalent to Z0(P∞C ). The

join extends to a map Z ∧ Z → Z which is equivariant.

Definition 3.6. Let σ : Cn+1 × Cn+1 → C2n+2 be the “shuffle” isomorphism

defined by σ(z, w) = (z0, w0, . . . , zn, wn). Consider the composition

Zp(PnC) ∧ Zp(Pn

C)#−→ Z2p+1(P2n+1

C ) σ∗−→ Z2p+1(P2n+1C ).

The compositions above are compatible with the inclusions Zp(PnC) → Zp(Pn+1

C )

and the suspension map. Thus they define a pairing µ : Z ∧ Z → Z. It is clear

that µ is equivariant.

Using the equivariant version of the Dold-Thom Theorem we can now compute

the equivariant homotopy type of Zp(PnC) and Z.

Theorem 3.7. The space Zp(PnC) is Z/2-homotopy equivalent to the following of


(3.1) Zp(PnC) '−→

n−p∏k=0

K(Z, Rk,k).

Furthermore, under these equivalences, Zp(Pn−1C ) includes in Zp(Pn

C) as a factor

in the product (3.1) with constant last coordinate. Hence, the space of stabilized

cycles, Z = colimn,p→∞Zp(PnC), is Z/2-homotopy equivalent to a weak product of


(3.2) Z '−→∞∏

k=0

K(Z, Rn,n).

Proof. By Theorem 3.4, Zp(PnC) ' Z0(Pn−p

C ) hence we need to show

Z0(PnC) '

n∏k=0

K(Z, Rk,k).

This equivalence is a consequence of Theorem 2.6 and the fact that PnC has an

equivariant cell decomposition with one cell of dimension Rk,k for k = 0, . . . , n; the

cell decomposition is given by the filtration P0C ⊂ P1

C ⊂ · · · ⊂ PnC of Pn

C (note that

PnC/Pn−1

C∼= Sn,n). Using this decomposition, it can be proved that (see Lemma 5.5)

(3.3) H∗,∗(PnC; Z) ∼=

n⊕k=0

H∗,∗(Sk,k; Z);


It follows that H∗,∗(PnC; Z) is a free module over the cohomology ring of a point,

R, with one generator xk in dimension (k, k), for k = 0, . . . , n. By the equivariant

version of Dold-Thom (2.6), the same is true of π∗,∗Z0(PnC). We proceed to make

an explicit choice of generators xk. Let x1 be the equivariant map

S1,1 3 x 7→ x−∞ ∈ Z0(P1C).

For k > 1 we observe that Sk,k ∼= S1,1 ∧ . . .∧S1,1 (k times) and set xk equal to the

composition

Sk,k x1#···#x1−−−−−−−→ Zk−1(P2k−1C )

Σ/ −k+1

−−−−→ Z0(PkC).

We claim that xk is a generator of πk,kZ0(PnC), for n ≥ k. To see this we observe

that its image under the functor F that forgets the Z/2-action is a generator of

π2kZ0(PnC) because it coincides with the choice of generator for π2kZ0(Pn

C) made

in [19]. Now, (3.3) and theorem 2.6 imply that πk,kZ0(PnC) is the Mackey functor

Z (see the proof of Proposition 3.10 for a more detailed explanation). Since the

map Z(pt) → Z(Z/2) induced by Z/2 → pt is the identity we conclude that xk is a

generator of πk,kZ0(PnC).

Define

F :n∏

k=0

Z0

(Sk,k

)→ Z0(Pn

C)

as the group homomorphism extension of x1 + · · ·+ xn. Since F is a group homo-

morphism it extends to a morphism of spectra HZ ∧∨n

k=0 Sk,k → HZ ∧ PnC. The

map π(F ) induced by F on the homotopy groups is a map between homology of∨nk=0 Sk,k and of Pn

C which are R-modules. Moreover, since F is the zero-th map of

a morphism of spectra, the induced map on homotopy groups π(F ) is a morphism of

R-modules. But the homologies of these two spaces are isomorphic free R-modules,

and F is defined so that π(F ) is a bijection on the generators. It follows that π(F )

is an isomorphism hence F is an equivariant homotopy equivalence.

By construction the equivalence F sends the inclusion Z0(Pn−1C ) ⊂ Z0(Pn

C) to

the inclusionn−1∏k=0

K(Z, Rk,k)× ∗ ⊂n∏

k=0

K(Z, Rk,k);

hence the statement about the inclusion of Zp(Pn−1C ) in Zp(Pn

C) is a consequence of

the commutative diagram

Zp(Pn−1C ) −−−−→ Zp(Pn

C)

Σ/

x xΣ/

Z0(Pn−p−1C ) −−−−→ Z0(Pn−p

C )


The equivalence (3.2) now follows from the fact that Z is a colimit of the spaces

Zp(PnC) w.r.t. the maps Zp(Pn−1

C ) ⊂ Zp(PnC) and the equivalences Σ/ : Zp(Pn

C) →

Zp+1(Pn+1C ).

Remark 3.8. The Z/2-homotopy equivalence F :∏

k≥0 Z0

(Sk,k

)→ Z is com-

pletely determined by a choice of generators xk of πk,kZ and the fact that it is an

abelian group homomorphism. Henceforth when we refer to the equivalence (3.2)

we mean the group homomorphism F determined by the choice generators xk made

in the proof above.

This equivalence agrees with the equivalence∏

k≥0 K(Z, 2k) ' Z of [19] when

we forget the Z/2-action. It is also possible to define an equivalence G : Z →∏k≥0 Z0

(Sk,k

)using the Z/2-homeomorphisms Pk

C∼= SP k(P1

C) ( see [18]).

We now analyze the equivariant pairing µ. In [19] it is proved that, non-

equivariantly, µ classifies the cup product in singular cohomology with integer co-

efficients. As mentioned before, there is a notion of cup product in H∗,∗(−; Z). In

Proposition 3.10 we show that µ classifies the cup product in equivariant cohomol-

ogy with Z coefficients.

Note 3.9. We will need to following fact: let G be a finite group and let Xα be

a family of G-spaces. Then Z0 (∨

α Xα) is G-homeomorphic to the weak product∏α Z0 (Xα). If iα0 denotes the inclusion Xα0 ⊂

∨α Xα the homeomorphism is

given by ⊕αiα∗; see [26] for a proof.

Proposition 3.10. Under the equivalence of Theorem 3.7 the map

µ : Z ∧ Z → Z

is Z/2-homotopic to the map

ν :

( ∞∏k=0

Z0

(Sk,k

))∧

( ∞∏k=0

Z0

(Sk,k

))→

∞∏k=0

Z0

(Sk,k

),

defined as the biadditive extension of the smash product of spheres, Sk,k ∧ Sk′,k′ →

Sk+k′,k+k′ . In particular, µ classifies the cup product in Z/2-equivariant cohomol-

ogy with Z coefficients.

Proof. Consider the inclusion of ik,k′ : Sk,k ∧ Sk′,k′ → Z ∧Z given by

Sk,k ∧ Sk′,k′ 3 x ∧ y 7→ (x−∞) ∧ (y −∞) ∈ Z0

(Sk,k

)∧ Z0

(Sk′,k′

)⊂ Z ∧ Z.


We start by showing that, for every k, k′, the compositions µ ik,k′ and ν ik,k′ are

Z/2-homotopic. In [19] it is proved that these compositions are non-equivariantly

homotopic. We will see that the forgetful map

(3.4) F : [Sn,n,Z]Z/2 −→ [Sn,n,Z]

is an isomorphism. Thus, the fact that µ ik,k′ and ν ik,k′ are homotopic implies

that they are also Z/2-homotopic. Note that F is the map induced in equivariant

cohomology by the projection p : Sn,n∧Z/2+ → Sn,n and the cofiber of p is Sn,n+1.

Since, for any Z/2-space X

[X,Z]Z/2∼=

∞⊕k=0

Hk,k(X; Z),

F fits into an exact sequence∞⊕

k=0

Hk,k(Sn,n+1; Z) → [Sn,n,Z]Z/2 → [Sn,n,Z] →∞⊕

k=0

Hk+1,k(Sn,n+1; Z).

From (2.1) it follows that the first and last groups on this sequence are both trivial

and hence F is an isomorphism.

We now use the Z/2-homeomorphism

∞∏k=0

Z0

(Sk,k

) ∼= Z0

( ∞∨k=0

Sk,k

)mentioned before. From what was said above we see that the restrictions of µ and

ν to∨∞

k=0 Sk,k ∧∨∞

k=0 Sk′,k′ are Z/2-homotopic. Let

H :∞∨

k=0

Sk,k ∧∞∨

k=0

Sk′,k′ ∧ I+ → Z

be an equivariant homotopy from the restriction of ν to the restriction of µ. Extend

H to an equivariant homotopy through biadditive maps

F : Z0

( ∞∨k=0

Sk,k

)× Z0

( ∞∨k=0

Sk′,k′

)× I → Z.

Since ∧ is biadditive we have F (−,−; 0) = ∧. Now, µ is biadditive up to Z/2-

homotopy, so F (−,−; 1) is Z/2-homotopic to µ. Since F (−,−; t) is biadditive, F

descends to

Z0

( ∞∨k=0

Sk,k

)∧ Z0

( ∞∨k=0

Sk′,k′

)∧ I+.

This completes the proof that ν and µ are Z/2-homotopic.

The statement regarding the cup product is a consequence of fact that the pair-

ing ∧ on the Z/2-Ω-prespectrum Rp,q 7→ Z0 (Sp,q) induces the cup product in

H∗,∗(−; Z). Indeed, the parings HZ ∧HZ → HZ are in bijective correspondence


with the pairings Z Z → Z — where denotes the box product of Mackey func-

tors defined by Lewis (see [21] and [27]). This correspondence sends a pairing

HZ ∧HZ → HZ to the map induced on π0, using the fact that, given two Mackey

functors M and M ′, π0(HM ∧ HM ′) = M M ′. But Z Z is just Z so, up to

equivariant homotopy, there is only one pairing HZ ∧ HZ → HZ which agrees

with the usual pairing on the non-equivariant Eilenberg-Mac Lane spectrum HZ,

when we forget the Z/2-action. Our product does this hence it classifies the cup

product in H∗,∗(−; Z).

One of the interesting features of the space Z is that the classifying space BU

maps naturally into it, as follows. We have

BU = limn→∞

Gn(C2n).

Linear spaces in C2n are degree one cycles on P2n−1C , thus BU maps to the compo-

nent of degree one, Z(1), of space of stabilized cycles and this map is equivariant.

Definition 3.11. Let P : BU → Z(1) be the equivariant map induced at the finite

level by the inclusions Gn(C2n) ⊂ Zn(P2n−1C ). The compatibility of these inclusions

with the maps used to define the limits BU and Z guarantees that P is well-defined.

The space BU with the action induced by complex conjugation is the classifying

space for the (connective) KR-theory of Atiyah [1]. Its Z/2-equivariant cohomology

can be easily computed using the equivariant cell decomposition coming from the

Schubert cells. Denoting byR the cohomology ring of a point H∗,∗(pt; Z), as before,

we get

H∗,∗(BU; Z) ∼= R[c1, . . . , cn, . . .],

where the cn’s are classes of dimension (n, n), n ∈ Z+, whose images under the

forgetful functor to singular cohomology are the Chern classes, cn; see Lemma 5.5

for a similar computation. The classes cn are universal characteristic classes for

real vector bundles. We call them equivariant Chern classes for real vector bundles.

We show that, as in the non-equivariant case, the map P : BU → Z classifies the

total equivariant Chern class, i.e., it classifies

1 + c1 + c2 + · · ·+ cn + · · ·

Proposition 3.12. Let ιn denote the universal (n, n)-dimensional class in the

cohomology of K(Z, Rn,n). Using the isomorphism of Equation (3.2), we consider


ιn has an element in the cohomology of Z. Then

P ∗(ιn) = cn.

Proof. The proof goes exactly as in the non-equivariant case [19]. One observes

that BU(n) = limk→∞Gk(Cn+k) maps to

limk→∞

Zk(Pn+kC ) ' Z0(Pn

C) 'n∏

k=0

Z0

(Sk,k

),

where the limit is defined using the map Σ/ : Zk(Pn+kC ) → Zk+1(Pn+k+1

C ). Recall

that under the isomorphism of Equation (3.2) the inclusion Z0(Pn−1C ) ⊂ Z0(Pn

C)

is the inclusion as a factor in the product. Thus P ∗(ιn)|BU(n − 1) = 0 and so

P ∗(ιn) = λcn, for some λ ∈ C. Let F denote the forgetful functor from equi-

variant cohomology to singular cohomology. Since F ιn = ιn (the generator of

H2nsing(Z0

(S2n

); Z)) and by [19] P ∗(ιn) = cn we conclude that λ = 1.

The equivariant product µ restricts to a product on the fixed point set ZRdef=

ZZ/2 giving a ring structure to the Z-graded homotopy groups of ZR. The compu-

tation of this ring is one of the main results of [18]. In view of Proposition 3.10 this

ring can be interpreted as a subring of the equivariant cohomology of a point:

Lemma 3.13. The ring (ZR, µ) is isomorphic to a subring of H∗,∗(pt; Z).

Proof. For k > 0, Theorem 3.7 gives,

πk(ZR) ∼=⊕n≥0

[Sk, Z0 (Sn,n)]Z/2∼=⊕n≥0

Hn,n(Sk,0; Z) ∼=⊕n≥0

Hn−k,n(pt; Z).(3.5)

So we see from (3.5) that π∗(ZR) is isomorphic as a group to the subring consisting

of the Hp,q(pt; Z) such that q ≥ 0. The fact that the product structure is the same

follows from Proposition 3.10.

From (2.1) we can also conclude that the homotopy type of K(Z, Rn,n)Z/2 '

Z0 (Sn,n)Z/2 is

(3.6)

K(Z, 2n)×K(Z/2, 2n− 2)×K(Z/2, 2n− 4)× · · · ×K(Z/2, n) n even,

K(Z/2, 2n− 1)×K(Z/2, 2n− 3)× · · · ×K(Z/2, n) n odd.


From this decomposition it follows that, for a space X with trivial Z/2 action

there is a natural equivalence

Hn,n(X; Z) ∼=

H2n

sing(X; Z)⊕n/2⊕k=1

H2n−2ksing (X; Z/2) n even,

(n−1)/2⊕k=0

H2n−2k−1sing (X; Z/2) n odd.

4. A version of Lawson homology for real varieties

In this section we propose a definition of Lawson homology for real algebraic

varieties. This definition is a natural equivariant generalization of Lawson homology

for projective varieties, and we show that it still carries all the basic properties which

make Lawson homology computable.

4.1. Lawson homology for complex varieties. We start by recalling the defini-

tion of the Lawson homology groups for complex projective varieties. These groups

are a hybrid of algebraic geometry and algebraic topology: for a projective variety

X, the Lawson homology groups of X, LpHk(X) are the following set of invariants

LpHk(X) def= πk−2pZp(X) for k ≥ 2p and p ≤ dim X.

In the case p = 0, it follows by the Dold-Thom theorem that L0Hk(X) ∼= Hsingk (X; Z).

For k = 2p, we get LpH2p(X) = π0Zp(X) and Friedlander has shown [7] that this is

isomorphic to the group of p-cycles on X modulo algebraic equivalence. We recall

that algebraic equivalence is generated by the following relation: c0, c1 ∈ Cp(X)

are equivalent if there exists a smooth curve C, a (p + 1)-cycle , Z, on X × C

equidimensional over C (i.e. Z meets each fiber X × t properly, t ∈ C), and

points t0, t1 of C, such that ci = Z • (PnC × ti), i = 0, 1. Here we assume that X

is embedded in PnC and • denotes intersection of cycles which is well defined as a

cycle since these cycles intersect properly [13].

The functorial properties of Lawson homology are a consequence of the following

basic facts (see [7]): let X, Y,W be projective varieties then

(i) a morphism f : X → Y induces a continuous map f∗ : Zp(X) → Zp(Y ), for

any 0 ≤ p ≤ dim X;

(ii) a flat morphism g : W → X of relative dimension r ≥ 0 induces a continuous

map g∗ : Zp(X) → Zp+r(W ), for any 0 ≤ p ≤ dim X.


The maps f∗ and g∗ are induced by push-forward and flat pull-back of cycles,

respectively. They induce maps f∗ : LpHn(X) → LpHn(Y ) and g∗ : LpHn(X) →

Lp+rRn+2r(Y ).

4.2. Real varieties. We will adopt the point of view that real varieties are complex

varieties with extra structure.

Definition 4.1. A real quasi-projective algebraic variety U is a quasi-projecti-

ve variety with an anti-holomorphic involution τ : U → U . A morphism of real

quasi-projective varieties (U ′, τ ′), (U, τ) is a morphism of quasi-projective varieties

f : U ′ → U such that f τ ′ = τ f .

The projective space PnC with τ(x0 : · · · : xn) = (x0 : · · · : xn) is an example

of a real variety. Any real quasi-projective variety has a real embedding into a

projective space, i.e., there is an embedding φ : U → PnC which is equivariant w.r.t

the action induced by complex conjugation on PnC ( see [29], for example).

If X is a real variety the anti-holomorphic involution τ induces a Z/2-action on

X. The fixed points XZ/2 are the real points of X ( i.e. defined over R) and are

denoted X(R).

Example 4.2. Let H = C ⊕ Cj denote the quaternions, and let PC(Hn) be the

projective space of complex lines in Hn. The anti-homolomorphic involution on

PC(Hn) given by multiplication by j from the left on Hn gives it a real structure.

As a complex variety, PC(Hn) is isomorphic to P2n−1C , but is clear these are very

different as real varieties since PC(Hn)(R) = ∅ and P2n−1C (R) = P2n−1

R . The real

varieties PC(Hn) are examples of Severi-Brauer varieties.

4.3. Lawson homology for real varieties. A version of Lawson homology for

real varieties should make use of the extra structure provided by the anti-holomorphic

involution in order to provide information about the real structure. In fact, if X

is a real variety the Chow varieties Cp,d(X) are also real varieties [28, Chapter I.9].

The corresponding anti-holomorphic involution Cp,d(X) → Cp,d(X) is induced from

the involution τ on X as follows: observe that, for a subvariety V , τ∗V is a subva-

riety of the same degree as V . Given a cycle c =∑

i niVi set τ∗(c) =∑

i niτ∗(Vi).

The real points of Cp,d(X) are the real p-cycles of degree d on X, i.e., of the form∑i niVi, where the Vi’s are irreducible real subvarieties of X. The Chow monoid

Cp(X) and the group of p-cycles Zp(X) are Z/2-spaces under the action of the

continuous involution τ∗.


Notation 4.3. The fixed points of Cp(X) and Zp(X) under the action of τ∗ will

be denoted by Cp(X)(R) and Zp(X)(R), respectively. We will refer to these spaces

as the monoid of real p-cycles and the group of real p-cycles on X, respectively.

It is important to note that the Z/2-homeomorphism type of Zp(X) is an invari-

ant of the real variety X. Indeed, suppose f : (X ′, τ ′) → (X, τ) is a real isomor-

phism. It follows that f induces a Z/2-equivariant homeomorphism f∗ : Zp(X ′) →

Zp(X) defined by f∗∑

i niVi =∑

i nif∗(Vi); see [7]. Thus the equivariant homeo-

morphism type of Zp(X), equipped with the action τ∗, is an invariant of the real

structure on X. From here onwards the groups Zp(X) are always considered as

Z/2-spaces with this action.

Going back to the problem of defining a version of the Lawson homology groups

for real varieties which are invariants of the real structure, it is natural to look

at spheres with Z/2-actions and consider equivariant homotopy classes. We are,

therefore, naturally led to the following definition.

Definition 4.4. Let X be a real projective variety. The real Lawson homology

groups of X, are the groups

LpRn,m(X) def= πn−p,m−pZp(X) for n, m ≥ p and p ≤ dim X.

The previous observations imply that the groups LpHn,m(X) are invariants of

the real structure on X.

Remark 4.5. It follows from Theorem 2.6 that, for cycles of dimension zero, the

real Lawson homology groups of a real projective variety X are the Z/2-equivariant

homology groups of X with coefficients in the Mackey functor Z, i.e.

L0Rn,m(X) ∼= Hn,m(X; Z).

In the case (n, m) = (p, p) we have LpRn,m(X) = π0(Zp(X)(R)) and it is natural

to expect that these groups should be related to the notion of algebraic equivalence.

The Proposition below answers this question. The answer is formulated in terms

of an equivalence relation for real cycles introduced by Friedlander and Walker in

[11], which uses the analytic topology on the set of real points of a real curve.

Proposition 4.6. Let X ⊂ PnC be a real variety. The real Lawson homology group

LpRp,p(X) is the group of real p-cycles modulo the equivalence relation generated

by: two cycles c0, c1 ∈ Cp(X)(R) are equivalent if there exists a smooth real curve

C, a real (p+1)-cycle Z on X×C equidimensional over C ( i.e. Z meets each fiber


X × t properly, t ∈ C), and points t0, t1 lying in the same connected component

of C(R), such that ci = Z • (PnC × ti), i = 0, 1.

Proof. Since π0(Zp(X)(R)) is the Grothendieck group of the monoid π0(Cp(X)(R))

it suffices to show that π0(Cp(X)(R)) is the quotient of Cp(X)(R) by the equivalence

relation ∼ defined above.

Let C be a smooth real curve, by [7, Cor.1.5] there is 1-1 correspondence between

morphisms C → Cp(X) of real varieties and real (p + 1)-cycles on X ×C which are

equidimensional over C. Under this correspondence, the morphism f corresponding

to a cycle Z ∈ Cp+1(X × C), as above, satisfies f(t) = Z • (PnC × t), t ∈ C. Thus

if c0, c1 are real p-cycles such that c0 ∼ c1 then c0 and c1 lie in the same connected

component of Cp(X)(R).

Suppose now that c0, c1 are real p-cycles which lie in the same connected compo-

nent of Cp(X)(R). Then c0, c1 lie in the same component of Cp,d(X) for some d. We

need to show that c0 ∼ c1. We will use following fact shown in [11, Prop.1.6]: given

a real quasi-projective variety T and points t0, t1 lying in the same connected com-

ponent of T (R) there exist smooth real curves C0, . . . , Ck and points ai, bi, lying in

the same connected component of Ci(R), for each i ∈ 0, . . . , k, and morphisms of

real varieties fi : Ci → T such that f0(a0) = t0, fi(bi) = fi+1(ai+1) and fk(bk) = t1.

Applying this with T = Cp,d(X), t0 = c0, t1 = c1 and the correspondence between

real morphisms Ti → Cp,d(X) and equidimensional (p + 1)-cycles on X × C we

conclude that c0 ∼ c1.

Remark 4.7. The computation of the real Lawson homology groups of a real

variety X completely determines the homotopy type of the spaces of real cycles

Zp(X)(R). This is an immediate consequence of that fact that these spaces are

topological abelian groups and hence products of Eilenberg-Mac Lane spaces and

so their homotopy type is completely determined by the homotopy groups. The ho-

motopy groups πk(Zp(X))(R) are the real Lawson homology groups LpRk+p,p(X).

This computation is important in itself as the Chow varieties Cp,d(X) are classical

objects in algebraic geometry about which not much is known.

The functorial properties of Lawson homology mentioned above also hold in the

real case: let X, Y and W be projective varieties, let f : X → Y be morphism of real

varieties and let g : W → X be a real flat morphism of relative dimension r. The

continuous maps f∗ : Zp(X) → Zp(Y ) and g∗ : Zp(X) → Zp+r(W ) are equivariant


and thus induce maps f∗ : LpRn,m(X) → LpRn,m(Y ) and g∗ : LpRn,m(X) →

Lp+rRn+r,m+r(Y ).

Next we establish the basic properties of real Lawson homology such as the

existence of relative groups and exact sequences for pairs. Following Lima-Filho’s

definition in the non-equivariant case, we define

Definition 4.8. Let (X, X ′) be a real pair, i.e. X ′ is a real subvariety of X. The

group of relative p-cycles is the quotient

Zp(X, X ′) def=Zp(X)Zp(X ′)

,

with the quotient topology. Note that Zp(X, X ′) is a Z/2-space with the action

induced from Zp(X).

The real Lawson homology groups of the pair (X, X ′) are

LpRn,m(X, X ′) def= πn−p,m−pZp(X, X ′)

where, as above, n, m ≥ p and 0 ≤ p ≤ dim X.

The next result is the main step in showing the existence of long exact sequences

in real Lawson homology. The proof is a simple generalization to the equivariant

context of [23, Thm.3.1].

Proposition 4.9. The short exact sequence of topological groups

(4.1) 0 −→ Zp(X ′) −→ Zp(X) −→ Zp(X, X ′) −→ 0

is an equivariant fibration sequence.

Proof. See Section 6.

Proposition 4.10. Let (X, X ′, X ′′) be a real triple. Then the short exact sequence

of topological groups

0 −→ Zp(X ′, X ′′) −→ Zp(X, X ′′) −→ Zp(X, X ′) −→ 0

is an equivariant fibration sequence.

As a consequence, there is a long exact sequence of real Lawson homology groups

→ LpRn,m(X ′, X ′′) → LpRn,m(X, X ′′) → LpRn,m(X, X ′) → LpRn−1,m(X ′, X ′′) →

Proof. As in Proposition 4.9 we only need to show that the exact sequence of

topological groups

(4.2) 0 −→ Zp(X ′, X ′′)(R) −→ Zp(X, X ′′)(R) −→ Zp(X, X ′)(R) −→ 0


is a fibration sequence. Just as in the non-equivariant case [23, Prop.3.1] , this

follows from Proposition 4.9 in a standard fashion. Since (4.2) is a sequence of

topological groups the result will follow if we can show that (4.2) has a local cross-

section at zero (see [30]). By the proof of Proposition 4.9 there is a neighborhood

U of zero in Zp(X, X ′)(R) and a section s : U → Zp(X)(R) to the projection π1 :

Zp(X)(R) → Zp(X, X ′)(R). Composing s with the projection π2 : Zp(X)(R) →

Zp(X, X ′′)(R) we get the desired section.

Finally, we recall a fundamental result of Lima-Filho that provides a definition

of Lawson homology for quasi-projective varieties. This also yields a localization

sequence which is an important computational tool.

Theorem 4.11. [23, Thm.4.3] A relative isomorphism Ψ : (X, X ′) → (Y, Y ′)

induces an isomorphism of topological groups:

Ψ∗ : Zp(X, X ′) → Zp(Y, Y ′)

for all p ≥ 0.

Remark 4.12. Our observation here is that, if Ψ : (X, X ′) → (Y, Y ′) is a relative

real isomorphism of real pairs then Ψ∗ : Zp(X, X ′) → Zp(Y, Y ′) is an equivariant

homeomorphism.

Following [23] we define:

Definition 4.13. Let U be a real quasi-projective variety. The group of p-cycles

on U is the topological group

Zp(U) def= Zp(X, X ′)

where (X, X ′) is a real pair such that X −X ′ is isomorphic to U as real varieties.

Such a pair is called a real compactification of U . The group Zp(U) is considered

as a Z/2-space with the action induced from the action on Zp(X). Note that such

compatification always exists: if U ⊂ PnC is a real quasi-projective variety then let

X be the Zariski closure of U in PnC and set X ′ = X − U .

The real Lawson homology groups of U are defined as the groups of the pair

(X, X ′):

LpRn,m(U) def= πn−p,m−pZp(U)

where, as before, n, m ≥ p and 0 ≤ p ≤ dim X.


Remark 4.14 (Independence of compactification and functoriality). Theorem 4.11

shows that the definition of LpRn,m(U) is independent of the compactification and

is covariant w.r.t proper maps of real quasi-projective varieties. In fact, a proper

map of real quasi-projective varieties, Ψ : U → V , induces a set-theoretic map Ψ∗ :

Zp(U) → Zp(V ). The question is whether Ψ∗ is continuous. We recall Lima-Filho’s

argument to show continuity of Ψ∗. Suppose U, V have real compactifications

(X, X ′) and (Y, Y ′), respectively. Let Γ ⊂ X × Y be the closure of the graph

Graph(Ψ), where X × Y is endowed with the product real structure. Set Γ′ =

Γ − Graph(Ψ). Let π1 and π2 denote the projections on the first and second

factors, respectively. Then π1 : (Γ,Γ′) → (X, X ′) is a relative real isomorphism

and π2 : (Γ,Γ′) → (Y, Y ′) is a map of real pairs because Ψ is proper. From

Theorem 4.11 it follows that π1∗ is an equivariant homeomorphism. The continuity

of Ψ∗ follows because it coincides with the composition

π2∗ π1−1∗ : Zp(X, X ′) → Zp(Y, Y ′).

If Ψ is an isomorphism of real quasi-projective varieties π2∗ is also equivariant

homeomorphism and hence so is Ψ∗.

The long exact sequence for triple now gives the localization sequence for real

Lawson homology: let V be a real closed subset of a real quasi-projective variety

U and let n, m ≥ p. Then there is a long exact sequence of real Lawson homology

groups

(4.3) · · · → LpRn,m(V ) → LpRn,m(U) → LpRn,m(U − V )

→ LpRn−1,m(V ) → · · ·

ending at

· · · → LpRp,m(V ) → LpRp,m(U) → LpRp,m(U − V ).

As a consequence we can now prove the real version of the “homotopy property”

for Lawson homology.

Proposition 4.15. Let U be a real quasi-projective variety and let Eπ−→ U be a

real algebraic vector bundle of rank k. Then the flat pull-back of cycles

π∗ : Zp(U) −→ Zp+k(E)

is an equivariant homotopy equivalence.


Proof. The proof goes exactly as in the non-equivariant context [8]: it suffices to

show that the map induced by π∗ on the Bredon homotopy groups is an isomor-

phism. Using localization and the 5-lemma we can reduce to the case where E is

trivial. At this point one can use induction on k to reduce to the case of k = 1.

Then one can further reduce to the case where U has a projective closure U such

that E → U is the restriction to U of O(1)|U → U . The result now follows from

the suspension Theorem.

4.4. Cycle map. An important feature of Lawson homology is the existence of a

natural map sp : LpHk(X) → Hsingk (X; Z) called cycle map (or cycle class map). In

in the case k = 2p, we have LpH2p(X) = π0(Zp(X)) and sp is the map which sends

the component of c ∈ Zp(X) to the 2p-homology class that c represents (see [25]).

This is the motivation for calling it the cycle map. Our next goal is to define an

appropriate version of this map for real Lawson homology. It will play an important

role in the examples of Section 5.

There are several ways to define the cycle map for Lawson homology (see [25], [12]

and [8]). The definition we give here is a natural modification of that of Friedlander

and Gabber [8]. The two main ingredients in this construction are intersection with

divisors to get a map s : Zp(X) → Ω2Zp−1(X), and the Dold-Thom theorem to

map to singular homology by composition with sp (p iterations of s). To apply this

construction to real Lawson homology we need to check that intersecting with a

real divisor is equivariant w.r.t the action of Gal(C/R). Once this is achieved we

apply Theorem 2.6 to get a cycle map which takes values in equivariant homology

with Z coefficients.

Let U be a real quasi-projective variety and let D be a real Cartier divisor, whose

inclusion of D in U is denoted by iD : D → U . This means that D is defined by

the vanishing of a real section, sD, of a real algebraic line bundle π : LD → U . Let

V be the complement of |D| in U (recall that if D =∑

i niV , the support of D is

the algebraic set |D| = ∪iVi). Since i∗DLD is closed in LD and LD|V = LD − i∗DLD

we have an exact sequence

0 −→ Zp(i∗DLD) −→ Zp(LD) −→ Zp(LD|V ) −→ 0,

which by Proposition 4.9 is an equivariant fibration sequence. Consider the com-

position

res sD∗ : Zp(U) → Zp(LD) → Zp(LD|V ).


We claim that res sD∗ is equivariantly homotopic to the zero map: in [8] a

homotopy H is defined such that Ht is multiplication by t in the fibres of LD, for

t ∈ [1,∞[, and as the zero map for t = ∞. It is clear that H is equivariant. This

equivariant homotopy determines an equivariant map

σD : Zp(U) → Zp(i∗DLD),

well defined up to equivariant homotopy.

Definition 4.16. The intersection with a real divisor D is defined as the compo-

sition

i!Ddef= (π∗)−1 σD : Zp(U) → Zp(i∗DLD) → Zp−1(|D|).

We now explain how, using intersection with divisors, we can define a version of

the map s of [8] in the context of real varieties.

Let U be a real quasi-projective variety. Consider the composition

(4.4) Zp(U) ∧ Z0

(P1

C) ω−→ Zp(U × P1

C)i!U−→ Zp−1(U)

where ω(V, t) = V × t and i!U denotes intersection with U × ∞ which is a

real divisor in U × P1C. This map is clearly equivariant for the diagonal action

on Zp(U) × Z0

(P1

C)

where P1C is equipped with the action induced by complex

conjugation.

Definition 4.17. For a real quasi-projective U variety let i!U and ω be as above.

Consider P1C embedded in Z0

(P1

C)

by t 7→ t−∞. We define

s : Zp(U) → Ω1,1Zp−1(U)

as the adjoint of the restriction of i!U ω to Zp(U)∧ P1C. The map s induces a map

in real Lawson homology groups

s∗ : LpRn,m(U) → Lp−1Rn,m(U).

We define the cycle map for the real Lawson homology of U as the map sp∗ :

LpRn,m(U) → L0Rn,m(U). Frequently we abuse notation and denote s∗ and sp∗ by

s and sp, respectively.

The cycle map is the motivation for the indexing in real Lawson homology. If U is

a projective variety, the group L0Rn,m(U) is isomorphic to Hn,m(U ; Z). We think

of the elements of LpRn,m(U) as having algebraic dimension p and homological

dimension (n, m).


The map s also induces a map in usual Lawson homology, LpHn(U)→ Lp−1Hn(U),

which we denote by s∗ as well. In [8] it is proved that , in usual Lawson homology,

s∗ can be defined by a different construction. Consider P1C embedded in Z0

(P1

C),

as above, by mapping t ∈ P1C to t−∞. The adjoint of the composition

Zp(U) ∧ P1C

#−→ Zp+1(U#P1C)

Σ/ −2

−−−→ Zp−1(U)

is another a map s′ : Zp(U) → Ω2Zp−1(U), which is homotopic to s. Therefore the

map s∗ can also be realized in the following way. The inclusion of P1C in Z0

(P1

C)

is

a generator x for π2Z0

(P1

C) ∼= Z. Joining with x gives a map

πn−2pZp(U) → πn+2−2pZp+1(U#P1C) Σ−2

−−−→ πn+2−2pZp−1(U)

that coincides with s∗ in usual Lawson homology.

All this works equivariantly but now x is seen as the generator of the group

π1,1Z0

(P1

C) ∼= Z, so joining with it gives a map

πn−p,m−pZp(U) → πn+1−p,m+1−pZp+1(U#P1C)

Σ/ −2

−−−→ πn+1−p,m+1−pZp−1(U)

that coincides with s∗ in real Lawson homology.

Remark 4.18. The following observation is often useful. Suppose the map s :

Zp(X) → Ω1,1Zp−1(X) is an equivariant homotopy equivalence. Assuming the

diagram

Zp(X) s−−−−→ Ω1,1Zp−1(X)

Σ/

y Ω1,1Σ/

yZp+1(Σ/ X) s−−−−→ Ω1,1Zp(Σ/ X)

is commutative, it follows that s : Zp+1(Σ/ X) → Ω1,1Zp(Σ/ X) is also an equivariant

homotopy equivalence.

To see that the diagram above commutes we use the definition of the adjoint

of s given by joining with a generator of π1,1Z0

(P1

C). The result follows from the

commutativity of the diagram

Zp(X)× Z0

(P1

C) #−−−−→ Zp+1(Σ/

2X)

Σ/ −2

−−−−→ Zp−1(X)

Σ/ ×id

y Σ/

y Σ/

yZp+1(Σ/ X)× Z0

(P1

C) #−−−−→ Zp+2(Σ/

3X)

Σ/ −2

−−−−→ Zp(Σ/ X)

up to Z/2-homotopy.


4.5. Relation to Lawson homology for complex varieties. To understand

the relation between real Lawson homology and usual Lawson homology we start

by observing that every complex variety gives rise to a real variety.

Example 4.19. Let U ⊂ PnC be a quasi-projective variety. Then U q U is a real

variety with the involution induced by complex conjugation (it can be embedded

as a real subvariety in PnC × A1, for example).

We have Zp(U q U) ∼= Zp(U)×Zp(U) and, under this isomorphism, the involu-

tion is given by

τ · (C1, C2) = (C2, C1).

Recall that, for pointed Z/2-spaces S and X, F (S, X) denotes the space of based

maps with the conjugation action: f 7→ τ f τ . It follows that, if we consider

Zp(U) equipped with the trivial Z/2-action, then

(4.5) Zp(U q U) ∼= F (Z/2+,Zp(U)).

A more elucidating explanation of this example can be given using the language of

schemes, as follows. The natural inclusion R ⊂ C determines a morphism Spec C →

Spec R. Hence any scheme X over C can be seen as a scheme over R. We will denote

this scheme by XR. For affine varieties this construction corresponds to thinking of

an algebra over C as an algebra over R, by restriction of scalars. The set of complex

points of XR is precisely X(C)∐

X(C), and the corresponding anti-involution sends

a point x in one component to x in the other component.

Thus, if U is a complex variety, we can view it as a real variety. The following

Proposition shows that real Lawson homology gives the usual Lawson homology

when applied to a complex variety.

Proposition 4.20. Let U be a quasi-projective complex variety. Then

LpRr,s(UR) ∼= LpHr+s(U).

Proof. By Equation (4.5) we have

LpRr,s(UR) ∼= [Sr−p,s−p ∧ Z/2+,Zp(UR)]Z/2∼= [Sr+s−2p,Zp(U)] = LpHr+s(U).

Suppose now that (U, τ) is a real variety. Forgetting the real structure we can

think of it as a complex variety U . The corresponding real variety UR is just U∐

U

with the involution which sends x of one component to the point τ(x) in the other


component. The folding map π : U∐

U → U is a proper flat map of real varieties

of relative dimension zero. Hence it induces maps

π∗ : Zp(U) → Zp(UR)

π∗ : Zp(UR) → Zp(U).

It is easy to check that π∗ π∗ is multiplication by 2. As a consequence we have

Proposition 4.21 (cf.[11, Cor.5.9]). Let U be a real quasi-projective variety. Then

there is a transfer map π∗ : LpRr,s(U) → LpHr+s(U) and a restriction map π∗ :

LpHr+s(U) → LpRr,s(U) such that the composition π∗ π∗ is multiplication by 2.

In particular, for any finitely generated module M over Z[ 12 ], LpRr,s(U) ⊗ M is

isomorphic to a submodule of LpHr+s(U)⊗ M .

Proof. Clear.

4.6. Relation to other theories. There is also a cohomological version of Law-

son homology, called morphic cohomology, defined by Friedlander and Lawson as

follows. For simplicity we only consider the case of normal varieties. Given normal

projective varieties X and Y , we let Mor(X, Cr(Y )) denote the abelian monoid of

algebraic morphisms from X to Cr(Y ), endowed with the compact open topology.

The monoid of algebraic cocyles of codimension t on X is the topological quotient

monoid

Ct(X) def= Mor(X, C0(PtC))/Mor(X, C0(Pt−1

C )).

Its associated Grothendieck group is called the topological group of codimension t

algebraic cocycles on X; it is denoted Zt(X). The morphic cohomology groups of

X, LpHk(X), are the bigraded groups defined by (see [10])

LtHk(X) def= π2t−kZt(X).

If X is a real variety then Zt(X) becomes naturally a Z/2-space and it is natural

to define a version of morphic cohomology for real varieties in terms equivariant

homotopy groups of Zt(X). This has been done recently by Friedlander and Walker

in [11]. There they introduce a set of invariants for real varieties called semi-

topological K-theory and denoted KRsemi(−), which interpolate between algebraic

K-theory and Atiyah’s KR-theory. Moreover, they show real morphic cohomology

interpolates between motivic cohomology and singular cohomology and use it to

define higher Chern classes for KRsemi(−). Using techniques similar to those of

[11] it would be possible to show that, for real varieties, real Lawson interpolates


between motivic homology and equivariant homology with Z coefficients. We will

not pursue this here as it would take to far afield.

A duality between Lawson homology and morphic cohomology has been estab-

lished by Friedlander and Lawson in [9]. In a forthcoming paper with Lima-Filho

we will show that for smooth real varieties real Lawson homology is dual to real

morphic cohomology. With this result, the computations performed in Section 5

concerning spaces of cycles yield the equivariant homotopy type of the correspond-

ing spaces of algebraic morphisms.

5. Examples and computations

In this section we compute the real Lawson homology groups for some real vari-

eties. The main tools in these computations are the localization sequence (4.3) and

the cycle map sp. We start with the fundamental example of affine space An with

its standard real structure.

Example 5.1 (The real Lawson homology of affine space An). Let An have the

real structure given by complex conjugation of its coordinates. We call this real

structure on An the standard real structure. By definition,

Zp(An) =Zp(Pn

C)Zp(Pn−1

C ).

Since An → An−p is a real algebraic bundle, it follows from Proposition 4.15

Zp(An) ' Z0(An−p) =Z0(Pn−p

C )Z0(Pn−1−p

C )' K(Z, Rn−p,n−p).

Here we used the following important property of the equivalence in Theorem 3.7:

under this equivalence the inclusion Zp(Pn−1C ) ⊂ Zp(Pn

C) corresponds to the inclu-

sionn−p−1∏

k=0

Z0

(Sk,k

)→

n−p∏k=0

Z0

(Sk,k

)as a factor. Thus, for 0 ≤ p ≤ n and r, s ≥ p,

LpRr,s(An) ∼= πr−p,s−pK(Z, Rn−p,n−p) ∼= Hn−r,n−s(pt; Z).

Moreover we see that the cycle map sp gives an equivariant homotopy equivalence

sp : Zp(An) −→ Ωp,pZ0(An).

Since sp will play a central role in the examples to follow we will explain this in

detail.


Recall the description of s on Zp(PnC) as the adjoint of the restriction of the

composition

(5.1) Zp(PnC) ∧ Z0

(P1

C) #−→ Zp+1(Pn+2

C )Σ/ −2

−−−→ Zp−1(PnC)

to Zp(PnC)∧P1

C. Where we consider P1C embedded in Z0

(P1

C)

by the map t 7→ t−∞.

Recall also that we have a complete description of the action of the join and suspen-

sion maps on the cycles of PnC: the suspension Theorem identifies Zp(Pn

C), Zp−1(PnC),

canonically, with Z0(Pn−pC ) and Z0(Pn+1−p

C ), respectively. The commutativity of

the diagram (up to equivariant homotopy)

Zp(PnC)× P1

C#−−−−→ Zp+1(Pn+2

C )Σ/ −2

−−−−→ Zp−1(PnC)

Σ/ −p×id

y yΣ/ −p

yΣ/ −(p−1)

Z0(Pn−pC )× P1

C#−−−−→ Z1(Pn+2−p

C )Σ/ −1

−−−−→ Z0(Pn+1−pC )

shows that the, under the above identifications, the map (5.1) is identified with the

product µ (see Proposition 3.10) restricted to Z0(Pn−pC ) ∧ P1

C.

By Theorem 3.7,

Z0(Pn−pC ) '

n−p∏k=0

Z0

(Sk,k

),

and by Proposition 3.10, µ restricted to Z0(Pn−pC ) ∧ P1

C is identified with the map

n−p∏k=0

Z0

(Sk,k

)∧ P1

C∧−→

n−p+1∏k=1

Z0

(Sk,k

)induced by the smash map Sk,k ∧ S1,1 ∧−→ Sk+1,k+1 (recall that P1

C∼= S1,1). It

follows that this map is one of the structural maps of the Ω-Z/2-prespectrum Rp,q →

Z0 (Sp,q) hence its adjoint is an equivariant homotopy equivalence

(5.2)n−p∏k=0

Z0

(Sk,k

)' Ω1,1

n−p+1∏k=1

Z0

(Sk,k

)This concludes the analysis of the map s : Zp(Pn

C) → Ω1,1Zp−1(PnC). To obtain

the result for the affine space An we observe that s is natural, so s : Zp(An) →

Ω1,1Zp−1(An) is obtained by passing to the quotient in (5.2) and we get that, for

An, s is the adjoint of

Zp(An) ∧ S1,1 ' Z0

(Sn−p,n−p

)∧ S1,1 ∧−→ Z0

(Sn+1−p,n+1−p

)' Zp−1(An)

which is an equivariant homotopy equivalence, as desired.

The following summarizes our conclusions regarding the real Lawson homology

of affine space An.


Lemma 5.2. The space Zp(An) is an equivariant Eilenberg-Mac Lane space of type

K(Z, Rn−p,n−p), for every 0 ≤ p ≤ n. Moreover, cycle map

sp : Zp(An) −→ Ωp,pZ0(An)


We will now make use of the cycle map and the localization sequence to prove

the following general result about the real Lawson homology of real varieties with

a real cell decomposition. The next definition is an adaptation to the real case of

[25, Definition 5.3]

Definition 5.3. Let (X, Y ) be a pair of real projective varieties. We say that X

is a real algebraic cellular extension of Y if there is a filtration

X = Xn ⊃ Xn−1 ⊃ · · ·X0 ⊃ X−1 = Y

by real projective subvarieties Xi such that Xi − Xi−1 is a union of affine spaces

Anij . If Y = ∅ we say that X has a real cell decomposition.

Theorem 5.4. Let X be a real quasi-projective variety with a real cell decomposi-

tion, then the map

sp : Zp(X) −→ Ωp,pZ0(X)

is an equivariant homotopy equivalence. In particular, the cycle map induces an

isomorphism

LpRn,m(X) ∼= Hn,m(X; Z),

so that LpRn,m(X) is independent of p in this case.

Proof. The result is proved by induction using the localization sequence and the

fact that, by Example 5.1, it holds for affine spaces. Assume that

sp : Zp(Xi−1) −→ Ωp,pZ0(Xi−1)

is an equivariant homotopy equivalence. Applying the localization sequence and

the cycle map sp we get a map of long exact sequences

LpRk+p,p(Xi)

sp

// LpRk+p,p(Xi −Xi−1)

sp

// LpRk+p−1,p(Xi−1)

sp

//

L0Rk+p,p(Xi) // L0Rk+p,p(Xi −Xi−1) // L0Rk+p−1,p(Xi−1) //


ending at

LpRp,p(Xi−1)

sp

// LpRp,p(Xi)

sp

// LpRp,p(Xi −Xi−1)

sp

// 0

L0Rp,p(Xi−1) // L0Rp,p(Xi) // L0Rp,p(Xi −Xi−1) // 0.

Exactness at the last group of the bottom row follows from the fact that, since

Xi−1 has a real cell decomposition, Hm−1+p,p(Xi−1; Z) = 0, for all m ∈ Z. This

follows from Lemma 5.5 since, Hr,s(pt; Z) ∼= R−r,−s = 0 for all r, s > 0.

By the assumptions and the 5-Lemma it follows that

sp∗ : LpRk+p,p(Xi) −→ L0Rk+p,p(Xi)

is an isomorphism for all k ≥ 0. Translating this into homotopy groups, it means

that

sp∗ : πk (Zp(Xi)(R)) −→ πk (Ωp,pZ0(Xi))

Z/2

is an isomorphism for all k ≥ 0. Since we already know that sp is a non-equivariant

homotopy equivalence [25], this implies that sp : Zp(Xi) → Ωp,pZ0(Xi) is an equi-

variant homotopy equivalence.

Lemma 5.5. Let X be a real variety with a real cell decomposition

X = Xn ⊃ Xn−1 ⊃ · · ·X0 ⊃ X−1 = ∅

such that Xi−Xi−1 is a union of affine spaces Anij . Let R denote the cohomology

ring of a point, H∗,∗(pt; Z). Then H∗,∗(X; Z) is an R-free module. Each cell Anij

gives rise to a generator xi,j in dimension Rnij ,nij .

Proof. Denote the unit disk of the representation Rnij ,nij by D(Rnij ,nij ). The real

cell decomposition gives X an equivariant cell decomposition with cells of type

D(Rnij ,nij ). The proof is by induction on the cells: by Definition 5.3, X0 is a

disjoint union of points fixed by the action, so the result holds. Assume it also

holds for Xi−1 and consider the cofibration sequence

Xi−1+ −→ Xi+ −→∨j

Snij ,nij .

There is a long exact sequence

(5.3) −→ Hr,s(Xi; Z) −→⊕

j Hr,s(Snij ,nij ; Z) δ−→ Hr−1,s(Xi−1; Z) −→


Observe that this is an exact sequence of R-modules and, by assumption, the

homology of Xi−1 is free on generators xk,j , k < i, of dimensions (nkj , nkj). Also

H∗,∗(Snij ,nij ; Z) ∼= Hnij−∗,nij−∗(pt; Z) = Rnij−∗,nij−∗

so, in particular, this R-module is free and generated by an element xi,j in dimen-

sion (nij , nij) (xi,j is sent to the identity element in R by the isomorphism above).

The connecting homomorphism δ in the sequence (5.3) is determined by the image

of the generators xi,j . But the induction hypothesis implies that this image is zero

because

Rr−1,r = 0

for all r ∈ Z; see Equation (2.1). This completes the proof.

The following are examples of real varieties with a real cell decomposition.

Example 5.6 (The Grassmannians Gq(Cn+1)). The variety Gq(Cn+1) has a real

structure given by the action induced by complex conjugation in Cn+1. The Schu-

bert cells give a real cell decomposition for Gq(Cn+1).

Example 5.7 (Products of varieties with real cell decompositions). Real varieties

with a real cell decomposition form a class which is closed under products. So,

for example, PnC × Pm

C has a real cell decomposition and we have that the group

LpRα(PnC × Pm

C ) is isomorphic to the α degree part of the RO(Z/2)-graded module

R[x, y]/(xn, ym) where R is the cohomology of a point and x, y have degree (1, 1).

Example 5.8 (Quadrics with signature zero). Any real smooth quadric in Pn−1C is

equivalent to a quadric of the form

Qn,kdef=(x1 : · · · : xn) ∈ Pn−1

C |x21 + · · ·+ x2

k − x2k+1 − · · · − x2

n = 0

where k ≤ n/2.

We consider the case, n = 2k, i.e. the quadratic form defining the quadric has

signature zero. We will show that the cycle map is an isomorphism. From now on

we use homogeneous coordinates (X : Y ) = (x1 : . . . : xn : y1 : . . . : yn) for the

points of P2n−1C . In these coordinates the quadratic form is XXT − Y Y T . The

point p0 = (X0 : Y0) = (0 : . . . : 0 : 1 : 0 : . . . : 0 : 1) is a real point of Q2n,n and the

tangent plane to Q2n,n through p0 is

H =(X : Y ) ∈ P2n−1

C |(X : Y ) · (X0 : −Y0) = 0


and

Q2n,n ∩H =(X : Y ) ∈ P2n−1

C |xn − yn = 0 and XXT − Y Y T = 0

.

Using coordinates x1, . . . , xn−1, y1, . . . , yn−1 and t = xn + yn for H we see that the

quadric Q2n,n∩H is given by the equation x21 + · · ·+x2

n−1−y21−· · ·−y2

n−1 = 0. Let

H ′ be the real hyperplane given by the equation t = xn + yn = 0. The intersection

Q2n,n ∩H ∩H ′ is a quadric Q2n−2,n−1 and we have

Q2n,n ∩H = Q2n−2,n−1#p0.

Thus Q2n,n ∩ H ∼= Σ/ Q2n−2,n−1. Assume that the cycle map is an isomorphism

LpRα(Q2n−2,n−1) → Hα(Q2n−2,n−1; Z). From Remark 4.18 it follows that the

same holds for Σ/ Q2n−2,n−1.

It is also easy to see that, if π : P2n−1C −p0 → H ′ is the projection onto H ′ centered

at p0, then π|Q2n,n − Q2n,n ∩ H is an isomorphism onto H ′ − H ∩ H ′ ∼= A2n−2.

Since everything is real, this is a real isomorphism. It is easy to check that Q4,2∼=

P1C × P1

C with the standard real structure, hence, by Example 5.7 the cycle map is

an isomorphism in this case. Using induction on n, the sequence for the real pair

(Q2n,n,Q2n,n ∩H) and the five Lemma, it follows that the cycle map

LpRr,s(Q2n,n) −→ Hr,s(Q2n,n; Z)

is an isomorphism. This reduces the computation of real Lawson homology to the

computation of the equivariant homology of Q2n,n.

The next example is a very simple case — albeit somewhat artificial — in which

the cycle map is an isomorphism but the variety doesn’t have a real cell decompo-

sition.

Example 5.9. Let U ⊂ PnC be a quasi-projective variety for which the non-

equivariant s map Zp(U) → Ω2Zp−1(U) is a homotopy equivalence. Let UR be the

real variety obtained from U by restriction of scalars. Recall from Example 4.19

that UR = U∐

U with the anti-holomorphic involution x 7→ x. We will show that

s : Zp(UR) → Ω1,1Zp−1(UR) is an equivariant homotopy equivalence. From (4.5)

we have Zp(UR) ∼= F (Z/2+,Zp(U)). Hence there is a commutative diagram

Zp(UR)(R)∼=−−−−→ Zp(U)

s

y s

yΩ1,1Zp−1(UR)

Z/2 ∼=−−−−→ Ω2Zp−1(U)


where the left and right vertical arrows denote the equivariant and the non-equiva-

riant s maps, respectively. It follows that the equivariant s map is a Z/2-homotopy

equivalence.

Example 5.10. Consider the variety PnC × Pn

C with the real structure given by

τ · (X, Y ) = (Y ,X) (X, Y ) ∈ PnC × Pn

C.(5.4)

We will show that the s map is an equivariant homotopy equivalence. In the

case n = 0 there is nothing to prove. Assume the result holds for n− 1. We have

(5.5) PnC × Pn

C = An × An ∪An × Pn−1

C ∪ Pn−1C × An

∪ Pn−1

C × Pn−1C .

Note that An × An (with the action of (5.4)) is a real subvariety and it is actually

isomorphic to A2n with the standard real structure. The isomorphism is

(X, Y ) 7→ (X + Y,√−1(X − Y )).

Also the second factor in the decomposition (5.5) can be written as the disjoint

union U q τ · U where U = An × Pn−1C . By Example 5.9 we know that the s map

Zp(U q τ · U) → Ω1,1Zp−1(U q τ · U) is a Z/2-homotopy equivalence. Finally, by

induction, the cycle map is also an equivalence in the case of the last factor, Pn−1C ×

Pn−1C . By localization and the 5-lemma it follows that the map s : Zp(Pn

C × PnC) →

Ω1,1Zp−1(PnC × Pn

C) is an equivariant homotopy equivalence.

Example 5.11 (The real Severi-Brauer curve). Let X be the Severi-Brauer curve

PC(H) defined in Example 4.2. If one identifies PC(H) with the two sphere S2,

the involution is the antipodal map. In particular, there are no fixed points so X

cannot have a real cell decomposition. The cycle map is very simple in this case

because there are no cycles above dimension 1 and Z1(X) ∼= Z.

The s map, in this case, sends Z1(X) to Ω1,1Z0(X) and we know from [23]

it is a non-equivariant homotopy equivalence. A direct homology computation

shows that the induced map s∗ : π0Z1(X) → π1,1Z0(X) is an isomorphism. Also

π1+n,1Z0(X) = 0, for n > 0 — because PC(H) has no homology in dimensions (r, s)

with r + s > 2 — hence s is an equivariant homotopy equivalence.

Example 5.12 (Quadrics with signature 3). It is easy to check that PC(H) is

isomorphic as a real variety to the plane quadric Q3,0. From Example 5.8 it follows

that the quadric of signature 3, Q2n−1,n−1, is obtained from Q3,0 by adding real


cells Ak and taking suspensions. Using exact sequences, the 5-lemma and the results

of the previous examples, it follows that the cycle map

sp : Zp(Q2n−1,n−1) −→ Ωp,pZ0(Q2n−1,n−1)


Example 5.13 (Quadrics with signature 2). From Example 5.8 it follows that the

signature 2 quadric, Q2n+2,n, is obtained from Q4,1 by adding real cells Ak and

taking suspensions. One can check that Q4,1∼= P1

C × P1C with the real structure

of Example 5.10. It follows that the cycle map is also an equivariant homotopy

equivalence in this case.

Of course the cycle map is not an isomorphism in general, otherwise Lawson

homology would not be very interesting. Products of elliptic curves (and abelian

varieties in general) provide examples for which the non-equivariant cycle map is

not an isomorphism. The reason is the following. The homology classes in the image

of the cycle map for usual Lawson homology, sp : LpH2p(−) → Hsing2p (−; Z), have

Hodge type (p, p) and abelian varieties have homology classes which are not of this

type, hence sp is not surjective (see [12] and [20] for details). The same argument

can be adapted to produce examples of real varieties for which the (equivariant)

cycle map is not an isomorphism.

Example 5.14. Let X be the product of elliptic curves C/Λ×C/Λ where Λ is the

lattice Z ⊕ Z ·√−1. Complex conjugation in C descends to an anti-holomorphic

involution on C/Λ, giving it a real structure. As a topological space, X is Z/2-

homeomorphic to S1,0×S0,1×S1,0×S0,1. We will show that the cycle map is not

an isomorphism from L1R1,1(X) to H1,1(X; Z). Let α ∈ H1,1(S1,0 × S0,1; Z) ∼= Z,

be the generator and set β = i∗(α), where i embeds S1,0 × S0,1 in X as the

subspace S1,0 × 0 × 0 × S0,1. If β were in the image of the cycle map, then

its image under the forgetful functor to singular homology, F(β), would be in

the image of the cycle map for usual Lawson homology. But this is impossible

because, as mentioned above, all classes in the image of the composite L1R1,1(X) s−→

H1,1(X; Z) F−→ Hsing2 (X; Z) are of Hodge type (1, 1) and F(β) is not of Hodge type

(1, 1): let z, w be the complex coordinates for C2. Then dz ∧ dw is a closed (2, 0)-

form on X. We have∫F(β)

dz ∧ dw 6= 0 hence F(β) is not a (1, 1)-cycle.


6. Proof of Proposition 4.9

The existence an of exact sequence associated to a pair of real varieties (X, X ′)

is one of the basic properties of real Lawson homology. We could have defined

the relative Lawson homology groups as relative homotopy groups of the pair

(Zp(X),Zp(X ′)) thus yielding a long exact sequence in real Lawson homology.

Definition 4.8 has the advantage of being very geometric giving us a lot of control

over Zp(X, X ′). Its disadvantage is that it is not obvious that it yields the desired

long exact sequence. The purpose of this Section is to prove this fact.

Proof of Proposition 4.9. The Proposition is a consequence of a result of Lima-Filho

[24, Thm.5.2], which shows that under certain conditions, a pair of abelian topo-

logical monoids (C,C ′) gives rise to a fibration sequence C ′ → C → C/C ′, where

C and C ′ are the Grothendieck groups of C,C ′, respectively, with the quotient

topology. We will check that this result applies.

Let (X, X ′) be a pair of real algebraic varieties. Recall that

Cp(X) def=∐d≥0

Cp,d(X)

is a monoid under addition of cycles. It is endowed with the disjoint union topology;

the algebraic sets Cp,d(X) are equipped with their analytic topology. This monoid

is filtered by

Cp,≤d(X) def=∐k≤d

Cp,k(X).

The real structures on X and X ′ induce real structures on the Chow varieties

Cp,d(X), Cp,d(X ′). Set C = Cp(X)(R) and C ′ = Cp(X ′)(R). Note that the

Grothendieck groups of C, C ′ are Zp(X)(R) and Zp(X ′)(R), respectively. The

monoid C is free and C ′ is freely generated by a subset of generators of C. In the

language of [24, Def.5.1(b)], (C,C ′) is a free pair.

The monoid C is filtered by Cd = Cp,≤d(X)(R). This is a filtration by compact

subsets that satisfy Cd + Cd′ ⊂ Cd+d′ . In the language of [24, Def.2.4(ii)], C is c-

filtered. We endow C×C with the product filtration: (C×C)d =⋃

n+m≤d Cn×Cm.

The last condition we need to check is that with

id(C × C)ddef= ((C × C)d−1 + ∆ + C ′ × C ′) ∩ (C × C)d,

(where ∆ denotes the diagonal) the inclusion id(C×C)d ⊂ (C×C)d is a cofibration.

This will show that the pair (C,C ′) is properly c-filtered [24, Def.5.(a)].


Observe that Cd and C ′d = Cd ∩ C ′ are the real points of the Chow varieties

Cp,≤d(X) and Cp,≤d(X ′) so, in particular, (Cd, C′d) is a pair of algebraic sets. Since

the sum operation

Cp(X)× Cp(X) +−→ Cp(X)

is algebraic [7] it follows that ((C × C)d,id (C × C)d) is also a pair of algebraic

sets and hence can be triangulated [14]. We conclude that [24, Thm.5.2] ap-

plies and so C ′ → C → C/C ′ is a principal fibration. We have C = Zp(X)(R),

C ′ = Zp(X ′)(R) and it is easy to check that C/C ′ = Zp(X, X ′)(R) (the natural

map Zp(X)(R)/Zp(X ′)(R) → Zp(X, X ′)(R) is a homeomorphism). Thus the exact

sequence of topological groups

0 −→ Zp(X ′)(R) −→ Zp(X)(R) −→ Zp(X, X ′)(R) −→ 0

is a principal fibration. Since the same holds for the sequence Zp(X ′) → Zp(X) →

Zp(X, X ′) [23], and the maps are all equivariant, it follows that this sequence is

actually a Z/2-fibration sequence.

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Department of Mathematics, Texas A&M University, USA

Current address: Department of Mathematics, Instituto Superior Tecnico, Portugal

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