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
4 PEDRO F. DOS SANTOS
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).
ALGEBRAIC CYCLES ON REAL VARIETIES 5
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.
6 PEDRO F. DOS SANTOS
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
ALGEBRAIC CYCLES ON REAL VARIETIES 7
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
8 PEDRO F. DOS SANTOS
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.
ALGEBRAIC CYCLES ON REAL VARIETIES 9
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.
10 PEDRO F. DOS SANTOS
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)
ALGEBRAIC CYCLES ON REAL VARIETIES 11
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
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, 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);
12 PEDRO F. DOS SANTOS
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 )
ALGEBRAIC CYCLES ON REAL VARIETIES 13
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.
14 PEDRO F. DOS SANTOS
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
ALGEBRAIC CYCLES ON REAL VARIETIES 15
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
16 PEDRO F. DOS SANTOS
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.
ALGEBRAIC CYCLES ON REAL VARIETIES 17
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.
18 PEDRO F. DOS SANTOS
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∗.
ALGEBRAIC CYCLES ON REAL VARIETIES 19
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
20 PEDRO F. DOS SANTOS
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
ALGEBRAIC CYCLES ON REAL VARIETIES 21
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
22 PEDRO F. DOS SANTOS
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.
ALGEBRAIC CYCLES ON REAL VARIETIES 23
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.
24 PEDRO F. DOS SANTOS
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 ).
ALGEBRAIC CYCLES ON REAL VARIETIES 25
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).
26 PEDRO F. DOS SANTOS
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.
ALGEBRAIC CYCLES ON REAL VARIETIES 27
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
28 PEDRO F. DOS SANTOS
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
ALGEBRAIC CYCLES ON REAL VARIETIES 29
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.
30 PEDRO F. DOS SANTOS
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.
ALGEBRAIC CYCLES ON REAL VARIETIES 31
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)
is an equivariant homotopy equivalence.
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) //
32 PEDRO F. DOS SANTOS
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) −→
ALGEBRAIC CYCLES ON REAL VARIETIES 33
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
34 PEDRO F. DOS SANTOS
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)
ALGEBRAIC CYCLES ON REAL VARIETIES 35
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
36 PEDRO F. DOS SANTOS
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)
is an equivariant homotopy equivalence.
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.
ALGEBRAIC CYCLES ON REAL VARIETIES 37
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)].
38 PEDRO F. DOS SANTOS
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