arXiv:0909.0569v1 [math.AG] 3 Sep 2009

VANISHING THEOREMS FOR REAL ALGEBRAIC CYCLESJEREMIAH HELLER AND MIRCEA VOINEAGU
Abstract. We establish the analogue of the Friedlander-Mazur conjecture
for Teh’s reduced Lawson homology groups of real varieties, which says thatthe reduced Lawson homology of a real quasi-projective variety X vanishes
in homological degrees larger than the dimension of X in all weights. As anapplication we obtain a vanishing of homotopy groups of the mod-2 topologicalgroups of averaged cycles and a characterization in a range of indices of the
motivic cohomology of a real variety as homotopy groups of the complex of
averaged equidimensional cycles. We also establish an equivariant Poincare
duality between equivariant Friedlander-Walker real morphic cohomology anddos Santos’ real Lawson homology. We use this together with an equivariant
extension of the mod-2 Beilinson-Lichtenbaum conjecture to compute somereal Lawson homology groups in terms of Bredon cohomology.

Contents
1. Introduction
2. Equivariant Homotopy and Cohomology
3. Topological Spaces of Cycles
4. Poincare Duality
5. Compatibility of Cycle Maps
6. Vanishing Theorem
7. Reduced Topological Cocycles
Appendix A. Topological Monoids
Appendix B. Tractable Monoid Actions
References

1
5
7
15
23
31
37
44
46
49

1. Introduction
Let X be a quasi-projective real variety. The Galois group G = Gal(C/R) acts
on Zq (XC ), the topological group of q-cycles on the complexification. Cycles on thereal variety X correspond to cycles on XC which are fixed by conjugation. Inside
the topological group of Zq (XC )G of cycles fixed by conjugation is the topologicalgroup Zq (XC )av of averaged cycles which are the cycles of the form α + α. The
space of reduced cycles on X is the quotient topological group
Rq (X) =Zq (XC )G.
Zq (XC )av

Homotopy groups of some of the above abelian topological groups are related
to classical topological invariants. For example for X a projective real variety
2000 Mathematics Subject Classification. 19E20, 19E15, 14F43.
1

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JEREMIAH HELLER AND MIRCEA VOINEAGU

we obtain the singular homology groups π∗ R0 (X) = H∗ (X(R), Z/2) [Teh05] andπ∗ Z0 (XC )av = H∗ (XC (C)/G, Z) [LLFM03], as well as Bredon homology π∗ Z0 (X) =H∗,0 (XC (C); Z) [LF97]. Other homotopy groups are related to classical algebraic
geometry invariants. For example π0 (Zr (XC )G ) computes the group of algebraic
cycles of dimension r on X modulo real algebraic equivalence [FW02a] and consequently with Z/n coefficients equals the Chow group CHr (X)⊗Z/n (see Proposition
5.1). However most of these homotopy groups remain a mysterious combination of
topological and algebraic information of the real variety X.
These homotopy groups are hard to compute and examples are few. Nonethelessan examination of existing computations shows that these homotopy groups are all
zero in large degrees. For example, in ([Lam90]) Lam proves that
Rq (PnR ) ≃n−q
YK(Z/2, i).

i=0In particular πk (Rq (PnR )) = 0 for k > n − q. Similar vanishing results are seenin the computations of [LLFM05] for a real variety X with the property that its
complexification is the quaternionic projective space (see Example 6.14).
In [Teh08] Teh proves a conditional Harnack-Thom type theorem for the homotopy groups of reduced algebraic cycles on X which holds under the assumptionthat these homotopy groups are all finitely generated and they are zero in highdegrees. In the case of divisors he shows that
πk Rd−1 (X) = 0
when k ≥ 3 for any smooth projective real variety X of dimension d.
The main theorem of this paper provides this vanishing in general and should beviewed as a massive generalization of both the classical vanishing of singular homology groups of a manifold in degree larger than the manifold and of the vanishing
results discussed above.
Theorem 1.1. Let X be a quasi-projective real variety. Then
πk Rq (X) = 0
for k ≥ dim X − q + 1.
In the case of divisors our result improves the previously known vanishing range.
The case of real projective space described above shows that the theorem’s vanishingrange is optimal.
The homotopy groups of reduced algebraic cycles Rq (X) define a homology theory for real quasi-projective varieties X introduced in [Teh05] which is defined by
RLq Hn (X) = πn−q (Rq (X)) for n ≥ q and called reduced Lawson homology. In thisnotation our vanishing result reads RLq Hn (X) = 0 for any n > dim(X). Thus ourvanishing result shows that the Friedlander-Mazur conjecture holds for the reduced
Lawson homology of real varieties.
The homotopy groups of Rq (X) fit into a long exact sequence· · · → πk+1 Rq (X) → πk Zq (XC )av → Lq HRq−k,q (X) → πk Rq (X) → · · ·where Lq HRq−k,q (X) = πk Zq (XC )G is the real Lawson homology introduced by
dos Santos in [dS03a]. As a consequence of Suslin rigidity the homotopy groups ofRq (X) are also related to motivic cohomology of X
2q−k
· · · → πk zequi (AqC , 0)(XC × ∆•C )av → HM
(X; Z(q)) → πk Rq (X) → · · · .

VANISHING THEOREMS FOR REAL ALGEBRAIC CYCLES

3

Thus an immediate corollary of the vanishing theorem is an identification of thehomotopy groups of the space of averaged cycles and of the complex of averaged
equidimensional cycles.
Corollary 1.2. Let X be a smooth quasi-projective real variety. Then for anyk ≥ dim X − q + 1
Lq HRq−k,q (X) = πk Zq (XC )av
and
2q−k
HM
(X; Z(q)) = πk zequi (AqC , 0)(XC × ∆•C )av .

Theorem 1.1 also implies that the mod-2 homotopy groups of the topologicalgroup of average cycles satisfy an optimal vanishing (see also Example 6.19).
Corollary 1.3. Let X be a smooth projective real variety of dimension d. Thenπn

Zp (XC )av=0
2Zp (XC )av

for n ≥ 2d − 2p + 1.
An essential ingredient in the proof of our vanishing theorem is the Milnor conjecture proved by Voevodsky in [Voe03]. The Milnor conjecture relates motivic
cohomology and etale cohomology while real morphic cohomology naturally compares with Bredon cohomology. We need to know that these cycle maps are suitably
related which is done in Theorem 5.9,
Theorem 1.4. Let X be a smooth quasi-projective real variety. The diagram commutes∼=

Lq HRq−k,q (X; Z/2) o

H2q−k,q(X; Z/2)
M
cyc

Φ



H q−k,q (XC (C); Z/2)

/ H 2q−k (XC (C); Z/2)
G


/ H 2q−k (X; µ⊗q ),et2pwhere H p−q,q (XC (C); Z/2) denotes Bredon cohomology and HG(XC (C); Z/2) denotes Borel cohomology.

This suggests that there are possible advantages in replacing the map on Chowgroups of real cycles into Borel cohomology with the map into Bredon cohomology
since in many respects Bredon cohomology behaves better than Borel cohomology.
An application of this idea will be given in a forthcoming paper.
Together with the mod-2 Beilinson-Lichtenbaum conjecture for real and complex
varieties (which is a consequence of the Milnor conjecture by [SV00a]) we conclude
an equivariant Beilinson-Lichtenbaum type theorem for an equivariant extension ofFriedlander-Walker’s real morphic cohomology groups (see Definition 3.13).
Theorem 1.5. Let X be a smooth quasi-projective real variety and k > 0. The
cycle map
Φ : Lq HRr,s (X; Z/2k ) → H r,s (XC (C); Z/2k )is an isomorphism if r ≤ 0 (and s ≤ q) and an injection if r = 1 (and s ≤ q).

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JEREMIAH HELLER AND MIRCEA VOINEAGU

Using Friedlander-Voevodsky duality for bivariant cycle theory we show in Corollary 4.20 that the equivariant morphic cohomology and real Lawson homology
groups are isomorphic through a Poincare duality. As a consequence the equivariant Beilinson-Lichtenbaum says that in a range we may compute the mod-2
real Lawson homology groups in terms of mod-2 Bredon cohomology. This allows
a computation for curves with integral coefficients.
Corollary 1.6. Let X be a smooth real curve. Then
Lq HRr,s (X; Z) → H r,s (XC (C); Z)
is an isomorphism for any q ≥ 0, r ≤ q, and s ≤ q.
The space of reduced cocycles on X, related via Poincare duality with the spaceof reduced cycles, is defined as
Rq (X) =Z q (XC )G
Z q (XC )av

where Z q (XC ) is the space of algebraic cocycles on XC and Z q (XC )G agrees with thespace of real cocycles introduced by Friedlander-Walker in [FW02a] (see Proposition
3.7). There is a natural comparison map
Rq (X) → Map(X(R), R0 (Aq ))and since R0 (Aq ) = K(Z/2, q) this provides a natural map
(1.7)

q−k
cyck : πk Rq (X) → Hsing(X(R); Z/2)

which is the cycle map for reduced morphic cohomology groups defined in [Teh05].
Via Poincare duality the vanishing theorem is equivalent to the statement that cyck
is an isomorphism for k > q.
Via the Milnor conjecture over C and over R we can deduce an isomorphism
πk Rq (X) → πk Rqtop (X) for k ≥ q. Here Rqtop (X) is the group of “reduced topological cocycles”. For a precise definition see Section 7, but essentially this is a versionof the quotient group Map(XC (C), Z0 (AqC ))G /Map(XC (C), Z0 (AqC ))av which has
reasonable homotopical properties (such as fitting into a homotopy fiber sequence
involving Map(XC (C), Z0 (AqC ))G and Map(XC (C), Z0 (AqC ))av ).
The final ingredient for our vanishing theorem is now provided by Corollary 7.14
which shows that for X projective,e : πk Rq (X) → πk Rqtop (X)Φq−kagrees with the cycle map πk Rq (X) → Hsing(X(R); Z/2) for k ≥ 2.
Here is a short outline of the paper. In the second section we review the equivariant homotopy used in the paper. The third section is dedicated to introducing
the topological spaces of cycles we study and proving some basic properties that
we use and for which we don’t find exact references in the literature. In the fourth
section we prove a Poincare Duality between equivariant morphic cohomology andreal Lawson homology. In the fifth section we discuss the cycle maps from equivariant morphic cohomology and Bredon cohomology and equivariant applications of
the Beilinson-Lichtenbaum conjecture. The sixth section is devoted to the proof ofour main vanishing Theorem. One of the main technical ingredients of this proof isleft for section seven where we reinterpret the cycle map 1.7 from reduced Lawson

VANISHING THEOREMS FOR REAL ALGEBRAIC CYCLES

5homology groups to the singular homology in a manner needed to prove our vanishing theorem. The paper ends with two appendixes where we prove and recollecta few results on topological monoids used in the paper.
The authors would like to thank Eric Friedlander, Christian Haesemeyer andMark Walker for helpful discussions.
Notation: By a quasi-projective k-variety we mean a reduced and separated
quasi-projective scheme of finite type over a field k. We write Sch/k for the category of quasi-projective k-varieties and Sm/k for the subcategory of smooth quasiprojective k-varieties. Except in section 2, G always denotes Gal(C/R) and σ ∈ Gdenotes the nontrivial element.

2. Equivariant Homotopy and Cohomology
We recall the basic definitions and theorems we need from equivariant homotopy
theory. For more details see [May96]. In this paper we will only work with G =Z/2, but since no simplification results in the basic definitions, we let G denotean arbitrary finite group. The category T opG of G-spaces consists of compactly
generated spaces equipped with a left G-action and morphisms are continuous Gequivariant maps. If X is a G-space and H ⊆ G is a subgroup write X H for thesubspace of all points fixed by H. The category T opG∗ of based G-spaces consists of
G-spaces X together with a G-invariant basepoint x ∈ X and maps are base-point
preserving equivariant maps. A space together with a disjoint, invariant base-point
will be denoted X+ .

Equivariant homotopy theory. Let I denote the unit interval with trivial Gaction. A G-homotopy between two equivariant maps f, g : X → Y is an equivariant
map F : X × I → Y such that F |X×{0} = f and F |X×{1} = g. An equivariant map
f : X → Y is an equivariant homotopy equivalence provided there is an equivariant
map g : Y → X such that both f ◦ g and g ◦ f are G-equivalently homotopic to theidentity. An equivariant map f : X → Y is a G-weak equivalence provided bothf H : X H → Y H is a non-equivariant weak equivalence for all subgroups H ⊆ G.
Formally inverting the G-weak equivalences gives the homotopy category of Gspaces. Similarly inverting the based G-weak equivalences between based G-spaces
we obtain the based G-homotopy category. Write [X, Y ]G for classes of based maps
in the homotopy category of based maps.
A G-CW complex X is a topological union X = ∪Xn of G-spaces such that X0
is a disjoint union of orbits G/H and Xn is obtained from Xn−1 by attaching cellsof the form Dn × G/H via attaching maps σ : S n−1 × G/H → Xn−1 .
The equivariant Whitehead theorem holds for G-CW complexes. That is, if
f : X → Y is a G-equivariant weak equivalence between G − CW -complexes thenf is a G-homotopy equivalence.
A map A → X is said to have the homotopy extension` property with respect toZ if for any equivariant partial homotopy H : X × {0} A×{0} A × I → Z there is

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JEREMIAH HELLER AND MIRCEA VOINEAGU

an equivariant map H ′ making the diagram below commute
X × {0}`

A×{0} A × I
∃H ′n

H

/
n7 Z
n
n

n

 n n
X ×I

.

An equivariant cofibration A ֒→ X is an equivariant map which has the homotopy extension property with respect to all Z in T opG. Inclusions of sub-GCW complexes A ⊆ X are equivariant cofibrations.
Let V be a real representation of G, write S V for the one-point compactificationof V . The V th homotopy group of a based G-space X isπV X = [S V , X]G .
Note that S V always has at least two fixed points, 0 and ∞.
When G = Z/2 and V = Rp,q , where V = Rp+q with G acting trivially on thefirst p-components and on the last q-components the G action is multiplication by−1 we use the notation
πp,q X = πRp,q X.
Borel homology and cohomology. The Borel-equivariant cohomology of X
with coefficients in an abelian group A is defined to be the ordinary singular cohomology of the homotopy orbit space of X:p
HG(X; A) = H p ((X × EG)/G; A).

Similarly the Borel-equivariant homology is defined to be
HpG (X; A) = Hp ((X × EG)/G; A).
When X has free G-action then (X × EG)/G → X/G is a homotopy equivp
alence and therefore when X has free G-action H p (X/G; A) ∼= HG (X; A) andG
Hp (X/G; A) ∼= Hp (X; A).
Mackey functors. Bredon homology and cohomology take Mackey functors as
coefficients. There are several equivalent ways to define a Mackey functor [May96].
Classically for G a finite group one defines a Mackey functor as follows . Let
FG denote the category of finite G-sets as objects and with equivariant set maps
as morphisms. A Mackey functor M consists of a pair of abelian-group valued
functors M = (M ∗ , M∗ ) on FG , with M ∗ contravariant and M∗ covariant. The
functors M ∗ and M∗ satisfy the following requirements.
(1) M ∗ ,M∗ take the same value on objects and convert disjoint unions of G-sets
into products of abelian groups.
(2) When
σ′

S

/T

β′


U

β
σ


/V

VANISHING THEOREMS FOR REAL ALGEBRAIC CYCLES

7

is a pull-back square of finite G-sets then
M (S)O
M ∗ (β ′ )M (U )M∗ (σ′ )/ M (T )O

M ∗ (β)M∗ (σ)/ M (V )is a commutative square of abelian groups.
Given an abelian group A, the constant Mackey functor A is the Mackey functor
which on objects A(G/K) = A and on a map f : G/H → G/K, M ∗ (f ) = id andM∗ (f ) is multiplication by the index [K : H].
Bredon homology and cohomology. Bredon cohomology (homology) with coefficients in a Mackey functor M is a cohomology (homology) theory H ∗ (−; M ))(H∗ (−; M ) graded by RO(G). For V ∈ RO(G) there is an equivariant EilenbergMaclane space K(M , V ) which represents the reduced cohomology H̃ V (X; M ) fora based G-space,
H̃ V (X; M ) = [X, K(M , V )]G .
When G = Z/2 then RO(G) = Z ⊕ Z with generators R1,0 and R0,1 . We use thep,qconvention that H p,q (X; M ) = H R (X; M ) (and similarly for homology).
p
If A is an abelian group (with trivial G-action) then H p,0 (X; A) = Hsing(X/G; A).
More generally, Borel and Bredon cohomology are related by the natural isomorphismp+q
H p,q (X × EG; A) ∼= HG (X; A(q))
qwhere A(q) is A with σ acting by (−1) (see [dSLF04, Proposition 1.15]).
Equivariant Dold-Thom theorem. Let X be a compactly generated Hausdorff
space.
The free abelian group on the points of X is defined to be Z0 (X) =`[ d SP d (X)]+ , where SP d (X) is the dth symmetric product on X and (−)+ denotes group completion of the displayed monoid which is topologized via the quotient topology.
PP
The degree homomorphism deg : Z0 (X) → Z is defined by deg( ni xi ) =ni
is a continuous homomorphism. Write Z0 (X)0 for the kernel of this map. Noticethat there is an isomorphism of topological groups Z0 (X) ∼= Z0 (X+ )0 .
If X is a G-space then the action on X induces a G-action on Z0 (X) and Z0 (X)0 .
By [LF97, Corollary 2.9] when X is a G-CW complex so is Z0 (X).
The classical Dold-Thom theorem says that πn Z0 (X)0 = H̃n (X; Z) and the equivariant Dold-Thom theorem proved by Lima-Filho [LF97] and dos Santos [dS03b]
says that
πV Z0 (X)0 = H̃V (X; Z).
In particular Z0 (S V )0 is an Eilenberg-Maclane space K(Z, V ).
3. Topological Spaces of Cycles
Group completions of monoids. Let M be a compactly generated Hausdorff
topological abelian monoid. The naive group completion of M is the quotient of
M × M by the monoid action of M where M acts by (a, b) 7→ (m + a, m + b).
Write M + for this abelian group, which is topologized as the quotient of M × M .
Recall that M is said to satisfy the cancellation property if a + m = b + m implies
that a = b for any a, b, m ∈ M . When M satisfies cancellation then the naive

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JEREMIAH HELLER AND MIRCEA VOINEAGU

group completion can be described as M + = M × M/ ∼, where (a, b) ∼ (c, d) if
a + d = b + c.
Naive group completion does not generally behave well topologically. For example it may happen that M + is not a Hausdorff topological group nor is it clear
how homotopy invariants of M and M + are related. Friedlander-Gabber [FG93]
and Lima-Filho [LF93] have studied conditions under which the naive group completion of a topological monoid is a Hausdorff group and M → M + is a homotopy
group completion. All of the topological monoids with which we work are tractablemonoids in the sense of Friedlander-Gabber (see Appendix B) and in particular thenaive group completion of these groups are homotopy group completions.
Our main objects of interest are the group completions of submonoids of the
Chow monoids of effective algebraic cycles on algebraic varieties. Let k be a fieldnof characteristic` 0 and j : Y ⊆ Pk be a projective k-variety. The Chow variety
Cq (Y, j) = d≥0 Cq,d (Y, j) of effective q-dimensional cycles on Y is an (infinite,
disjoint) union of projective k-varieties. See [Fri91] for details.
Cycle-spaces over C. Let Z be a complex variety. Denote the set of complex
points equipped with the analytic topology by Z(C)an . Since there will be little
chance for confusion we will often simply write this space as Z(C) with the topology
understood. If j : Y ⊆ PnC is a projective variety then Cq (Y, j)(C) is a topologicalmonoid and we will generally omit j from the notation since the homeomorphism
type of this space is independent of j.
The monoid Cq (Y )(C)an is tractable and therefore the naive group completionis a homotopy group completion. Write
Zq (Y ) = (Cq (Y )(C))+
for the naive group group completion of this monoid. Define the filtration {0} ⊆· · · ⊆ Zq,≤d (Y ) ⊆ Zq,≤d+1 (Y ) ⊆ · · · ⊆ Zq (Y ) byaCq,d1 (Y )(C) × Cq,d2 (Y )(C) / ∼⊆ Zq (Y ).
Zq,≤d (Y ) = 
d1 +d2 ≤dBy [LF93] each Zq,≤d (Y ) is a closed, compact Hausdorff space and Zq (Y ) has the
weak topology with respect to this filtration.
When U is quasi-projective with projectivization U ⊆ U then define Zq (U ) =Zq (U )/Zq (U∞ ) where U∞ = U\U . The images of Zq,≤k (U ) in the quotient Zq (U )give a filtration by compact subspaces (and Zq (U ) has the weak topology withrespect to this filtration). This definition is independent of choice of projectivization[LF92], [FG93].
Cycle-spaces over R. Suppose that Z is a real variety. Write the set of R-points
equipped with the analytic topology as Z(R)an or simply Z(R) with the topology
understood.
Let Y be a projective real variety. Consider the topological monoid Cq (Y )(R).
As explained in the proof of [FW02a, Proposition 8.2] (see Proposition B.5), thetopological monoid Cq (Y )(R) is tractable and therefore its naive group completionis a homotopy group completion. Write
Zq (Y ) = (Cq (Y )(R))+
for the naive group completion.

VANISHING THEOREMS FOR REAL ALGEBRAIC CYCLES

9Suppose that U is a quasi-projective real variety with projectivization U ⊆ U.
Define the topological group of q-cycles on the quasi-projective real variety U to be
Zr (U ) = Zr (U )/Zr (U \U ).
If X is a real variety and π : XC → X is its complexification then G acts on
XC (C) and induces a homeomorphism∼=

X(R) −→ XC (C)G .
In particular if X is a projective real variety then by [Fri91, Proposition 1.1]
Cq (XC ) = Cr (X)C and so we have the isomorphism of topological monoids∼=

Cr (X)(R) −→ Cr (XC )(C)G .
Proposition 3.1. Let U be a quasiprojective real variety. Then∼=

Zr (U ) −→ Zr (UC )G
is an isomorphism of topological abelian groups. In particular the group Zr (U ) is
independent of projectivization U ⊆ U .
Proof. For U projective this follows immediately from Proposition A.3. The quasiprojective case now follows by a comparison of short exact sequences of topologicalabelian groups0/ Zr (U \U )/ Zr (U )/ Zr (U )/00/ Zr (U C \UC )G/ Zr (U C )G
/ Zr (UC )G/ 0,

where the exactness of the bottom row is a consequence of the Lemma 3.3.



Remark 3.2. Let U be a quasi-projective real variety. Since + : Zk (UC )×Zk (UC ) →Zk (UC ) is closed we see by taking G-fixed points that + : Zk (U ) × Zk (U ) → Zk (U )is a closed map for any real variety U .
Lemma 3.3. Let Y ⊆ X be a closed subvariety of a real projective variety. Then∼=Zr (XC )G /Zr (YC )G −→ (Zr (XC )/Zr (YC ))G
and

∼
=

Zr (XC )av /Zr (YC )av −→ (Zr (XC )/Zr (YC ))av
are isomorphisms of topological groups.
Proof. Consider the quotient maps π : Zr (XC ) → Zr (XC )/Zr (YC ) and q : Zr (XC )G →Zr (XC )G /Zr (YC )G . Consider the filtration {(πZr (XC )≤d })G of (Zr (XC )/Zr (YC ))GG
G
and the filtration {q(Zr (XC )G≤d )} of Zr (XC ) /Zr (YC ) . These spaces have theweak topology given by these filtrations so it is enough to see that
→ (πZr (XC )≤d )G
q(Zr (XC )G≤d ) −
is a homeomorphism for all d.
First we show that Zr (XC )G(X ) )G is surjective. If [η] ∈ (πZr (XC )≤d )G ,≤d → (πZ
Pr C ≤dwe can choose a representative η =nV V ∈ Zr (XC )≤d such that each V * YC .
Since η − η ∈ Zr (YC ) (and each V * YC ) we see that η = η and therefore the map
Zr (XC )G→ (πZr (XC )≤d )G
≤d −

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JEREMIAH HELLER AND MIRCEA VOINEAGU

is surjective. The map
qZr (XC )G→ (πZr (XC )≤d )G≤d −is easily seen to be injective and since Zr (XC )G≤d is compact this map is closed so
is a homeomorphism.
The second statement about the topological group of averaged cycles is provedin a similar fashion.
Spaces of algebraic cocycles. In this section we recall the construction of topological monoids of algebraic cocycles [FL92, Fri98]. Let X, Y be quasi-projective
real varieties over k = C or R. Write Mork (X, Y ) for the set of continuous algebraic
maps between X and Y . When X is semi-normal then Mork (X, Y ) = Hom (X, Y ).
Friedlander-Walker construct “analytic” topologies on Mork (X, Y ) in [FW01b] fork = C and in [FW02a] for k = R. The set of continuous algebraic maps with this
topology will be written Mork (X, Y )an . By [FW02a, Lemma 1.2] MorR (X, Y )an =(MorC (XC , YC )an )G .
When X,Y are projective real varieties and W ,Z are projective complex varieties
then this topology coincides with the subspace topology induced by the inclusions
MorC (W, Z) ⊆ Map(W (C), Z(C))
andMorR (X, Y ) ⊆ Map(XC (C), YC (C))G ,where Map(−, −) denotes the space of continuous maps is with compact-open topology. When the domain is only quasi-projective then the analytic topology on thealgebraic mapping spaces is no longer the compact-open topology but rather the
topology of convergence with bounded degree (see [FL97, Appendix A]).
Let W be a quasi-projective complex variety and Z be a projective complex
variety. Write d = dim W . Let Cr (Z)(W ) denote the monoid of effective cycles
on W × Z equidimensional or relative dimension r on W . This is made into a
topological monoid via the subspace topology induced by the inclusiondef

Cr (Z)(W ) ⊆ Cd+r (W × Z) =Cd+r (W × Z)
Cd+r (W∞ × Z)

where W ⊆ W a projective closure with closed complement W∞ = W \W . This
topology may also be described as follows. Let
Er (Z)(W ) ⊆ Cr+d (W × Z)
denote the constructable submonoid consisting of effective cycles whose restriction
to W × Z is equidimensional of relative dimension r over W . By [Fri98, Proposition
1.8] the topology on Cr (Z)(W ) (given by the subspace topology above) coincides
with the quotient topology given by
Cr (Z)(W ) =Er (Z)(W ).
Cr+d (W∞ × Z)

Define the topological group of equidimensional cycles of relative dimension rover W as Zr (Z)(W ) = [Cr (Z)(W )]+ (where as usual the naive group completionis given the quotient topology). Since Cr (Z)(W ) is a tractable monoid the naive
group completion is a homotopy group completion.
In [Fri91] it is shown that a morphism of varieties f : W → Cr (Z) has an
associated graph in Zf ∈ Cr (Z)(W ). By [FL97, Proposition A.1] this defines an

VANISHING THEOREMS FOR REAL ALGEBRAIC CYCLES

11

isomorphism of topological monoids Γ : MorC (W, Cr (Z)) → Cr (Z)(W ) for any
normal, quasi-projective complex variety W by [FL97, Proposition A.1]. Therefore
the graph map Γ also induces an isomorphism of topological abelian groupsΓ : MorC (W, C0 (Z))+ → Z0 (W )(Z),
for any normal, quasi-projective variety W and any projective variety Z. The
composite of Γ and the continuous inclusion Z0 (W )(Z) ⊆ Zdim W (W × Z) defines
the duality map
Γ→ Z0 (W )(Z) ⊆ Zd (W × Z).
D : MorC (W, C0 (Z))+ −While this is a continuous injective homomorphism it is not a topological embedding(see [FL97]).
Lemma 3.4. (c.f. [Teh05, Proposition 2.9])(1) If W is a normal quasi-projective complex variety and Z1 ⊂ Z2 is a closed
subvariety of a complex projective variety then
MorC (W, C0 (Z1 ))+ ⊆ MorC (W, C0 (Z2 ))+is a closed subspace.
(2) If U is a normal quasi-projective real variety and Y ⊆ Z is a closed subvariety of a projective real variety then
MorR (U, C0 (Y ))+ ⊆ MorR (U, C0 (Z))+
is a closed subspace.
Proof. For the first statement it is equivalent to show that Zr (Z1 )(W ) ⊆ Zr (Z2 )(W )is closed. Using Lemma A.1 we see that Cr+d (W × Z1 ) ⊂ Cr+d (W × Z2 ) is closed
and since Cr (Z1 )(W ) = Cr+d (W × Z1 ) ∩ Cr (Z2 )(W ) we conclude that Cr (Z1 )(W ) ⊂Cr (Z2 )(W ) is a closed subspace. Write π : Cr (Z2 )(W )×2 → Zr (Z2 )(W ) for thequotient. Then π −1 Zr (Z1 )(W ) = Cr (Z1 )(W )×2 + ∆(Cr (Z2 )(W )) (where ∆ denotes
the diagonal) is closed by step (1) in Proposition 3.6. Therefore Zr (Z1 )(W ) ⊆Zr (Z2 )(W ) is closed.
The second statement follows immediately from the first statement together with
Proposition A.3 and [FW02a, Lemma 1.2].
Definition 3.5.
(1) Let W be a quasi-projective complex variety. The spaceof algebraic q-cocyles is defined to be
Z q (W ) =MorC (W, C0 (PqC ))+q−1 +MorC (W, C0 (PC))(2) Let U be a quasi-projective real variety. The space of real algebraic qcocyles is defined to be
Z q (U ) =MorR (U, C0 (PqR ))+q−1 +MorR (U, C0 (PR))

Proposition 3.6. Let U be a normal quasi-projective real variety then
!G
(MorC (UC , C0 (PqC ))an,+ )G ∼MorC (UC , C0 (PqC ))an,+=→
q−1 an,+ G −q−1 an,+(MorC (UC , C0 (PC)))
MorC (UC , C0 (PC))

12JEREMIAH HELLER AND MIRCEA VOINEAGU

and(MorC (UC , C0 (PqC ))an,+ )av∼=→
q−1 an,+ av −(MorC (UC , C0 (PC)))
are isomorphisms of topological groups.

MorC (UC , C0 (PqC ))an,+
q−1 an,+MorC (UC , C0 (PC))!av

Proof. By Lemma A.2 and Proposition A.3 it is enough to show that for Y ′ ⊆ Ya closed subvariety of a projective real variety and U a quasiprojective real variety
that
G

Cr (YC )(UC )GCr (YC )(UC )−→Cr (YC′ )(UC )GCr (YC′ )(UC )is an isomorphism of topological monoids.
We proceed in several steps.
(1) The map+ : Cr (YC )(UC ) × Cr (YC )(UC ) → Cr (YC )(UC )is a proper map. Observe that if α + β is equidimensional then both α andβ are equidimensional and therefore
Cr (YC )(UC ) × Cr (YC )(UC )
Cr+k (UC × YC ) × Cr+k (UC × YC )++/ Cr (YC )(UC )/ Cr+k (UC × YC )is a pull-back square. Since addition is a proper map on effective cycles we
see that it is a proper map for effective cocycles as well.
(2) The map
G
Cr (YC )(UC )GCr (YC )(UC )−→Cr (YC′ )(UC )GCr (YC′ )(UC )is easily seen to be a continuous bijection by an argument similar to theone used in Lemma 3.3.
(3) Finally since Cr (YC )(UC ) → Cr (YC )(UC )/Cr (YC′ )(UC ) is a closed map by
Lemma A.1 we conclude that the continuous bijectionG
Cr (YC )(UC )GCr (YC )(UC )−→Cr (YC′ )(UC )GCr (YC′ )(UC )is a closed map and therefore a topological isomorphism.
The second statement for average cocycles is proved in a similar fashion, using
Proposition 3.10 and that Cr (YC )(UC )av ⊆ Cr (YC )(UC ) is closed.

As with the topological group of cycles on a real variety X we may view the
topological group of real cocycles as the topological group of cycles on the complexification which are fixed by the Galois action.
Proposition 3.7. Let X be a normal quasi-projective real variety. ThenZ q (X) = Z q (XC )G .

Proof. This follows from Proposition A.3 together with the previous proposition
since MorR (X, C0 (PqR )) = MorC (XC , C0 (PqC ))G .
VANISHING THEOREMS FOR REAL ALGEBRAIC CYCLES

13

Remark 3.8. The space Z q (XC ) has the equivariant homotopy type of a G-CW complex (see Corollary B.6).
When W = XC is the complexification of a quasi-projective real variety and
Z = YC is the complexification of a projective real variety the graph map is anequivariant morphism. In particular the duality map
D : Z q (XC ) → Zd (XC × AqC )is an equivariant continuous map.
Definition 3.9.
(1) Let X be a projective real variety. Define the topologicalgroup of averaged cocycles to be
Z q (XC )av = {f + σ · f |f ∈ Z q (XC )} ⊆ Z q (XC ).
(2) Let X be a normal projective real variety. Define the topological group of
reduced cocycles to be the quotient topological group
Rq (X) =Z q (XC )G.
Z q (XC )av

Lemma 3.10 shows that Rq (X) is a Hausdorff topological group.
In [Teh05] Teh defines
MorC (XC , C0 (YC ))+,G
MorC (XC , C0 (YC ))+,av
and defines the reduced cocycles are defined as
R0 (Pq )(X)
Rq (X) =R0 (Pq−1 )(X)
for any real normal projective variety X and real projective variety Y.
By Proposition 3.6 this definition and the one above give isomorphic topologicalgroups.
R0 (Y )(X) =Lemma 3.10. (c.f. [Teh05, Proposition 2.4]) Let X be a real projective variety.
The subset of averaged cocycles Z q (XC )av ⊆ Z q (XC ) is a closed subgroup.
Proof. Write f for σ · f and V for σ · V .
Suppose that {[fn ]+[fn ]} is a sequence in Z q (XC )av which converges in Z q (XC ).
Write [γ] = limn→∞ [fn ] + [fn ] for its limit. We need to conclude that [γ] is anaveraged cocycle.
The set {[fn ] + [fn ]} ∪ {[γ]} ⊆ Z q (XC ) is compact. Applying the duality map tothis set yields the compact subset{Γ([fn ]) + Γ([fn ])} ∪ {Γ([γ])} ⊆ Zd (XC × AqC ).
Since this is a compact subset it lies in Zd,≤k (XC × AqC ) for some k.
The sequence {[gn ]} ⊆ Zd,≤k (XC × AqC ) has a convergent subsequence. Write{[gni ]} for this convergent subsequence and write limni →∞ [gni ] = [g] ∈ Zd,≤k (XC ×
AqC ) for its limit. Note that [g] satisfies [g] + [g] = Γ(γ). Since Γ is injective andits image consists precisely of equidimensional cycles, we are done if we can find an
equidimensional cycle [g ′ ] such that [g ′ ] + [g ′ ] = [g] + [g].
Choosea representative γ ∈ MorC (XC , C0 (PqC ))+ of [γ]. Choose a representativePg=nV V ∈ Zd (XC × PqC ) of [g] such that if nV 6= 0 then V * Pq−1 × X. SinceP[g] + [g] = Γ([γ]) ∈ Zd (XC × AqC ) we see that g + g = (nV + nV )V = Γ(γ) + h

14

JEREMIAH HELLER AND MIRCEA VOINEAGU

q−1
where h ∈ Zd (XC × PC). Write h = ΣmW W . Since V * Pq−1 whenever nV 6= 0
we see that if mW 6= 0 then a term of −mW W must appear in Γ(γ). In particular
h is equidimensional. Consequently g + g is equidimensional.
If nV + nV 6= 0 then V is equidimensional. Define
X
g′ =
nV V.
nV +nV 6=0

Since g is an equidimensional cycle there is an f ∈ (MorC (XC , C0 (PqC ))+ )G suchthat Γ(f ) = g ′ . Since Γ([f ] + [f ]) = [g ′ ] + [g ′ ] = [g] + [g] = Γ([c]) and Γ is injective,
we conclude that [c] = [f ] + [f ].
′A continuous algebraic map f : W → V between two complex varieties induces
a continuous map f : W (C) → V (C). Friedlander-Lawson [FL92, Proposition 4.1]
show that this defines a continuous map(3.11)

Φ : Z r (W ) → Map(W (C), Z0 (ArC )),where the mapping space between two topological spaces is given with the compactopen topology. If Y is a real variety this provides a continuous equivariant comparison map
Φ : Z r (YC ) → Map(YC (C), Z0 (ArC ))of topological abelian groups.
Definition 3.12. (Real Morphic Cohomology) Friedlander-Walker [FW02a] define
real morphic cohomology of a quasi-projective real variety by
Lq HRn (X) = π2q−n Z q (X)for 2q − n ≥ 0.
We will be using an equivariant extension of their theory for normal quasiprojective real varieties defined below.
Definition 3.13. (Equivariant Morphic Cohomology) Let X be a normal quasiprojective variety. Then the equivariant morphic cohomology is (in equivariant
homotopy indexing notation)
Lq HRk,r (X) = πq−k,q−r Z q (XC ),for q − k, q − r ≥ 0.
By Proposition 3.7 we see that
Lq HRn (X) = π2q−n Z q (XC )G
so Friedlander-Walker’s real morphic cohomology groups are a part of the equivariant morphic cohomology, Lq HRq−r,q (X) = πr,0 Z q (XC ) = Lq HR2q−r (X).
In [dS03a] dos Santos defines real Lawson homology.
Definition 3.14. (Real Lawson Homology) For any quasi-projective real variety
X, the real Lawson homology is defined by
Lq HRn,m (X) = πn−q,m−q Zq (XC ),for n − q, m − q ≥ 0.
Definition 3.15. Let X be a quasi-projective real variety. The following definitions
are taken from [LLFM03].

VANISHING THEOREMS FOR REAL ALGEBRAIC CYCLES

15(1) Define the space of averaged cycles Zq (X)av to be
Zq (XC )av = Im(N ) ⊆ Zq (XC )G ,so Zq (XC )av ⊆ Zq (XC )G is the subgroup generated by cycles of the form
Z + Z and given the subspace topology. By Remark 3.2 this is a closed
subgroup. Here N : Zq (XC ) → Zq (XC ) is defined by N (Z) = Z + Z.
(2) Define the space of reduced cycles Rq (X) to be the quotient group
Rq (X) =Zq (XC )G.
Zq (XC )av

Remark 3.16. These spaces all have the homotopy type of a CW -complex (seeCorollary B.6).
Teh [Teh05] defines the reduced real Lawson homology of X to be
RLq Hn (X) = πn−q Rq (X),for n ≥ q. According to Lemma 3.3 this definition coincides with the definition
given in [Teh05] in the case of a quasi-projective variety.
Example 3.17. Let X be a projective real variety.
(1) [LLFM03, Lemma 8.4] The space of averaged zero-cycles computes the
singular homology of the quotient of analytic space of complex points
πk Z0 (XC )av = Hk (X(C)/G; Z).
(2) By the equivariant Dold-Thom theorem [dS03b] the space of fixed zerocycles computes (a portion of) Bredon cohomology
πk Z0 (XC )G = Hk,0 (X(C); Z).
(3) [Teh05, Proposition 2.7] The space of reduced real cycles computes the
singular homology with Z/2 coefficients of the analytic space of real points
πk R0 (X) = Hk (X(R); Z/2).
4. Poincare Duality
In this section we use the duality for bivariant cycle homology in [FV00] toestablish a duality between Lawson homology of a real variety and real morphic
cohomology. This together with the duality between Lawson homology and morphic
cohomology [FL97] gives an equivariant duality between the algebraic cocycle spaces
and algebraic cycle spaces for the complexification of a real variety.
The material and methods used here closely parallel [FW03, Section 3] whereFriedlander-Walker reformulate Lawson homology and morphic cohomology forcomplex varieties.
Recognition Principle. Let F (−) be a presheaf sets (respectively simplicial sets,
or abelian groups) on Sch/R. If T is a topological space then define F (T ) by thefiltered colimit
F (T ) = colim F (V ).
T →V (R)In particular we obtain a simplicial set (respectively a bisimplicial set, or simplicial abelian group) by(4.1)

d 7→ F (∆dtop ).

16

JEREMIAH HELLER AND MIRCEA VOINEAGU

We record an analogue of the recognition principle [FW03, Theorem 2.3] which isneeded to move the duality for bivariant cycle homology to a duality for real Lawson
homology and morphic cohomology. Friedlander-Walker’s proof in the complex caseuses the uad-topology which is essentially due to Deligne.
Definition 4.2.
(1) A continuous map of topological spaces f : S → T is said
to satisfy cohomological descent if for any sheaf A of abelian groups on T
the natural map
H ∗ (T, A) → H ∗ (NT (S), f ∗ A)
is an isomorphism. Here NT (S) → T is the Cech nerve of f , i.e. NT (S) is
the simplicial space which in degree n is the n + 1-fold fiber product of S
over T . A map f : S → T is said to be of universal cohomological descentprovided the pullback S ×T T ′ → T ′ along any continuous map T ′ → T isagain of cohomological descent.
(2) The uad-topology on SchR is the Grothendieck topology associatedto the`pretopology generated by collections {Ui → X} such that
Ui (R)an →X(R)an is a surjective map of universal cohomological descent.
Example 4.3.
(1) A proper and surjective map of real varieties X → Y which
induces a surjective map of real points is a uad-cover. Indeed, in this case
X(R)an → Y (R)an is a proper surjective map of topological spaces, andtherefore is a map of universal cohomological descent (see [Del74, 5.3.5]).
(2) Any Nisnevich cover is a uad-cover. Any cdh-cover is a uad-cover. In
particular every real variety X is locally smooth in the uad topology because
resolution of singularities implies there is a cdh-cover X ′ → X, with X ′smooth.
(3) Unlike the complex case not every etale-cover is a uad-cover (e.g. Spec C →Spec R is an etale cover but not a uad-cover).
Here is the recognition principle.
Theorem 4.4 ([FW03, Theorem 2.2]). Suppose that F → G is a natural transformation of presheaves of abelian groups on SchR . If Fuad −→ Guad is an isomorphismof uad-sheaves, thenF (∆•top ) → G(∆•top )is a homotopy equivalence of simplicial abelian groups.
Proof. Friedlander-Walker’s proof given in [FW03] works by changing the space
X(C)an associated with a complex variety with the space Y (R)an associated to areal variety together with the fact that Y (R)an may be triangulated.

Corollary 4.5. Suppose that f : F → G is a map of presheaves of simplicial
abelian groups such that F (V ) → G(V ) is a homotopy equivalence for any smooth
V . Then the map of simplicial abelian groups diag F (∆•top ) → diag G(∆•top ) is a
homotopy equivalence.
Poincare Duality. Let X be a variety over a field k of characteristic zero. Recall the presheaf zequi (X, r)(−) of equidimensional r-cycles. This is the uniqueqf h-sheaf on Sch/k such that for a normal variety U the group zequi (X, r)(U ) is
the free abelian group on closed, irreducible subvarieties V ⊆ U ×k X which areequidimensional of relative dimension r over some irreducible component of U .

VANISHING THEOREMS FOR REAL ALGEBRAIC CYCLES

17

If X and Y are real varieties then G = Gal(C/R) acts on the group zequi (XC , r)(UC )by σ · [V ⊆ UC ×C XC ] = [σV ⊆ UC ×C XC ].
Lemma 4.6. Let X and U be real varieties. Thenπ∗zequi (X, r)(U ) −→ (zequi (XC , r)(UC ))G
is a natural isomorphism where π : (U ×R X)C → U ×R X.
Proof. It suffices to check this for U normal, since normalization is a qf h-cover. By[SV00b, Lemma 2.3.2] π ∗ : Cycl(U × X) → Cycl((U × X)C )G is an isomorphism,
where Cycl(W ) denotes the group of cycles on W . We are done if we see thatf : V → U is equidimensional if and only if f˜ : VC → UC is equidimensional. By[Gro66, Proposition 13.3.8] if f is equidimensional then so is f˜. Suppose that f˜ :VC → UC is equidimensional. Since UC is normal, f˜ : VC → UC is an open mappingand for all v ′ ∈ V the local rings OVC ,v′ are equidimensional by [Gro66, Corollaire
14.4.6]. By [Gro65, Corollaire 2.6.4, Proposition 7.1.3] the map f : V → U is
open and OV,v is equidimensional for all v ∈ V since UC → U is faithfully flat andtherefore f is equidimensional.
In the proof of [FW02a, Proposition 2.4] it is shown that for any presheaf F (−)
of sets on Sch/C and any topological space T the natural map∼=

colim F (VC ) −→ colim F (U )T →V (R)T →U(C)is an isomorphism. In the first indexing set V ranges over real varieties and in thesecond U ranges over complex varieties.
In particular zequi (XC , r)(YC ×C T ) may be computed via the filtered colimit
zequi (XC , r)(YC ×C T ) = colim zequi (XC , r)(YC ×C VC ),T →V (R)which equips zequi (XC , r)(YC ×C T ) with an action of G. Filtered colimits commute
with fixed points and sozequi (X, r)(Y ×R T ) → (zequi (XC , r)(YC ×C T ))G = colim (zequi (XC , r)(YC ×C VC ))G
T →V (R)is an isomorphism.
For X projective we have the natural isomorphism of presheaves (in fact of qf hsheaves) of abelian groups on SchR (see [SV00b, Lemma 4.4.14]∼ MorR (−, Cr (X))+ .
zequi (X, r)(−) =The following is the real analogue of [FW03, Proposition 3.1].
Proposition 4.7. Let T be a compactly generated Hausdorff topological space andX a quasi-projective real variety. There is a natural map of abelian groups
zequi (X, r)(T ) → Homcts (T, Zr (X))
given by sending (f : T → U (R), α ∈ zequi (X, r)(U )) to the function t 7→ α|f (t) .
This map is contravariant for continuous maps of compactly-generated Hausdorff
spaces T ′ → T , covariant for proper maps X → X ′ and contravariant for flat maps
X ′ → X (with a shift in dimension).
When X is a projective real variety the induced map of simplicial abelian groups
zequi (X, r)(∆•top ) → Sing• Zr (X)18

JEREMIAH HELLER AND MIRCEA VOINEAGU

is the natural homotopy equivalence+ ≃[Sing• (Cr (X)an )] −→ Sing• Zr (X).
More generally, for a quasi-projective real variety X with projectivization X ⊂ X
this map fits into a comparison of homotopy fiber sequences(4.8)

zequi (X r X, r)(∆•top )/ zequi (X, r)(∆•top )/ zequi (X, r)(∆•top )Sing• Zr (X r X)/ Sing Zr (X)/ Sing• Zr (X).

•

Therefore the map
zequi (X, r)(∆•top ) → Sing• Zr (X)is a natural weak equivalence for any quasi-projective real variety X.
Proof. The map
zequi (XC , r)(T ) = colim zequi (XC , r)(W ) → Homcts (T, Zr (XC ))T →W (C)given sending (f : T → W (C), α ∈ zequi (XC , r)(W )) to the function t 7→ α|f (t)is shown to be well-defined in [FW03, Proposition 3.1] and to satisfy the statednaturality properties. Observe that if W = VC is the complexification of a realvariety then α|f (t) = α|f (t) . Therefore composing with the natural isomorphism∼=

colim zequi (XC , r)(VC ) −→ colim zequi (XC , r)(W )T →V (R)

T →W (C)gives a well-defined equivariant map
zequi (XC , r)(T ) → Homcts (T, Zr (XC ))which by taking fixed points induces the map
Gzequi (X, r)(T ) = zequi (XC , r)(T )G → Homcts (T, Zr (XC )) = Homcts (T, Zr (X)) ,which is the map of the proposition and satisfies the stated naturality properties.
When X is a projective real variety and T is a compact Hausdorff space, the map
ef f
zequi
(XC , r)(T ) → Homcts (T, Cr (XC )) is an isomorphism by [FW02b, Corollary4.3]. Since this is an equivariant map, taking fixed points yields the isomorphismof monoids∼=
ef f(X, r)(T ) −zequi→ Homcts (T, Cr (X)) .
Therefore the map∼=zequi (X, r)(∆•top ) −→ [Homcts ∆•top , Cr (X)an ]+ → Sing• Zr (X)is a homotopy equivalence by Quillen’s theorem [FM94, App Q] on homotopy groupcompletions of simplicial abelian monoids.
Finally the diagram (4.8) commutes by the naturality properties of the map
zequi (X, r)(T ) → Homcts (T, Zr (X)) . By [FV00, 5.12,8.1], Proposition 4.12 (homotopy invariance), and Theorem 4.4 (recognition principle) the upper row of the
diagram (4.8) is a homotopy fiber sequence. Comparing the upper and lower homotopy fiber sequence yields the final statement of the proposition.
VANISHING THEOREMS FOR REAL ALGEBRAIC CYCLES

19

Proposition 4.9. For a quasi-projective real variety U , projective real variety Y ,and compact Hausdorff space T , there is a natural map of abelian groupszequi (Y, 0)(U ×R T ) → Homcts T, MorR (U, C0 (Y ))+given by sending (f, α) to the function t 7→ α|f (t) .
Proof. The map

zequi (YC , 0)(UC ×C T ) → Homcts T, MorC (UC , C0 (YC ))+

from [FW03, Proposition 3.3] is equivariant and therefore taking fixed points induces the natural map of abelian groupszequi (Y, 0)(U ×R T ) → Homcts T, MorR (U, C0 (Y ))+ .


Let Y be a projective real variety and U a normal quasi-projective real variety
of dimension d with projectivization U ⊆ X and closed complement X∞ = X \ U .
Write Er (Y )(U ) ⊆ Cr+d (Y ×R X) for the submonoid consisting of those cycles ofdimension r + d on Y × X whose restriction to U is equidimensional of relative
dimension r over U . This is a constructable embedding. This can be seen byarguing as in [Fri98] for the complex case. The subspace topology on this monoid
agrees with the quotient topology Cr (Y )(U ) = Er (Y )(U )/Cr+d (Y ×X∞ ) by the same
reasoning as in [Fri98, Proposition 1.8]. The topological group of equidimensional
cycles is the naive groups completion Zr (Y )(U ) = Cr (Y )(U )+ . Since these aretractable monoids, they are related to equidimensional cocycles via the homotopy
fiber sequence
Zr+d (Y × X∞ ) → (Er (Y )(U ))+ → Zr (Y )(U ).
Define the presheaf e(U, Y, r)(−) to be the pull-back of presheaves/ z ef f (Y × X, r + d)(−)
e(U, Y, r)(−)
equi


ef f
zequi
(Y, r)(U × −)/ z ef f (Y × U, r + d)(−).
equi

We have for each quasi-projective real variety V the short exact sequence of abeliangroups0 → zequi (Y × X∞ , r + d)(V ) → (e(U, Y, r)(V ))+ → zequi (Y, r)(U × V ) → 0.
Proposition 4.10. Let Y be a projective real variety, U a normal quasi-projective
real variety, and T a compact Hausdorff space. Then∼=e(U, Y, r)(T ) −→ Homcts (T, Er (Y )(U ))is an isomorphism.
Proof. Observe that if V is a quasi-projective real variety then the isomorphism
ef f
MorR (V, Cr+d (Y × X)) ∼= zequi (Y × X, r + d)(V ) restricts to give the isomorphism
MorR (V, E(Y )(U )) ∼= e(U, Y, r)(V ). Here if E ⊆ W is a constructable subset thenMor(V, E) ⊆ Mor(V, W ) is the subset consisting of those continuous algebraic
maps whose image is contained in E.
∼=
The isomorphism e(U, Y, r)(T ) −→ Homcts (T, Er (Y )(U )) now follows as in [FW01b,
Corollary 4.3].
20

JEREMIAH HELLER AND MIRCEA VOINEAGU

Proposition 4.11. Let U be a normal quasi-projective real variety and Y a projective real variety. The map of simplicial abelian groups from Proposition 4.9
zequi (Y, 0)(U × ∆•top ) → Sing• (MorR (U, C0 (Y ))+ )is a homotopy equivalence.
Proof. By proposition 4.10 we have e(U, Y, r)(∆•top ) ∼= Sing• Er (Y )(U ). Now by taking group completions, tractability of the monoid Er (Y )(U ) and Quillen’s theorem[FM94, App Q] we conclude that≃

→ Sing• (Er (Y )(U )+ )e(U, Y, r)(∆•top )+ −is a homotopy equivalence.
We conclude the proposition by comparing homotopy fiber sequences of simplicial
abelian groups
zequi (Y × X∞ , r + d)(∆•top )/ (e(U, Y, r)(∆•top ))+/ zequi (Y, r)(U × ∆•top )Sing• Zr+d (Y × X∞ )/ Sing (Er (Y )(U )+ )/ Sing• Z r (Y )(U ).

•The left arrow is a homotopy equivalence by Proposition 4.7, we have just seenthat the middle map is a homotopy equivalence, the right horizontal maps induce asurjection on π0 and so we conclude that zequi (Y, r)(U × ∆•top ) → Sing• Z r (Y )(U )is a homotopy equivalence.
Proposition 4.12. The presheaves zequi (X, r)(∆•top × −) are homotopy invariant
in the sense that the map of complexes
zequi (X, r)(∆•top ) → zequi (X, r)(∆•top ×R ∆1R )is a quasi-isomorphism.
Proof. The same argument as in [FW01a, Lemma 1.2].

The duality theorem for bivariant cycle theory [FV00, Theorem 7.4] says thatfor real varieties X, U with U smooth of dimension d, the natural inclusion(4.13)

D : zequi (X, r)(U ×R −) ֒→ zequi (X ×R U, r + d)(−)

induces a quasi-isomorphism of complexes≃→ zequi (X ×R U, r + d)(∆•R ).
D : zequi (X, r)(U ×R ∆•R ) −Proposition 4.14. For a smooth real variety U and a quasi-projective real variety
X the mapDzequi (X, r)(U ×R ∆•top ) −→ zequi (X ×R U, r + d)(∆•top )is a quasi-isomorphism.
Proof. Consider the commutative diagram
zequi (X, r)(U ×R ∆•top )


π∗ y

D

−−−−→

D

zequi (X ×R U, r + d)(∆•top )

 ∗
yπ

zequi (X, r)(U ×R ∆•R ×R ∆•top ) −−−−→ zequi (X ×R U, r + d)(∆•R ×R ∆•top ).

VANISHING THEOREMS FOR REAL ALGEBRAIC CYCLES

21

The vertical arrows are quasi-isomorphisms by homotopy invariance. The bottom
right arrow is a quasi-isomorphism by Corollary 4.5 sincezequi (X, r)(U ×R ∆•R × W ) → zequi (X ×R U, r + d)(∆•R ×R W ).
is a quasi-isomorphism for all smooth real varieties W by [FV00, Theorem 7.4] andtherefore the top horizontal map is a quasi-isomorphism as well.
Lemma 4.15. Let Y be a projective real variety and U a smooth real variety. The
following diagram commutes
Dzequi (Y, 0)(U ×R ∆•top ) −−−−→ zequi (U ×R Y, d)(∆•top )




y
y
D

Sing• MorR (U, C0 (Y ))+ −−−−→

Sing• Zd (U ×R Y )where the vertical maps are the ones from Proposition 4.9 and Proposition 4.7.
Proof. By [FW03, Proposition 3.3] the diagram of equivariant maps of simplicial
sets
zequi (YC , 0)(UC ×C ∆•top ) −−−−→ zequi (UC ×C YC , d)(∆•top )
D




y
y
Sing• MorC (UC , C0 (YC ))+ −−−−→

Sing• Zd (UC ×C YC )D

commutes. Taking fixed points yields the result.



Write
q/q−1

zequi (PRq−1
, 0)(U ) → zequi (PqR , 0)(U )), 0)(U ) = coker(zequi (PRq−1⊆ PqR .
for the cokernel of the map of presheaves induced by PR

Proposition 4.16. Let U be a smooth real variety of dimension d. The sequenceof natural maps of complexes below consist of quasi-isomorphisms.
q/q−1(4.17) zequi (AqR , 0)(U ×R ∆•top ) ← zequi (PR→, 0)(U ×R ∆•top ) →Sing• (MorR (U, C0 (PqR ))+q−1 +Sing• (MorR (U, C0 (PR))→ Sing• Z q (U ).

Proof. That the first map of diagram (4.17) is a quasi-isomorphism follows fromconsideration of the comparison diagram/ z(Pn , r)(U × ∆•top )/ z(An , r)(U × ∆•top )D× U, r + d)(∆•top )D/ z(Pn × U, r + d)(∆•top )D/ z(An × U, r + d)(∆•top )Sing• Zr+d (Pn−1 × U )/ Sing• Zr+d (Pn × U )/ Sing• Zr+d (An × U ).

z(Pn−1 , r)(U × ∆•top )
z(P

n−1

The vertical arrows are all quasi-isomorphisms by Proposition 4.14 and by Proposition 4.7. Because Ck (V ) is a tractable monoid, the bottom row is homotopy
equivalent to a short exact sequence of simplicial abelian groups and therefore the
top rows are as well. It follows immediately that the first arrow of diagram 4.17 is aquasi-isomorphism. The second arrow of diagram (4.17) is a quasi-isomorphism by

22

JEREMIAH HELLER AND MIRCEA VOINEAGU

Proposition 4.11 and the last arrow of the diagram is a quasi-isomorphism because
MorR (U, C0 (PnR )) is a tractable monoid.
Definition 4.18. If k < 0 then define Zk (X) to be Z0 (X × A−k ).
We can now conclude the duality for real morphic cohomology and real Lawson
homology.
Corollary 4.19. Let U be a smooth real variety of dimension d. Then≃D

− Zd−q (UC )G→ Zd (AqC ×C UC )G ←Z q (UC )G −is a natural homotopy equivalence.
In particular it induces the natural isomorphism∼=

Lq HRn (U ) −→ Ld−q HRd−n,d (U ).
Proof. This follows from Proposition 4.14, Lemma 4.15, Proposition 4.7, Proposition 4.16, and homotopy invariance [dS03a, Proposition 4.15]. Indeed these showthat the following diagram is commutative and the left hand maps are homotopy
equivalences,zequi (AqR , 0)(U × ∆•top ) oDzequi (AqR × U, d)(∆•top ) oq/q−1

zequi (PR, 0)(U × ∆•top )
/

D

q/q−1
zequi (PR× U, d)(∆•top )/Sing• (MorR (U, C0 (PqR ))+Sing• (MorR (U, C0 (Pq−1))+
R/ Sing• Z q (U )

/

D

q
R ×U)
/ Sing• Zd (Pq−1Sing• Zd (PR ×U)

D/ Sing• Zd (Aq × U ).

Therefore the right hand map is also a homotopy equivalence

Combining Friedlander-Lawson’s duality between Lawson homology and morphic
cohomology over C and the duality over R immediately gives an equivariant duality
theorem.
Corollary 4.20. Let U be a smooth real variety of dimension d. The sequence ofmaps
Z q (UC ) → Zd (UC ×C AqC ) ← Zd−q (XC )consists of G-equivariant homotopy equivalences. In particular∼=

Lq HRn,m (U ) −→ Ld−q HRd−n,d−m(U ).
for all smooth quasi-projective real varieties U .
Remark 4.21. A smooth G-manifold M equipped such that the action of G on
its tangent bundle makes it into a real n-bundle satisfies an equivariant Poincare
duality,∼=→ Hn−p,n−q (M ; Z).
P : H p,q (M ; Z) −In a forthcoming paper we prove that the duality D is compatible under the cycle
maps with the duality P.

VANISHING THEOREMS FOR REAL ALGEBRAIC CYCLES

23

5. Compatibility of Cycle Maps
Generalized cycle maps. Let X be a smooth real variety. The generalized cycle
map relates motivic cohomology and etale cohomology,2q−kcyc : H2q−k,q(X; Z/2) → Het(X; µ⊗q
2 ).
M

By [Cox79] the etale cohomology of a real variety is equal to the Borel equivariant
cohomology of its space of complex points,
2q−k∼ 2q−kHet(X; µ⊗q2 ) = HZ/2 (XC (C); Z/2).

On the other hand morphic cohomology and motivic cohomology agree withfinite coefficients (see Proposition 5.1).
Combining the generalized cycle map in morphic cohomology and the comparison
map between Bredon and Borel equivariant cohomology
2q−k
Lq HRq−k,q (X; Z/2) → H q−k,q (XR (C); Z/2) → HZ/2
(X(C); Z/2)∼=

together with the isomorphism H2q−k,q(X; Z/2) −→ Lq HRq−k,q (X; Z/2) gives anM
other map
2q−k
⊗q
2q−k
H2q−k,q(X; Z/2) ∼= Het (X; µ2 ).
= Lq HRq−k,q (X; Z/2) → HZ/2 (XC (C); Z/2) ∼M

In this section we verify that these two potentially different cycle maps are equal
and we explore a few consequences. In particular this allows compatibility of cycle
maps allows us to conclude that Lq HRq−p,q (X; Z/2k ) → H q−p,q (XR (C); Z/2k ) is
an isomorphism for p ≥ q and for any smooth X.
Before continuing, we show that motivic cohomology and morphic cohomology
for real varieties agree with finite coefficients. This is a well-known to the experts,
but because of the lack of a good reference we prove it below.
Proposition 5.1. Let X be a smooth real variety. Then for any n > 0∼=

H2q−k,q(X; Z/n) −→ Lq HRq−k,q (X; Z/n).
M
Proof. We show that the natural map of simplicial abelian groups
zequi (Aq , 0)(X × ∆•R ) ⊗ Z/n → zequi (Aq , 0)(X × ∆•R × ∆•top ) ⊗ Z/n
is a quasi-isomorphism which implies the result by Proposition 4.11 and Proposition
4.12. Write F (U ) for the presheaf
U 7→ πk (zequi (Aq , 0)(X × ∆•R × U ) ⊗ Z/n)
on Sch/R and F0 (U ) for the constant presheaf
U 7→ πk (zequi (Aq , 0)(X × ∆•R ) ⊗ Z/n).
Restricted to Sm/R these are homotopy invariant presheaves with transfers.
Recall [FW02a, Lemma 3.8] that if F (−) is a homotopy invariant presheaf withhtransfers, Y is smooth, and y ∈ Y (R) then F (Spec OY,y
) → F (R) is an isomorh
phism, where OY,y is the Henselization of the local ring OY,y .
Let H(−) denote either the kernel or the cokernel of the natural transformation
F0 (−) → F (−). Let Y be a quasi-projective real variety and γ ∈ H(Y ). Let
Ỹ → Y be a cdh-cover with Ỹ smooth (in particular it is a uad-cover). Since

24

JEREMIAH HELLER AND MIRCEA VOINEAGU

H(Spec OỸh ,y ) = 0 for any y ∈ Ỹ (R) there are finitely many etale maps Ỹk → Ỹ`such that γ|Ỹk = 0 and Ỹk → Ỹ is a uad-cover.
Therefore Huad = 0 and (F0 )uad → Fuad is an isomorphism. By Theorem 4.4 weconclude thatπk (zequi (AqR , 0)(X × ∆•R ) ⊗ Z/n) → πk (zequi (AqR , 0)(X × ∆•R × ∆•top ) ⊗ Z/n)
is an isomorphism.
An application of the Bousfield-Friedlander spectral sequence finishes the proof.
Friedlander-Walker introduce in [FW02a] the equivalence relation of real algebraic equivalence. Briefly two cycles α, β on a real variety X are real algebraicallyequivalent provided there is a smooth real curve C, two real points c0 , c1 in thesame analytic connected component of C(R), and a cycle γ on X × C such thatα = γ|c0 and β = γ|c1 . Since Lq HRq,q (X) is the group of codimension q cycles on
X modulo real algebraic equivalence we obtain the following corollary.
Corollary 5.2. Let X be a smooth real variety and 0 ≤ r ≤ dim(X). Rational
equivalence and real algebraic equivalence yield the same equivalence relation on thegroup of r−cycles on X with finite coefficients.
Recall that zequi (Pq/q−1 , 0)(U ) = zequi (Pq , 0)(U )/zequi (Pq−1 , 0)(U ). Write
q/q−1

Z/2(q)(X) = (zequi (PR, 0)(X ×R ∆•R ) ⊗ Z/2)[−2q]Z/2(q)sst (X) = Sing• (Z q /2(XC )G )[−2q]G
Z/2(q)top (X) = Homcts XC (C) × ∆•top , Z/20 (AqC ) [−2q]G
Z/2(q)Bor (X) = Homcts XC (C) × EG × ∆•top , Z/20 (AqC ) [−2q]where we identify a simplicial abelian group with its associated bounded above
cochain complex. These form presheaves of cochain complexes on Sm/R. These
chain complexes compute respectively motivic cohomology, real morphic cohomology, Bredon cohomology, and Borel cohomology. Note that Z/2(q), Z/2(q)top and
Z/2(q)Bor are in fact complexes of etale sheaves on (Sm/R).
There are natural maps between these complexes,ρΦψ

Z/2(q)(X) −→ Z/2(q)sst (X) −→ Z/2(q)top (X) −→ Z/2(q)Bor (X)obtained as follows. From Proposition 4.9 and the projection ∆•R → Spec R we
obtainzequi (Pq/q−1 , 0)(X ×R ∆•R ) → Sing Z q (X ×R ∆•R ) = Sing Z q (XC ×C ∆•C )G
which induces Z/2(q)(X) → Z/2(q)sst (X). The second map Φ is the map (3.11)
and the third map ψ is induced by the projection XC (C) × EG → XC (C).
Nisnevich hypercohomology and descent. These cohomology theories may becomputed as Nisnevich hypercohomology groups. This allows us to view these cycle
maps as maps in a derived category where we can use a computation of [SV00a].

VANISHING THEOREMS FOR REAL ALGEBRAIC CYCLES

25

Say that a cartesian square
(5.3)

/Y

V

f

 
U

i


/X
f

is a distinguished Nisnevich square provided the map Y −→ X is etale, i : U ⊆ X
is an open embedding, and f : (Y \V ) → (X\U ) is an isomorphism. The Nisnevich
topologyis the Grothendieck topology on Sm/k generated by covers of the form`
U Y → X where U ⊆ X and f : Y → X form part of a distinguished square as
above.
Given a presheaf of chain complexes F and a closed i : A ⊆ B and open complement j : U ⊆ B definej∗

F (B)A = cone(F (B) −→ F (U ))[−1],which fits into the exact trianglej∗

F (B)A → F (B) −→ F (U ).
Say that a presheaf F (−) of chain complexes satisfies Nisnevich descent provided
that for a distinguished square as in (5.3) the square
F (X)/ F (Y )F (U )

/ F (V )is homotopy cartesian. Recall this means that this square induces the MayerVietoris exact triangle (in the derived category of abelian groups):
F (X) → F (Y ) ⊕ F (U ) → F (V ).
Equivalently, it means that F (Y )Z ′ → F (X)Z is an isomorphism in the derived
category of abelian groups where Z = X\U and Z ′ = Y \V .
When a presheaf of chain complexes F (−) (with F (∅) = 0) satisfies Nisnevich
descent then the Nisnevich hypercohomology of a smooth X with coefficients in F
is computed as
H p (F (X)) = H p (FN is (X)) = HpN is (X; FN is )(see for example [CTHK97, Theorem 7.5.1] for presheaves of chain complexes,[Nis89] for descent in the case of presheaves of spectra, [BG73] for descent in the
Zariski topology).
Note thatsst
H2q−p)N is ) = Lq H q−p,q (X; Z/2),
N is (X; (Z/2(q)
top
H2q−p
) = H q−p,q (XR (C); Z/2),
N is (X; Z/2(q)
2q−p
Bor
H2q−p
) = HZ/2(X(C); Z/2).
N is (X; Z/2(q)In the first case this follows because the motivic complex Z/2(q) satisfies Nisnevich
descent and Z/2(q)(X) → Z/2(q)sst (X) is a quasi-isomorphism of chain complexes
for all smooth X.

26

JEREMIAH HELLER AND MIRCEA VOINEAGU

Given i : A ⊆ B a closed subvariety with open complement j : U ⊆ B and write
C(j) for the mapping cone of j : U (C) ⊆ B(C). Then by a comparison of exact
triangles we see that
G
Z/2(q)top (B)A ≃ Homcts∗ C(j) ∧ ∆•top,+ , Z/20 (AqC ) [−2q]andG
Z/2(q)Bor (B)A ≃ Homcts∗ C(j) ∧ EG+ ∧ ∆•top,+ , Z/20 (AqC ) [−2q].

Let F (−) denote either Z/2(q)top (−) or Z/2(q)Bor (−) and let
V

U

j′

j

/Y

/X

be a distinguished Nisnevich square in Sm/R then
V (C)

U (C)

j′

j

/ Y (C)/ X(C)

is an equivariant homotopy pushout diagram of G-spaces (see for example [DI04]≃). Therefore C(j ′ ) −→ C(j) is an equivariant homotopy equivalence. Consequently
F (X)Z → F (Y )Z ′ is an isomorphism in the derived category of abelian groups andtherefore/ F (U )F (X)F (Y )

/ F (V )is homotopy cartesian. This means that both Z/2(q)top (−) and Z/2(q)Bor (−) satisfy Nisnevich descent.
Compatibility of cycle maps. We are now ready to show that two cycle maps
discussed in the beginning of this section are the same map.
Lemma 5.4. Suppose that V is a quasi-projective complex variety considered as a2q−kreal variety. Then πk Z/2(q)Bor (V ) = Hsing(V (C); Z/2).
Proof. If V is a complex variety then (V ×R C)(C) = V (C) ∐ V (C) and G acts
by interchanging the factors. In particular G acts freely on VC (C) and (VC (C) ×
EG)/G → VC (C)/G = V (C) is a vector-bundle which immediately implies that2q−kπk Z/2(q)Bor (V ) = Hsing(V (C); Z/2).

Write π0 : (Sm/R)et → (Sm/R)N is for the canonical map of sites.
Proposition 5.5. The complex of etale sheaves π0∗ Z/2(q)Bor on (Sm/R)et is canonically quasi-isomorphic to µ⊗q
2 .

VANISHING THEOREMS FOR REAL ALGEBRAIC CYCLES

27

Proof. Write Hi for the etale sheafification of the ith cohomology presheaf of
Z/2(q)Bor . First we show that Hi = 0 for i 6= 0. It is enough to show thatfor each real variety X and γ ∈ H i ([XC (C) × EG]/G; Z/2) that we can find an
etale covering (Uj → X) such that γ|Uj = 0 for each j. The map Y = XC → X
is an etale cover for any real variety X. Write γ ′ = γ|Y . By the previous Lemmai
H i (YC (C) × EG]/G; Z/2) = Hsing(Y (C); Z/2). Since Y has an etale cover Uj → Y′such that γ |Uj = 0 for each j (see e.g. [Mil80, Lemma III.3.15] ) we conclude that
Hi = 0 for i 6= 0.
When XC is connected then H 0 ([XC (C) × EG]/G; Z/2) = Z/2. More generally
if X = ∐Xi is the disjoint union of c connected real varieties then H 0 ([XC (C) ×
EG]/G; Z/2) = Z/2×c . This shows H0 = Z/2 = µq2 .
Finally since Hi = 0 for i 6= 0 we have canonical isomorphisms= HomD− ((Sm/R)et ) π ∗ Z/2(q)Bor , µ2⊗q
Z/2 = Homet H0 , µ⊗q2•Recall [SV00a, Section 6] that there is an injective etale resolution 0 → µ⊗q
2 →J•such that π0∗ J is a complex of Nisnevich sheaves with transfers with homotopy invariant cohomology sheaves. Proposition 5.5 gives a canonical map π0∗ Z/2(q)Bor →J • and by adjointness we obtain a map•
Z/2(q)Bor → R(π0 )∗ µ⊗q
2 = (π0 )∗ J .

Consider the following sequence of maps of complexes of Nisnevich sheaves
(5.6)

Φ

→ Z/2(q)top → Z/2(q)Bor → R(π0 )∗ µ⊗q
Z/2(q) → (Z/2(q)sst )N is −2 .

The complex of Nisnevich sheaves with transfers B2 (q) is defined in [SV00a,
Section 6] to be the truncation
B2 (q) = τ≤q (π0 )∗ J • = τ≤q (Rπ0∗ µ⊗q
2 ),p
in particular HpN is (X; B2 (q)) = Het(X; µ⊗q2 ) for p ≤ q and all smooth X. Since thecohomology sheaves of Z/2(q) (and therefore of Z/2(q)sst as well) vanish in degrees
i > q and so the sequence of maps (5.6) factors through the truncations,(5.7)

Z/2(q) → (Z/2(q)sst )N is → τ≤q Z/2(q)top → τ≤q Z/2(q)Bor → B2 (q).

Remark 5.8. It is important to note that the composites 5.6 and 5.7 are non-trivial.
This can be seen, for example, by evaluating on Spec C. The map Z/2(q)(X) →(Z/2(q)sst )N is (X) is a quasi-isomorphism for any smooth real variety X by Proposition 5.1. The comparison map (Z/2(q)sst )N is (C) → Z/2(q)top (C) is an equality.
By Proposition 5.17 below, for any X, the map Z/2(q)top (X) → Z/2(q)Bor (X) induces an isomorphism on cohomology in degrees p ≤ q. Finally since Z/2(q)Bor →Rπ0∗ µ⊗q
2 is obtained as the adjoint of a quasi-isomorphism and Spec C is an etale
point (of (Sm/R)et ) the map Z/2(q)Bor (C) → Rπ0∗ µ⊗q
2 (C) cannot be zero.
Write D− (N is) (respectively D− (N SwT /R)) for the derived category of boundabove complexes of Nisnevich sheaves (respectively Nisnevich sheaves with transfers). Write DM − (R) ⊆ D− (N SwT /R) for the full subcategory consisting of complexes with homotopy invariant Nisnevich cohomology sheaves.

28

JEREMIAH HELLER AND MIRCEA VOINEAGU

Theorem 5.9. Let X be a smooth real variety. The diagram commutes(5.10)∼=

Lq HRq−k,q (X; Z/2) o

H2q−k,q(X; Z/2)
M
cyc

Φ



H q−k,q (XC (C); Z/2)

/ H 2q−k (XC (C); Z/2)
G


/ H 2q−k (X; µ⊗q ),et2

Proof. By [SV00a, Corollary 6.11.1] and the vanishing of the cohomology sheavesof Z/2(q) above degree q,
Z/2 = HomDM − (R) (Z/2(q), B2 (q)) = HomDM − (R) (Z/2(q), π0∗ J • ) .
Also by [SV00a, Lemma 6.5] the inclusion of bi-complexes
HomN is (Z/2(q), π0∗ J • ) ⊆ HomN SW T (Z/2(q), π0∗ J • )is an equality. Therefore
Z/2 = HomDM − (R) (Z/2(q), π0∗ J • ) = HomD− (N is) (Z/2(q), π0∗ J • ) .
Finally, the map Z/2(q) → π0∗ J • obtained from (5.6) is not trivial by Remark 5.8
and so we conclude that it must be the cycle map.
Applications and computations. As a result of the compatibility of cycle maps
we can conclude some Beilinson-Lichtenbaum type theorems for morphic cohomology which we need to prove the vanishing theorem. We also use these to make a
few computations of equivariant morphic cohomology.
Corollary 5.11. Let X be a smooth real variety. The map
Φ : Lq HRq−k,q (X; Z/2n ) → H q−k,q (XC (C); Z/2n )is an isomorphism for q ≤ k and a monomorphism for q = k + 1.
Proof. Consider the commutative diagram (5.10). By [SV00a] the Milnor conjecture, proved by Voevodsky [Voe03], implies that cyc is an isomorphism for k ≥ q and
an injection for k = q − 1. This immediately implies the statement for injectivity.
2q−k
If k ≥ q then since cyc is an isomorphism we conclude that HG(XC (C); Z/2) →2q−k
Het(X; µ⊗q)isasurjective
map
betweenfinitely
dimensional
Z/2-vector
spaces.
2
By [Cox79] these are isomorphic Z/2-vector spaces and therefore the map is an iso2q−kmorphism. Since H q−k,q (XC (C); Z) → HG
(XC (X); Z/2) is also an isomorphism
for k ≥ q by Proposition 5.17 we conclude that Φ is also an isomorphism for k ≥ q.
This yields the result for Z/2-coefficients.
We have the following diagram of distinguished triangles in D− (N is) :Z/2(q)sst
N is/ Z/4(q)sst/ Z/2(q)sst/ Z/2(q)sst [1]τ≤q Z/2(q)top/ τ≤q Z/4(q)top/ τ≤q Z/2(q)top/ τ≤q Z/2(q)top [1],

N is

N is

N is

To see that the bottom row is a triangle in D− (N is) it is enough to check that themap on cohomology sheaves Hq (τ≤q Z/4(q)top ) → Hq (τ≤q Z/2(q)top ) is a surjection.
This follows from the surjectivity of the composition
Hq (Z(q)) → Hq (τ≤q Z/4(q)top ) → Hq (τ≤q Z/2(q)top ).

VANISHING THEOREMS FOR REAL ALGEBRAIC CYCLES

29

which is a consequence of the local vanishing of Z(q) and the quasi-isomorphisms
Z/2(q) → B2 (q) and τ≤q Z/2(q)top → B2 (q). Now the conclusion follows fromthe long exact sequence in hypercohomology associated to the diagram. Using
induction on n we conclude that Φ : (Z/2n (q)sst )N is → τ≤q Z/2n (q)top is a quasiisomorphism.
Corollary 5.12. Let X be a smooth complex variety. For any n > 0 the map
2q−kΦ : Lq H 2q−k (X; Z/2n ) → Hsing(X(C); Z/2n )is an isomorphism for any q ≤ k and a monomorphism for q = k + 1.
Proof. Follows immediately from Corollary 5.11 by viewing X as real variety.



Corollary 5.13. Let X be a smooth real variety and k > 0. The cycle map
Φ : Lq HRr,s (X; Z/2k ) → H r,s (XC (C); Z/2k )is an isomorphism if r ≤ 0 (and s ≤ q) and an injection if r = 1 (and s ≤ q).
Proof. Write Fq = hofib(Z q /2k (XC ) → Map(XC (C), Z/2k0 (AqC ))) for the homotopy
fiber of the cycle map. The homotopy fiber construction is equivariant and yields
an equivariant homotopy fiber sequence
Fq → Z q /2k (XC ) → Map(XC (C), Z/2k0 (AqC )).
By Corollary 5.12 and Corollary 5.11 both πk (Fq ) = 0 and πk (FqG ) = 0 for k ≥ q−1.
Therefore Ωq−1 Fq is equivariantly weakly contractible for q ≥ 1 and if q = 0 then F0
is equivariantly contractible. The result follows now from the long exact sequenceof homotopy groups applied to the equivariant homotopy fiber sequenceΩq−1 Fq → Ωq−1 Z q /2k (XC ) → Ωq−1 Map(XC (C), Z/2k0 (AqC )).
Corollary 5.14. Let X be a smooth real curve. Then
Lq HRr,s (X; Z) → H r,s (XC (C); Z)
is an isomorphism for any q ≥ 0, r ≤ q, and s ≤ q.
Proof. By Poincare duality for real Lawson homology and equivariant morphic∼=
cohomology and Remark 4.21, Lq H r,s (X; Z) −→ H r,s (XC (C); Z) for q ≥ 1. By
Corollary 5.13, L0 HRr,s (X; Z/2k ) → H r,s (X; Z/2k ) is an isomorphism for r, s ≤ 0.
When A is an abelian group and 2 is invertible in A then a transfer argument showsthat∼=
L0 HRr,s (X; A) → H r,s (X; A).
This isomorphism and the one with mod-2k coefficients give the result of the corollary.

Corollary 5.15. Let X be a smooth real surface. Then for any k > 0
Lq HRr,s (X; Z/2k ) → H r,s (XC (C); Z/2k )is an isomorphism for q = 0 and r, s ≤ 0 and it is an injection for r = 1 and s ≤ 1.
Moreover L1 HR1,s (X; Z/2k ) = 0 for s ≤ −2.
Recall that π0 Z q (XC )G is the group of codimension q cycles on X modulo real
algebraic equivalence.

30

JEREMIAH HELLER AND MIRCEA VOINEAGU

Corollary 5.16. Let X be a smooth real variety of dimension d. Then for any
k>0
L1 HRr,s (X; Z/2k ) → H r,s (X; Z/2k )is an isomorphism for any r ≤ 0 and s ≤ 1 and it is an injection for r = 1 ands ≤ 1. Moreover
L1 HR1,1 (X; Z/2k ) = CH 1 (X) ⊗ Z/2k ⊆ H 1,1 (XC (C); Z/2k )L1 HR1,s (X; Z/2k ) = 0

for s ≤ −2

Proof. All statements follow immediately from Corollary 5.13 except the last one.
The first part of the last statement follows from Proposition 5.1. For rest of thelast statement, by Corollary 5.13 together with Proposition 5.17 we have
1+s
(XC (C); A) = 0
L1 HR1,s (X; Z/2k ) ֒→ H 1,s (XC (C); Z/2k ) ֒→ HG

for 1 + s < 0.

We finish this section with the computation used in Corollary 5.11 that Bredonand Borel cohomology agree in the range relevant to the Beilinson-Lichtenbaum
conjecture.
Proposition 5.17. Let W be a G-CW complex, M a G-module, and M the associated constant Mackey functor. The map
H m,q (W ; M ) → H m,q (W × EG; M )is an isomorphism for m ≤ 0 and it is an injection for m = 1.
In particular for q ≤ p the map
2q−p
H q−p,q (W ; Z/2) → HZ/2
(W, Z/2)

is an isomorphism and an injection for p = q − 1.
Proof. Define ẼG = colimn S 0,n . This space fits into a homotopy cofiber sequence
EG+ → S 0 → ẼG
and ẼG/G ≃ S 1,0 ∧ BG.
First we consider the case that G acts trivially on W . From the previous homotopy cofiber sequence we obtain the homotopy cofiber sequence(5.18)

W+ ∧ EG+ → W+ → W+ ∧ ẼG.

Since G acts trivially on W we have that (W+ ∧ẼG)/G = W+ ∧ẼG/G ≃ W+ ∧S 1,0 ∧k−1
BG and therefore H̃ k,0 (W+ ∧ ẼG; M ) = H̃sing(W+ ∧ BG; M ) = 0 if k ≤ 1. Notice
0,1
0,1
0,n
0,n+1 ∼
that S ∧ ẼG = S ∧ colimn S= colimn S= ẼG. Since ẼG ∼= S 0,1 ∧ ẼG
is an equivariant equivalence this induces an isomorphismH̃ k,s (W+ ∧ ẼG; M ) ∼= H̃ k,0 (W+ ∧ ẼG; M )for all s and therefore H̃ k,s (W+ ∧ ẼG, M ) = 0 for k ≤ 1 for all s. Now from thelong exact sequence associated to the cofiber sequence (5.18) it follows that for all
q the map H m,q (W ; M ) → H m,q (W × EG; M ) is an isomorphism for m ≤ 0 and
an injection for m = 1.
Consider now a general G-CW complex W and consider the quotient W/W G .
Since G acts freely on the based space W/W G we have the isomorphism∼=H̃ s,t (X/X G; M ) −→ H̃ s,t (X/X G ∧ EG+ ; M ).

VANISHING THEOREMS FOR REAL ALGEBRAIC CYCLES

31

Applying the five lemma to the comparison of long exact sequences obtained fromthe cofiber sequences W+G → W+ → W/W G and (W G × EG)+ → (W × EG)+ →W/W G ∧ EG+ yields the result.

6. Vanishing Theorem
Let X be a real variety. Continue to write G = Z/2 and σ ∈ G for the nontrivial
element. Recall that the reduced real cycle group is defined to be the quotient
topological group
Zq (XC )G
Rq (X) =.
Zq (XC )av
This is the free Z/2-module generated by closed subvarieties Z ⊆ X such that both
Z and ZC are irreducible. In particular, if X is a complex variety viewed as avariety over R then Rq (X) = 0.
Reduced real Lawson homology of X is defined by the homotopy groups of thisspace,RLq Hn (X) = πn−q Rq (X).
In this section we prove our main theorem which we state now.
(Theorem 6.10). Let X be a quasi-projective real variety. Then
πk Rn (X) = RLn Hk+n (X) = 0
for k ≥ dim X − n + 1.
To avoid difficulties with point-set topology below we work simplicially. Notethat if X is a G-space then Sing• X is a G-simplicial set and Sing• (X G ) = (Sing• X)G .
G
If A• is a G-simplicial set then |AG• | = |A• | (see for example [Dug05, Lemma A.5]).
Definition 6.1.

and(1) Let W be a G-space. Write

q
e
Ztop
(W ) = Homcts W × ∆•top , Z0 (AqC ) ,
Zeq /2top (W ) = Homcts W × ∆•top , Z/20 (AqC ) .
These are simplicial abelian groups and G acts on them by the standard
formula (σf )(x) = σf (σx).
(2) Let X be a normal quasi-projective real variety. WriteandWe haveandZeq (XC ) = Sing• Z q (XC )Zeq /2(XC ) = Sing• Z q /2(XC ).

q(XC (C))G ∼πk Zetop= H q−k,q (XC (C), Z)

πk Zeq (XC )G ∼= Lq HR2q−k (X; Z)
and similarly for the mod-2 groups. In` particular if X is a complex variety viewed
as a real variety then XC (C) = X(C) X(C) (with G-action switching the factors)
and so2q−kπk Zeq (XC (C))G ∼(X(C), Z) and πk Zeq (XC )G ∼= Lq H 2q−k (X; Z)
=H
top

sing

32

JEREMIAH HELLER AND MIRCEA VOINEAGU

and similarly for the mod-2 groups.
The comparison maps (3.11) of simplicial abelian groupsqΦ : Zeq (XC ) → Zetop(XC (C))andΦ : Zeq /2(XC ) → Zeq /2top (XC (C))

are G-equivariant for any quasi-projective real variety X.
If M• is a simplicial G-module write N = 1 + σ : M• → M• and define M•av tobeM•av = Im(N ) = Im(1 + σ : M• → M• ).
Definition 6.2. Let W be a G-space. Define the group of reduced topologicalcocycles (of codimension q) to be the quotient simplicial abelian group
G
Homcts W × ∆•top , Z/20 (AqC )Zeq /2top (W )Gqe
Rtop (W ) =.
q av = eq•
Homcts W × ∆top , Z/20 (AC )Z /2top (W )av

To relate the space of reduced algebraic cocycles with the reduced topologicalcocycles we introduce the following auxiliary simplicial set for X a quasi-projective
normal real variety:Gfq
e q (X) = Z /2(XC ) .
R
Zeq /2(XC )av

Proposition 6.3. Let X be a normal quasi-projective real variety. The following
diagrams commute and the horizontal rows are short exact sequences of simplicial
abelian groups (and therefore in particular the horizontal rows are homotopy fiber
sequences of simplicial sets)(6.4)

Zeq /2(XC )G
ΦG

and
(6.5)

C


/ Zeq /2top (XC (C))

Zeq /2(XC )av/ Zeq /2(X )GΦav/ Zeq /2(X )av
C

Φav

Φ


q
e
Z /2top (XC (C))G
Zeq /2top (XC (C))av

N/ Zeq /2(X )


N /
q
e
Z /2top (XC (C))av ,

C

ΦG

/R
e q (X)Φ/ Zeq /2top (XC (C))G/Re qtop (XC (C)).

Proof. These diagrams commute because Φ is a G-homomorphism.
Whenever M is a G-module whose underlying abelian group is 2-torsion thenN
the sequence of abelian groups 0 → M G → M −→ M av → 0 is exact.
In particular the underlying sequences of simplicial abelian G-modules in thefirst diagram form short exact sequences of simplicial abelian groups.
In the second diagram the horizontal rows form short exact sequences by definieq (−) and R
eq (−).
tion of Rtop
By definition we have

N

(Sing• Z q /2(XC ))av = Im(Sing• Z q /2(XC ) −→ Sing• Z q /2(XC )).

VANISHING THEOREMS FOR REAL ALGEBRAIC CYCLES

33

There is a natural map i : (Sing• Z q /2(XC ))av → Sing• (Z q /2(XC )av ) which issimplyi(f + f ) = f + f
dfor a continuous map f : ∆top → Z q /2(XC ). The map i : (Sing• Z q /2(XC ))av →Sing• (Z q /2(XC )av ) induces a map
e q (X) → Sing• Rq (X).
i:R

(6.6)

Lemma 6.7. Let X be a normal real projective variety. The map (6.6) of simplicial
abelian groupse q (X) → Sing Rq (X)R•
is a homotopy equivalence.

Proof. By Proposition A.5 the maps Z q /2(XC )/Z q /2(XC )G −→ Z q /2(XC )av and
q
av
q
G
q
av
Z (XC ) /2Z (X) −→ Z /2(XC ) are isomorphisms of topological groups. Therefore both
0 → Z q /2(XC )G → Z q /2(XC ) → Z q /2(XC )av → 0
and
0 → Z q /2(XC )av → Z q /2(XC )G → Rq (X) → 0
are short exact sequences of topological abelian groups. These groups all have thehomotopy type of a CW -complex and therefore these sequences are homotopy fiber
sequences [Teh05]. Applying Sing• to these homotopy fiber sequence and comparing
with the homotopy fiber sequences of the top rows of 6.4 and 6.5 gives commutative
diagrams of homotopy fiber sequences of simplicial sets:(6.8)

Zeq /2(XC )G/ Zeq /2(XC )≃≃Sing• Z/2q (XC )G/ Sing Z/2q (XC )•Zeq /2(XC )av/ Zeq /2(XC )G/ Zeq /2(XC )avi/ Sing Z/2q (XC )av ,
•

and
(6.9)

≃

i


Sing• Z/2(XC )av/ Sing Z/2q (XC )G•/R
e q (X)i/ Sing• Rq (X).

≃
From the first diagram we see that i : Zeq /2(XC )av −→ Sing• (Z q /2(XC )av ) is aweak equivalence of simplicial sets and consequently from the second diagram weconclude that≃e q (X) −R→ Sing• Rq (X)is a weak equivalences of simplicial abelian groups and therefore is a homotopy
equivalence of simplicial sets.


We now prove our main theorem.
Theorem 6.10. Let X be a quasi-projective real variety of dimension d. ThenRLn Hn+k (X) = πk Rn (X) = 0
for k ≥ d − n + 1.

34

JEREMIAH HELLER AND MIRCEA VOINEAGU

Proof. We first consider the case when X is a smooth projective real variety.
In case n = d = dim(X) we have thatRd (X) =Zd (XC )G= Z/2×c ,Zd (XC )av

where c denotes the number of irreducible components of X which are not definedover C.
Therefore π0 (Rd (X)) = Z/2×c and πi (Rd (X)) = 0 for i > 0.
Consider the comparison of homotopy fiber sequences (6.4) for q > 0.
(6.11)

Zeq /2(XC )GN/ Zeq /2(X )
C

ΦG

C

Φav

Φ


/ Zeq /2top (XC (C))


Zeq /2top (XC (C))G/ Zeq /2(X )av
N / eq
Z /2qtop (XC (C))av ,By the Milnor conjecture (see Corollary 5.12) the comparison map Φ induces anisomorphism on πk for k ≥ q and induces an injection for k = q − 1. By Corollary
5.11 the map ΦG induces an isomorphism on πk for k ≥ q and induces an injection
for k = q − 1. We now conclude by the 5-lemma that Φav induces an isomorphismon πk for k ≥ q + 1. When k = q we have the comparison diagram:(6.12)

πq Zeq /2G/ πq Zeq /2/ πq Zeq /2av

πq Zeq /2G
top/ πq Zeq /2top/ πq Zeq /2av

∼
=



∼
=



Φav



top

/ πq−1 Zeq /2G_/ πq−1 Zeq /2G

top

and so Φav induces an injection for k = q.
Considering now the comparison diagram (6.5) and using the five-lemma we havethat Φ induces an isomorphism on πk for k ≥ q + 2 and an injection for k = q + 1.
e qtop (XC (C)) = H q−k (X(R), Z/2) for k ≥ 2. In particBy Corollary 7.14, πk R
e q (X) = 0 for k ≥ q + 1, when q ≥ 1. By the homotopy equivalences
ular πk R
e q (X) ≃ Sing• Rq (X) (see Lemma 6.7) and the duality [Teh05, Theorem 5.14]
R≃
between reduced cycle and reduced cocycle spaces Rq (X) −→ Rd−q (X) the vanishe q (X) = 0 for k ≥ q + 1 is equivalent to the vanishing πk Rn (X) = 0 foring πk R
k ≥ dim X − n + 1.
Now let X be a smooth quasi-projective variety and let X ⊆ X be a projective
closure with closed complement Z = X\X. The result follows from the projective
case and the long exact sequence in homotopy groups induced by the homotopy
fiber sequence
Rn (Z) → Rn (X) → Rn (X).
Finally let X be an arbitrary quasi-projective variety. There is an increasing
filtration of closed subvarieties∅ = X−1 ⊆ X0 ⊆ X1 ⊆ · · · ⊆ Xd = X
such that Xi+1 \Xi is smooth and dim Xi = i. We proceed by induction, the casei = 0 is done. Consider the long exact sequence which arises from the homotopy
fiber sequence
Rn (Xi ) → Rn (Xi+1 ) → Rn (Xi+1 \Xi ).

VANISHING THEOREMS FOR REAL ALGEBRAIC CYCLES

35

Since the result holds for Xi by induction and for Xi+1 \Xi because it is smooth we
obtain the result for Xi+1 .
Remark 6.13. If X is a projective smooth real variety of dimension d it is provedin [Teh05, Theorem 6.7] that πk (Rd−1 (X)) = 0 for any k ≥ 3. Theorem 6.10 incase n = d − 1 improves this vanishing bound.
Example 6.14. If n = 0 and if X has no real points then R0 (X) = 0 and so
RL0 H∗ (X) = π∗ R0 (X) = H∗sing (X(R), Z/2) = 0.
Let P(H) denote the space of complex lines in the quaternions H = C ⊕ jC wherej 2 = −1. Multiplication by j defines an involution on P(H) and write Q for thecorresponding 1-dimensional real curve. We know that Q is the smooth real curve
X 2 + Y 2 + Z 2 = 0 in P2R . This is the Severi-Brauer variety corresponding to the
non-trivial element of Br(R) = Z/2 and has no real points. This means R0 (Q) = 0
and R1 (Q) = Z/2. Thus in this case,
0 = RL0 H0 (Q) = RL0 H1 (Q) = H0 (Q(R), Z/2)
and
Z/2 = RL1 H1 (Q).
2d+1

Let X = SP
(Q) be the smooth projective real variety given by an odd
symmetric power of Q. Because XC = PC (Hd+1 ), we have R2q (X) = Z/2 andR2q+1 (X) = 0 for any 2q ≤ 2d + 1 (see [LLFM05, Theorem 2.3]). This implies thatthe only nonzero reduced Lawson homology groups of X are RL2r+1 H2r+1 (X) =Z/2 for any r ≤ d. Notice that in this case dim(X) = 2d + 1.
These computations show that the vanishing in the above theorem is best possible, even in the case of a real variety with no real points.
Example 6.15. According to [Lam90], RLr Hn (PdR ) = Z/2 for any 0 ≤ r ≤ n ≤ d
and RLr Hn (PdR ) = 0 for any n > d.
We also obtain the following vanishing result.
Corollary 6.16. Let X be a smooth projective real variety of dimension d. Thenπn

Zp (XC )av=0
2Zp (XC )av

for n ≥ 2d − 2p + 1.
Proof. By the Corollaries 5.12, 5.11, A.5 and 4.20 we conclude thatπnZp (XC )av=0
2Zp (XC )G

for n ≥ 2d − 2p + 1 from the long exact sequence in homotopy groups induced by
the short exact sequence [Teh08, Proposition 4.3]
(6.17)0→Zp (XC )Zp (XC )av
Zp (XC )G→→ 0.
→
2Zp (XC )G
2Zp (XC )2Zp (XC )G

Consider the short exact sequences of topological abelian groups0→

Zp (XC )av
Zp (XC )av
2Zp (XC )G→→→ 0.
2Zp (XC )av
2Zp (XC )av
2Zp (XC )G36

JEREMIAH HELLER AND MIRCEA VOINEAGU

Multiplication by 2 induces a homeomorphism
Rp (X) =Zp (XC )G ∼ 2Zp (XC )G=Zp (XC )av
2Zp (XC )av

and plugging the vanishing for homotopy groups of Rp (X) into the above exact
sequence yields the result.
Remark 6.18. Using the same arguments as in Theorem 6.10 shows that the vanishing in Corollary 6.16 holds for any quasi-projective real variety.
Example 6.19. Let X = PdR . Then
Zp (PdC )av=0πn2Zp (PdC )av
for any n ≥ 2d − 2p + 1. If p = d, then
Zd (PdC )av= Z/2
π2d−2p2Zd (PdC )av
If p = 0 then, for any real projective variety X,Zp (XC )avπ∗= H∗ (X(C)/G, Z/2).
2Zp (XC )av
These computations show that the vanishing bound of Corollary 6.16 is the best
possible. For these computations see [LLFM05].
The following corollary shows that in a range the morphic cohomology of a realvariety X can be computed by the homotopy groups of average cycles on X.
Corollary 6.20. Let X be a real quasi-projective variety. Then
πq (Zp (XC )G ) ≃ πq (Zp (XC )av )for any q ≥ dim(X) − p + 1.
Proof. This follows from Theorem 6.10 together with the long exact sequence of
homotopy groups associated to the homotopy fiber sequence
0 → Zp (XC )av → Zp (XC )G → Rp (X) → 0.

Example 6.21.
(1) In case of divisors p = dim(X) − 1, Corollary 6.20 and[Teh08, Proposition 6.2] show that
πq (Zp (XC )av ) = 0
for any q ≥ 2.
(2) In the case of zero-cycles p = 0, we get
Hk,0 (X(C), Z) ≃ Hk (X(C)/G, Z)
for any k ≥ dim(X) + 1.
We conclude this section by observing that the vanishing theorem also shows that
motivic cohomology of a real variety can be computed in a range via the complex
of averaged equidimensional cycles on the complexification.

VANISHING THEOREMS FOR REAL ALGEBRAIC CYCLES

37

Let X and Y be a quasi-projective real varieties. The group of reduced equidimensional cycles is defined to be the quotient grouprequi (Y, r)(X) =zequi (YC , r)(XC )G.
zequi (YC , r)(XC )av

It is essentially a consequence of Suslin rigidity that the complex of reduced
equidimensional cycles computes the reduced Lawson homology.
Proposition 6.22. Let X be a quasi-projective real variety.
(1) The diagram≃≃

requi (Aq , q)(X × ∆•top ) ←− requi (Pq/q−1 , q)(X × ∆•top ) −→ Sing• Rq (X)consists of homotopy equivalences of simplicial sets.
(2) The map≃

requi (Pq/q−1 , 0)(X × ∆• ) −→ requi (Pq/q−1 , 0)(X × ∆• × ∆•top )is a homotopy equivalence of simplicial sets.
Proof. The proof is similar to other proofs in this paper so we only provide a sketch.
First observe that the simplicial abelian group of reduced equidimensional cycles
may be computed as
requi (Y, r)(X) =(zequi (YC , r)(XC ) ⊗ Z/2)G.
(zequi (YC , r)(XC ) ⊗ Z/2)av

Using Proposition 4.11 and the appropriate analogues of the homotopy fiber sequences (6.4) and (6.5) we see that
requi (Pq , 0)(X × ∆•top ) →(Sing• MorC (XC , C0 (PqC ))+ /2)G(Sing• MorC (XC , C0 (PqC ))+ /2)av

is a homotopy equivalence. The first part follows now in a similar fashion as Proposition 4.16. The second part follows from the fact that both over C and over R withfinite coefficients motivic cohomology agrees with morphic cohomology.

Corollary 6.23. Let X be a quasi-projective real variety. Thenp
HM
(X; Z(q)) = π2q−p zequi (AqC , 0)(XC ×C ∆•C )av

for q − 1 ≥ p
7. Reduced Topological Cocycles
For typographical simplicity throughout this section we write Z = Z/20 (S q,q )0 .
e qtop (W ) = H q−k (W G ; Z/2), forThis section is devoted to the computation that πk R
singe qtop (W ) is the quotient simplicial abelian groupk ≥ 2, where R
G
Homcts∗ W ∧ ∆•top,+ , Zqe
av
Rtop (W ) =Homcts∗ W ∧ ∆•top,+ , Z

and W is a based finite G-CW complex.
The idea is to reduce to the case of trivial action. Before doing this we sketchwhat happens when G acts trivially on W . By [LLFM03, Proposition 8.3] the short
exact sequence
q,q G
0 → Z/20 (S q,q )av)0 → R0 (S q,q )0 → 0
0 → Z/20 (S38

JEREMIAH HELLER AND MIRCEA VOINEAGU

is a fibration sequence (in fact principle
fibration sequence) of topological spaces.
Applying Homcts∗ W ∧ ∆•top,+ , − to this sequence yields a homotopy fiber sequence of simplicial sets. Now we compare the homotopy fiber sequences of simplicial abelian groups
av
G
/R
/ Hom W ∧ ∆•e qtop (W )Hom W ∧ ∆•top,+ , Z
top,+ , Z
Hom W ∧ ∆•top,+ , Z av∼

=
/ Hom W ∧ ∆•top,+ , Z G/ Hq (W ),where Hq (W ) = Homcts∗ W ∧ ∆•top,+ , R0 (S q,q )0 . We will see that when W has
trivial G-action then the left vertical arrow induces an isomorphism on πk for k ≥ 1
e qtop (W ) = πi Hq (W ) = H q−i (W ; Z/2) for i ≥ 2(see Corollary 7.9). Therefore πi R
singwhen W has trivial G-action.
For a based G-CW complex W and a topological G-module Z, write
Homcts∗ (W, Z)G0
for the set of based equivariant maps which are equivariantly homotopic to the
0-map (via a based homotopy).
Lemma 7.1. Let W be a based G-CW complex and let Z be a topological G-module.
The simplicial set
G
d 7→ Homcts∗ W ∧ ∆dtop,+ , Z 0
Gis the path-connected component of the vertex 0 ∈ Homcts∗ W ∧ ∆•top,+ , Z .
Gis in the same path component as
Proof. A vertex g ∈ Homcts∗ W ∧ ∆•top,+ , ZG
such
the 0-map if and only if there is a 1-simplex F ∈ Homcts∗ W ∧ ∆1top,+ , Z
Gthat F (0) = 0 and F (1) = g. This happens exactly when g ∈ Homcts∗ (W, Z)0 .
A d-simplex, f : W ∧ ∆dtop,+ → Z is in the path-component of 0 if and only if its
restriction to a vertex is in the path component of 0. Since ∆dtop,+ is equivariantly
contractible and the restriction f |W ∧{v}+ to a vertex is equivariantly homotopic tothe constant map 0 we conclude that f itself is equivariantly homotopic to 0.
Definition 7.2. Let W be a based G-CW complex.
av(1) Define Homcts∗ W ∧ ∆•top,+ , Z 0 to be the path-connected component of
av
the vertex 0 in the simplicial set Homcts∗ W ∧ ∆•top,+ , Z .
(2) Define
G
Homcts∗ W ∧ ∆•top,+ , Z 0
q
e
av ,
Rtop (W )0 =Homcts∗ W ∧ ∆•top,+ , Z 0
here the quotient is in the category of simplicial abelian groups.

Restricting to W G gives rise to the comparison map

e qtop (W )0 → R
e qtop (W ) → Homcts∗ W G ∧ ∆•top,+ , Z/20 (S q ) .
Re qtop (W ) is an isomorphism for i ≥ 2 and an injection
e qtop (W )0 → πi R
Note that πi R
e qtop (W ) we will show that R
e qtop (W )0 → R
e qtop (W G )0 is
for i = 0, 1. To compute πi R
an isomorphism. The surjectivity is easy but the injectivity takes some work.

VANISHING THEOREMS FOR REAL ALGEBRAIC CYCLES

39

Proposition 7.3. Let i : A ֒→ W be an equivariant cofibration between based
G-CW -complexes and let Z be a topological G-module. ThenG

Gi∗ : Homcts∗ (W, Z)0 → Homcts∗ (A, Z)0
is surjective.

Proof. Suppose that f : A → Z is a based equivariant map which is based equivariantly homotopic to the 0-map. Let H : A ∧ I+ → Z be an equivariant homotopy
such that H(−, 0) = 0 and H(−, 1) = f .
By the homotopy extension property of cofibrations, an equivariant map H ′making the diagram below commute exists`
W ∧ {0}+ A∧{0}+ A ∧ I+ 0∐Hm/6 Z
_
m
m m
m
′
m H
m m

.
W ∧ I+The restriction of f ′ = H ′ (−, 1) to A is equal to f and H ′ is an equivariant homotopy between f ′ and the 0-map.
Corollary 7.4. Let i : A ֒→ W be an equivariant cofibration between based G-CW complexes. The induced map
e q (W )0 → R
e q (A)0i∗ : R
top
top

is a surjection.
Proof. Consider the square

G
Homcts∗ W ∧ ∆•top,+ , Z 0e q (W )0R
top/ / Homcts∗

GA ∧ ∆•top,+ , Z 0/Re q (A)0 .
top

By the previous proposition, the top horizontal arrow is surjective. The verticalarrows are surjective by definition and therefore the bottom horizontal arrow is also
surjective.
For a based CW -complex W and a topological abelian group Z we will write
Homcts∗ (W, Z)0 for the set of based continuous maps which are based homotopic
to the 0-map. Note that the simplicial set

d 7→ Homcts∗ W ∧ ∆dtop,+ , Z 0is the path-connected component of the vertex 0 ∈ Homcts∗ W ∧ ∆•top,+ , Z (forexample consider W and Z with trivial G-action and apply Lemma 7.1). If Z and
W have a G-action write (Homcts∗ (W, Z)0 )av for the image of N = 1 + σ. Thisset consists of maps h : W → Z which can be written as h = f + f where f is acontinuous map which is nonequivariantly homotopic to 0.
We now justify the use of similar notation for two potentially different simpliav
cial sets. Previously we wrote Homcts∗ W ∧ ∆•top,+ , Z 0 for the path-component

40

JEREMIAH HELLER AND MIRCEA VOINEAGU

avof the vertex 0 ∈ Homcts∗ W ∧ ∆•top,+ , Z . We now verify that this pathcomponent can be described explicitly as the image under N = 1 + σ of the path- av
component of 0 ∈ Homcts∗ W ∧ ∆•top,+ , Z . In otherwords Homcts∗ W ∧ ∆•top,+ , Z 0 =(Homcts∗ W ∧ ∆•top,+ , Z 0 )av . This explicit description will be fundamental to ourproof of Proposition 7.6 below.
Proposition 7.5. Let W be a based G-CW -complex. The simplicial set

d 7→ (Homcts∗ W ∧ ∆dtop,+ , Z 0 )av
av
is the path-connected component of 0 ∈ Homcts∗ W ∧ ∆•top,+ , Z

Proof. First we identify (Homcts∗ (W, Z)0 )av as the set of vertices of the pathconnected component of 0 ∈ Homcts∗ (W, Z)av . Any f + f ∈ (Homcts∗ (W, Z)0 )av
av
is in the path component of 0. Suppose the vertex h + h ∈ Homcts∗ (W, Z) is in
the same path component as 0. This means there is a map of simplicial setsav
F : I → Homcts∗ W ∧ ∆•top,+ , Z
such that F (0) = 0 and F (1) = h + h. Consider the diagram of simplicial sets,0/ Homcts∗ W ∧ ∆•top,+ , Z
{0}
_
6
m m
′
m
F
N
≃
m m

 m mF mav/ Homcts W ∧ ∆•.
I
top,+ , Z∗

A surjection between simplicial abelian groups is a fibration and therefore an F ′exists to make the above square commute.
The map F ′ (1) : W → Z is in Homcts∗ (W, Z)0 and satisfies
F ′ (1) + F ′ (1) = N (F ′ (1)) = F (1) = h + h.
We conclude that (Homcts∗ (W, Z)0 )av is the set of vertices of the path-connectedavcomponent of the 0 ∈ Homcts∗ (W, Z) .

Now to conclude that the simplicial set d 7→ (Homcts∗ W ∧ ∆dtop,+ , Z 0 )av isthe path-connected component of 0 we need to see that if the restriction g|W ∧{v}+
av
of g ∈ Homcts∗ W ∧ ∆ntop,+ , Z
to a vertex v ∈ ∆ntop lies in (Homcts∗ (W, Z)0 )av
 av
av
n
then g ∈ (Homcts∗ W ∧ ∆top,+ , Z 0 ) . That is, if g ∈ Homcts∗ W ∧ ∆ntop,+ , Z
and that there is a map f : W → Z which is homotopic to 0 such that the restrictionof g to some vertex v ∈ ∆ntop satisfies g|W ∧{v}+ = f + f then we need to see that g
can be written g = f ′ + f ′ for some f ′ : W ∧ ∆ntop,+ → Z which is homotopic to 0.
For this we consider the lift f ′ of g,f/ Homcts∗ W ∧ ∆•top,+ , Z
{v}
_
6m
m
f′ m mN≃m
m
 m mgav/ Homcts W ∧ ∆•.
∆n∗
top,+ , Z

The map f ′ : W ∧ ∆ntop,+ → Z satisfies f ′ + f ′ = N (f ′ ) = g, the restriction of f ′ tov ∈ ∆ntop is homotopic to the 0-map and, since ∆ntop is contractible, f ′ is homotopic
to the 0-map as well.

VANISHING THEOREMS FOR REAL ALGEBRAIC CYCLES

41

Therefore we conclude thatd 7→ (Homcts∗ W ∧ ∆dtop,+ , Z 0 )av
av
is the path-component of 0 in Homcts∗ W ∧ ∆dtop,+ , Z .

Proposition 7.6. Let i : A ֒→ W be an equivariant cofibration. Then∗ie q (A)0 ].
e q (W/A)0 ։ ker[Re q (W )0 −
→R
R
top
top
top
∗

i
e qtop (A)0 ) is a d-simplex. Then [f ]e qtop (W )0 −→ R
Proof. Suppose that [f ] ∈ ker(Ris represented by an equivariant map f : W ∧ ∆dtop,+ → Z which is equivariantlyavhomotopic to 0. Since i∗ [f ] = 0 this means that i∗ f ∈ Homcts∗ A ∧ ∆dtop,+ , Z 0 .
Thus there is a continuous map h : A ∧ ∆dtop,+ → Z, (nonequivariantly homotopicto 0), such that i∗ f = f |A∧∆dtop,+ = h + h. Since h is (nonequivariantly) homotopic

to the 0-map, h : A∧∆dtop,+ → Z extends to a continuous map h′ : W ∧∆dtop,+ → Z
which is (nonequivariantly) homotopic to the 0-map.
Explicitely, let H : A ∧ ∆dtop,+ ∧ I+ → Z be a homotopy such that H(−, 0) = 0
and H(−, 1) = h. By the homotopy extension property for cofibrations, the dottedarrow exists in the diagram`A ∧ ∆dtop+ ∧ I+ A∧∆d ∧{0}+ W ∧ ∆dtop,+ ∧ {0}+ H∐0i/4 Z
top,+ i i_
i i
H′ i iii i

i i
W ∧ ∆dtop,+ ∧ I+ .
Now H ′ (−, 1) = h′ is the desired extension of h, H ′ is a homotopy between h′def

and the 0-map and F = f − (h′ + h′ ) represents the same class as [f ]. Since
F |A∧∆dtop,+ = 0 the map F defines the map F ′ : W/A ∧ ∆dtop,+ → Z such thatp

F = p∗ F ′ : W −→ W/A ∧ ∆dtop,+ → Z.
Thereforee qtop (W/A)0 ։ ker(i∗ : R
e qtop (W )0 → R
e qtop (A)0 ),R
because p∗ [F ′ ] = [p∗ F ′ ] = [F ] = [f ].

Lemma 7.7. (c.f. [LLFM03, Lemma 8.8]) Suppose that A• is a simplicial Gmodule. Thenav
|Aav
• | = |A• |Proof. Let f• : B• → C• be a map between simplicial sets, then | Im f• | = Im |f• |.
The lemma follows since (−)av is defined to be the image of the map N = 1 + σ. 
Proposition 7.8. Suppose that Y = |Y• | is the realization of a based G-simplicial
av
set. Then Z0 (Y )0 → Z0 (Y )av
0 and Z/20 (Y )0 → Z/20 (Y )0 are Serre fibrations.
Proof. If Y is a based set and A is an abelian group then define A⊗Y = ⊕y∈Y \{∗} A.
If Y• is a based G-simplicial set then A ⊗ Y• is a G-simplicial set. In case A =Z or A = Z/2 we have Z(|Y• |)0 = |Z ⊗ Y• | and Z/20 (|Y• |)0 = |Z/2 ⊗ Y• | (see[dS03b, McC69]). The map Z ⊗ Y• → (Z ⊗ Y• )av is a surjection between simplicial
abelian groups and so is a fibration of simplicial sets and similarly for Z/2 ⊗ Y• →(Z/2 ⊗ Y• )av .

42

JEREMIAH HELLER AND MIRCEA VOINEAGU

The realization of a Kan fibration is a Serre fibration and therefore both Z0 (Y )0 =NN|Z ⊗ Y• | −→ |(Z ⊗ Y• )av | = Z0 (Y )av→ |(Z/2 ⊗ Y• )av | =0 and Z/20 (Y )0 = |Z/2 ⊗ Y• | −av
Z/20 (Y )0 are Serre fibrations of topological spaces.
Below we apply this proposition to the cases Y = S q,q and Y = S q,q ∧ Z/2+
so we make explicit that these are realizations of G-simplicial sets. We consider
Z/2 as a simplicial set in the usual way. The simplicial set S 1,0 is the ordinary S 1with trivial action. The simplicial set S 0,1 is the simplicial whose nondegenerate
simplices are two vertices {0} and {∞} and two 1-simplices. The G-action fixes
the vertices and switches the 1-simplices. The realization of this simplicial set isthe usual S 0,1 . Now S p,q is the G-simplicial set S p,q = (S 1,0 )∧p ∧ (S 0,1 )∧q and its
realization is the usual S p,q .
Lemma 7.9. Let W be a based G-space with trivial action. Suppose that Z is aN
topological G-module such that Z −→ Z av is a Serre fibration. Thenav
Homcts∗ W ∧ ∆•top,+ , Z 0 = Homcts∗ W ∧ ∆•top,+ , Z av 0 .
Proof. Since W and ∆dtop have trivial actions,av
Homcts∗ W ∧ ∆•top,+ , Z/20 (Y )0 0 ⊆ Homcts∗ W ∧ ∆•top,+ , Z/20 (Y )av
0 0,

and we wish to see that it is onto.
Suppose that f : W ∧ ∆dtop,+ → Z av is a map which is homotopic to 0. Let H
be a homotopy between 0 and f and let H ′ be a lift of H,0/
W ∧ ∆dtop,+p7 Z
 _ ∧ {0}+
p
′
H p
≃
N
pp
p

p

H
/ Z av ,W ∧ ∆dtop,+ ∧ I+

which exists because the right-hand vertical map is a fibration. Finally the map′
f ′ (−) = H ′ (−, 1) satisfies f ′ + f = f and H ′ is a homotopy between 0 and f ′ . 
Proposition 7.10. Suppose that W has trivial G-action. Then for all n ≥ 0 and
all q ≥ 0,
e qtop (W ∧ Z/2+ )0 = {0}.
RProof. First recall [dS03b, Lemma 2.4] that given a finite G-set Z then there is a
G-homeomorphism∼=

defined by f 7→P

Hom∗ (Z+ , Z0 (Y )0 ) −→ Z0 (Y ∧ Z+ )0z∈Z f (z) ∧ z. This yields

G
Homcts∗ W ∧ Z/2+ ∧ ∆•top,+ , Z/20 (S q,q )0 0
av =Homcts∗ W ∧ Z/2+ ∧ ∆•top,+ , Z/20 (S q,q )0 0
G
Homcts∗ W ∧ ∆•top,+ , Z/20 (S q,q ∧ Z/2+ )0 0
av ==Homcts∗ W ∧ ∆•top,+ , Z/20 (S q,q ∧ Z/2+ )0 0

Homcts∗ W ∧ ∆•top,+ , Z/20 (S q,q ∧ Z/2+ )G
0 0
 ,
=
Homcts∗ W ∧ ∆•top,+ , Z/20 (S q,q ∧ Z/2+ )av
0 0

e q (W ∧ Z/2+ )0 =R
top

VANISHING THEOREMS FOR REAL ALGEBRAIC CYCLES

43

where the last equality follows from Lemma 7.9 and Proposition 7.8 because W
has trivial G-action. But since the action of G on S q,q ∧ Z/2+ is free we have theisomorphism∼=

Z/20 (S q,q ∧ Z/2+ )av→ Z/20 (S q,q ∧ Z/2+ )G0 −0and therefore
Homcts∗e qtop (W ∧ Z/2+ )0 =R
Homcts∗S n ∧ ∆•top,+ , Z/20 (S q,q ∧ Z/2+ )G0 0
 = {0}.
W ∧ ∆•top,+ , Z/20 (S q,q ∧ Z/2+ )av
0 0Recall that the action of G on a based set (Y, ∗) is said to be free if Y G = ∗.
Corollary 7.11. Suppose that W is a based finite G-CW complex with free Gaction. Thene qtop (W )0 = {0}.
Re qtop (Wn+1 )0 → R
e qtop (Wn )0 is an isomorphism for any
Proof. First we observe that R
n. Indeed since W is a free G-CW complex Wn /Wn−1 is a wedge of spheres of thee qtop (∨S n+1 ∧e qtop (Wn+1 /Wn )0 = R
form S n ∧ Z/2+ . By the previous proposition R
Z/2+ )0 = {0}.
e q (Wn+1 /Wn )0 ։ ker(Re q (Wn+1 )0 → R
e q (Wn )0 ) is
By Proposition 7.6, R
top
top
top
e q (Wn+1 )0 ⊆ R
e q (Wn )0 . By Corollary 7.4 this map issurjective and therefore R
toptope q (Wn+1 )0 = R
e q (Wn )0 . Since W = WN for large N
onto as well and therefore R
toptope q (W )0 = R
e q (W0 )0 = {0}.
we conclude that R

top
top

Corollary 7.12. Suppose that W is a finite G-CW -complex. Then∼= eq
e qtop (W )0 −R
→ Rtop (W G )0

is an isomorphism of simplicial abelian groups.

Proof. Consider the cofibration sequence W G ֒→ W → W/W G . The space W/W G
has a free G-action and so Proposition 7.6 and Corollary 7.11 imply thate q (W/W G )0 ։ ker[Re q (W )0 → R
e q (W G )0 ]{0} = R
top
top
top

e qtop (W )0 ⊆ R
e qtop (W G )0 . Since it is also a surjection byis surjective. Therefore R
Corollary 7.4 it is an isomorphism.
For a based G-CW complex W define
Hq (W ) = Homcts∗ W G ∧ ∆•top,+ , R0 (S q,q )0 .

The homotopy groups of Hq (W ) compute singular cohomology of the fixed point
q−kspace, πk Hq (W ) = Hsing(W G , Z/2).
Theorem 7.13. Let W be a finite G-CW -complex. Thene q (W )0 → πi Hq (W ) = H q−i (W G ; Z/2)
πi R
top
sing

is an isomorphism for i ≥ 2 and an injection for i = 0, 1.

44

JEREMIAH HELLER AND MIRCEA VOINEAGU

e q (W )0 = R
e q (W G )0 by Corollary 7.12
Proof. Since Hq (W ) = Hq (W G ) and R
top
top
G
we immediately reduce to the case that W = W . Since G acts trivially on W ,Lemma 7.9 and Proposition 7.8 giveG
Homcts∗ W ∧ ∆•top,+ , Z/20 (S q,q )0 0
q
e
av =
Rtop (W )0 =Homcts∗ W ∧ ∆•top,+ , Z/20 (S q,q )0 0

Homcts∗ W ∧ ∆•top,+ , Z/20 (S q,q )G
0 0
 .
=
Homcts∗ W ∧ ∆•top,+ , Z/20 (S q,q )av
0 0
By [LLFM03, Proposition 8.3] the short exact sequence

q,q G
0 → Z/20 (S q,q )av)0 → R0 (S q,q )0 → 0
0 → Z/20 (Sis a principle fibration sequence.
Finally comparing homotopy fiber sequences of simplicial abelian groups/ Hom W ∧ ∆•top,+ , Z G/Re q (W )0
Hom W ∧ ∆•top,+ , Z av 0
top
0


Hom W ∧ ∆•top,+ , Z av

yields the result.

/ Hom W ∧ ∆•top,+ , Z G/ Hq (W )Corollary 7.14. Let W be a finite G-CW -complex. Thene qtop (W ) → πi Hq (W ) = H q−i (W G ; Z/2)
πi R
sing

is an isomorphism for i ≥ 2.

e qtop (W ) is an isomorphism for i ≥ 2 and ane qtop (W )0 → πi R
Proof. The map πi R
injection for i = 0, 1Appendix A. Topological Monoids

In this appendix we collect a few simple results on topological monoids. By topological monoid we will mean a compactly generated Hausdorff topological abelianmonoid (and similarly for the phrase topological group). An abelian monoid M issaid to have the cancellation property if for any n, m, p ∈ M n + p = m + p implies
m = n.
Lemma A.1. Suppose that M is a topological monoid with the cancellation property. If + : M × M → M is closed and N ⊆ M is a closed submonoid then thequotient map π : M → M/N is closed.
Proof. Suppose that V ⊆ M is closed. Then since π : M → M/N is a quotient
map to see that πV is closed it is enough to see that π −1 πV ⊆ M is closed. But
π −1 πV = (V + N ) ∩ (M + N ) which is closed.

Lemma A.2. Let M be a topological monoid with the cancellation property andlet N ⊆ M be a submonoid. Suppose that M/N is a topological monoid, M + is a
topological group and N + is closed. Then the isomorphism of groups
 +
M+
M
→ +
N
N
is an isomorphism of topological groups.

VANISHING THEOREMS FOR REAL ALGEBRAIC CYCLES

45

Proof. The map M → M + → M + /N + sends N to 0 and so we obtain the monoid
homomorphism M/N → M + /N + which is continuous. This induces the continuous
group homomorphism φ : (M/N )+ → M + /N + .
On the other hand the topological monoid quotient map M → M/N induces thecontinuous group homomorphism M + → (M/N )+ . Since N + is mapped to 0 it
induces the continuous group homomorphism ψ : M + /N + → (M/N )+ .
The continuous maps ψ and φ are easily seen to be inverse to each other.

Recall that if A is a topological monoid with G-action we write Aav ⊆ A for theimage of N = 1 + σ, so Aav ⊆ A is the topological submonoid consisting of elementsof the form a + σa.
Proposition A.3. Suppose that M is a Hausdorff topological abelian monoid withthe cancellation property and that M + is a Hausdorff group. Suppose that G acts
on M . Then the isomorphism of groups∼=(M G )+ −→ (M + )G
is an isomorphism of topological groups. If (M + )av ⊆ M + is closed then∼=

(M av )+ −→ (M + )av
is an isomorphism of topological groups.
Proof. We just have to show that the “identity” map(M + )G → (M G )+is continuous.
The group completion M + is topologized as the quotientq

M ×M −→ M +.
where q(a, b) = a − b. The map q : q −1 (M + )G → (M + )G is again a quotient map
since (M + )G is closed.
id×σπ

∆×id

+

2
M × M × M −−−→ M ×4 −→ M × M which
Consider the map M × M −−−−−→
sends (a, b) 7→ (a, b, σb, σb) 7→ (a + σb, b + σb). This is a continuous map. Its
restriction to q −1 (M + )G is a continuous map q −1 (M + )G → M G × M G which
induces the identity map on quotients(M + )G → (M G )+ ,and therefore this is a continuous map.
The second statement for averaged cycles is proved in a similar fashion.

Lemma A.4. Let M be a topological monoid with the cancellation property. Suppose that M + is a topological group and 2M + is closed in M + . Then
M+
M
→
2M
2M +
is an isomorphism of topological abelian groups.
Proof. For m ∈ M write [m] for its image in M/2M . This quotient monoid is agroup since [m] + [m] = 0.
For (m, n) ∈ M + write [m, n] for its image in M + /2M +. The map M/2M →+M /2M + which sends [m] to [m, 0] is continuous because M → M/2M is a quotient. It is an injection because if [m, 0] = [0, 0] then there is (2n, 2n′ ) such that

46

JEREMIAH HELLER AND MIRCEA VOINEAGU

m+2n′ = 2n which says that [m] = 0. It is surjective since [m, n] = [m+n, n+n] =[m + n, 0]. The inverse M + /2M + → M/2M is continuous since it is the map[m, n] 7→ [m + n, 0] = [m + n, n + n].
Proposition A.5. Let X be a normal quasi-projective real variety.
(1) The continuous homomorphism N : Z q /2(XC ) → Z q /2(XC )av induces anisomorphism of topological groups
Z q /2(XC )→ Z q /2(XC )av .
Z q /2(XC )G(2) The continuous homomorphism Z q (XC )av → Z q /2(XC )av induces an isomorphism of topological groups
Z q (XC )av→ Z q /2(XC )av .
2Z q (X)G
Proof. Addition is a closed map for the monoid C0 (PqC )(XC ) (see the proof of Proposition 3.6) and therefore we conclude by Lemma A.1 that addition is also closed
q−1
for both the effective cocycles C q (XC ) = C0 (PqC )(XC )/C0 (PC)(XC ) and closed for
C q /2(XC ) .
∼=By Lemma A.4 the map C q /2(XC ) −→ Z q /2(XC ) is an isomorphism of topologicalgroups and therefore addition is closed for Z q /2(XC ). In particular N = 1 +σ : Z q /2(XC ) → Z q /2(XC ) is closed. Since Z q /2(XC ) is 2-torsion ker(N ) =Z q /2(XC )G . It now follows thatZ q /2(XC ) N−→ Z q /2(XC )av
Z q /2(XC )G
is an isomorphism of topological groups. For the second statement we need toconclude that the continuous bijection Z q (XC )av /2Z q (XC )G → Z q /2(XC )av has acontinuous inverse. Write g for the inverse. ThenZ q (XC )
Z /2(X)avq/ Z q (XC )av

g


q
av
Z
(X
C)
/
q
2Z (XC )G

is commutative and each map except possibly g is continuous. By the first part ofN
the proposition the composition Z q (XC ) → Z q /2(XC ) −→ Z q /2(XC )av is a quotient
map and therefore g is continuous. We conclude that the map is an isomorphismof topological groups.
Appendix B. Tractable Monoid Actions
We recall Friedlander-Gabber’s notion of tractability for a topological monoid.
Definition B.1. [FG93](1) If M is a Hausdorff topological monoid which acts on a topological space T ,the action is said to be tractable if T is the topological union of inclusions∅ = T−1 ⊆ T0 ⊆ T1 ⊆ · · ·VANISHING THEOREMS FOR REAL ALGEBRAIC CYCLES

47such that for each n ≥ 0 the inclusion Tn−1 ⊆ Tn fits into a push-out of
M -equivariant maps (with R0 empty)(B.2)

Rn × M −−−−→ Sn × M




y
y
Tn−1

−−−−→

Tn ,

where the upper horizontal map is induced by a cofibration Rn ֒→ Sn of
Hausdorff spaces.
The monoid M is said to be tractable if the diagonal action of M on
M × M is tractable.
(2) If in addition M , T are G-spaces say that the action of M on T is equivariantly tractable if the action map is G-equivariant, the Rn ֒→ Sn areequivariant cofibrations between G-spaces, and the pushout squares (B.2)
are G-equivariant.
Fixed points of an equivariant cofibration is a cofibration and fixed points preserve pushouts along a closed inclusion. Therefore if T is an equivariantly tractableM -space then it is in particular a tractable M -space and T G is a tractable M G space.
The most important feature of tractability is that the naive group completion
M → M + of a tractable monoid is a homotopy group completion [FG93].
It is useful to know that all of our topological groups have reasonable equivariant
homotopy types. Below we observe that this is the case by using essentially the same
reasoning as in [FW01b, Proposition 2.5]. The essential topological property usedhere is that Hironaka’s triangulation theorem implies that the complexification ofa real variety may be equivariantly triangulated (see for example [KW03, Theorem1.3]).
Proposition B.3. Suppose that T is a tractable M -space. If Rn , Sn have the homotopy type of a CW -complex then so does T /M Suppose that T is an equivariantly
tractable M -space. If Rn , and Sn have the equivariant homotopy type of a G-CW
complex then T /M has the equivariant homotopy type of a G-CW complex.
Proof. We prove the second statement, the first follows in the same manner by
discarding equivariant considerations. Modding out by the M -action in (B.2) we
obtain equivariant pushout-squares
Rn


y

−−−−→

Sn


y

Tn−1 /M −−−−→ Tn /M.
By induction and homotopy invariance of pushouts along G-cofibrations we see that
Tn /M has the homotopy type of a G-CW complex and that Tn−1 /M → Tn /M isa G-cofibration. By [Wan80, Theorem 4.9] we conclude that colimn Tn /M has thehomotopy type of a G-CW complex. We are done since T /M = colimn Tn /M by
the proof of [Fri98, Lemma 1.3].
Proposition B.4. Let E ⊆ X be a constructable subset of a real projective variety.
(1) The space EC has the homotopy type of a G-CW complex.

48

JEREMIAH HELLER AND MIRCEA VOINEAGU(2) Suppose that F ֒→ E is a closed constructable embedding. Then FC ֒→ EC
is an equivariant cofibration.
Proof. Let EC , FC be closures (in XC ) of EC and FC . There is an equivariant
triangulation of E C so that F C and E C \EC are subcomplexes [KW03, Theorem
1.3]. Then EC and FC are unions of open simplices. The deformation retract of EC
onto a subsimplicial complex given in the proof of [FW01b, Proposition 2.5] is anequivariant retract onto a G-simplicial complex (and similarly for FC ) which gives
the first statement. The construction of a deformation retract onto FC of an open
neighborhood U of FC in EC given in [FL97, Lemma C1] works equivariantly whichshows that FC ֒→ EC is a cofibration.

As shown in [FG93, Proposition 1.3] the Chow monoids associated to complex
varieties are tractable and in [FL97, Proposition C.3] these results are extended tocertain constructable submonoids of Chow monoids. Their proofs work equivariantly to give the equivariant analogue of their result. A submonoid N ⊆ M is said
to be full if whenever m + m′ ∈ N then both m, m′ ∈ N . The condition below
on E ⊆ Ck (X) in the proposition is satisfied if E is Zariski closed or if it is a full
submonoid.
Proposition B.5. Let X be a projective real variety and E ⊆ Ck (X) by a constructable submonoid such that + : E × E → E is a Zariski closed mapping. Then
(1) EC is an equivariantly tractable monoid.
(2) EC+ has the homotopy type of a G-CW complex.
(3) If F ⊆ E is a closed constructable embedding then EC is tractable as an
FC -space and EC /FC is an equivariantly tractable monoid.
(4) Suppose that F ֒→ E is a closed constructable embedding. Then F + ⊆ E +is closed and the sequence
0 → FC+ → EC+ → (EC /FC )+ → 0
is an equivariant short exact sequence of groups of spaces of the homotopy
type of a G-CW complex.
Proof. In [FL97, Proposition C.3] the monoid EC is shown to be tractable as follows.
Write E(d) = EC ∩ Ck,d (XC ) and let ν : N2 → N be a bijection such that if a ≤ c,b ≤ d then ν(a, b) ≤ ν(c, d). DefineandRn = Im 

Sn = E(an ) × E(bn ), where ν(an , bn ) = n
[

c≥0



E(an − c) × E(bn − c) × E(c) → E(an ) × E(bn ) ⊆ SnTn = Im ([ν(a,b)≤nE(a) × E(b)) × EC → EC × EC  .

The spaces Rn , Sn , and Tn are G-spaces, fit into the appropriate pushout squares,
and Rn ֒→ Sn is a closed constructable embedding since addition is closed on EC
therefore by Proposition B.4 the Rn ֒→ Sn are equivariant cofibrations. This showsthat E is equivariantly tractable. The third item follows from similar considerationof [FL97, Proposition C.3]. The second item follows by applying Proposition B.3.

VANISHING THEOREMS FOR REAL ALGEBRAIC CYCLES

49

For the last part write Rn′ , Sn′ , and Tn′ for the spaces above giving the tractabilityof FC . Then Rn′ ⊆ Rn and Sn′ ⊆ Sn are cofibrations. Considering the comparisonof pushouts[[′
Tn′ /FC = Tn−1
/FC Sn′ → Tn /EC = Tn−1 /EC Sn
R′nRn

we see by induction that Tn′ /FC ֒→ Tn /EC is a cofibration and in particular isclosed. Therefore FC+ = colimn Tn′ /FC ⊆ colimn Tn /EC = EC+ is a closed subspace[FP90, Proposition A.5.5]. Finally EC /FC )+ = EC+ /FC+ by Lemma A.2 which gives
the displayed exact sequence.

Spaces of algebraic cycles and algebraic cocycles on complex varieties are shownto have CW -structures or homotopy type of CW -spaces in [LF92, FW01b] and in[Teh05] for real varieties.
Corollary B.6. Let U be a quasi-projective real variety. Then the spaces Zk (UC ),Z/ℓk (UC ), Z q (UC ), Z q /ℓ(UC ) all have the homotopy type of a G-CW complex.
The spaces Zk (UC )av , Z/ℓk (UC )av , Rk (U ), Z q (UC )av , Z q /ℓ(UC )av , and Rq (UC )all have the homotopy type of a CW -complex.
Proof. That Zk (UC ) has the homotopy type of a G-CW complex follows immediately from the previous proposition. Let U ⊂ U be a projectivization. Write
q−1
q−1
F0 (PC)(UC ) = E0 (PC)(UC ) + Cd (PqC × U C ). This is a closed constructable subq−1
q−1
monoid F0 (PC )(UC ) ⊆ E0 (PqC )(UC ) and (E0 (PqC )(UC )/F0 (PC)(UC ))+ ∼= Z q (UC )has the equivariant homotopy type of a G-CW complex. Since (ℓCk (UC ))+ ⊆Ck (UC )+ is closed we easily see that (ℓCk (UC ))+ = ℓ(Ck (UC ))+ ⊆ Ck (UC )+ . Therefore Z/ℓk (UC ) ∼= (Ck (UC )/ℓCk (UC ))+ has the equivariant homotopy type of a Gq
q−1
CW complex. Similarly one sees that Z q /ℓ(UC ) ∼= (E0 (PC )(UC )/F0 (PC )(UC ) +qℓE0 (PC )(UC ))+ has the equivariant homotopy type of a G-CW complex.
The monoid inclusions ℓCk (UC ) ∩ Ck (UC )av ⊆ Ck (UC )av ⊆ Ck (UC )G ⊆ Ck (UC ) are
all closed and so Zk (UC )av , Rk (U ) ∼== (Ck (UC )G /Ck (UC )av )+ , and Z/ℓk (UC )av ∼avav +(Ck (UC ) /ℓCk (UC ) ∩ Ck (UC ) ) all have the homotopy type of a CW -complex.
qq−1
qSimilarly Z q (UC )av ∼== (E0 (PC )(UC )av /F0 (PC )(UC )∩E0 (PC )(UC ))+ , Z q /ℓ(UC )av ∼
q
q−1
q
q
av
(E0 (PC )(UC ) /(F0 (PC )(UC ) + ℓE0 (PC )(UC )) ∩ E0 (PC )(UC )av )+ , and Rq (UC ) ∼=q−1(E0 (PqC )(UC )G /F0 (PC)(UC )G + E0 (PqC )(UC )av )+ all have the homotopy type of a
CW complex.

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