QUATERNIONIC ALGEBRAIC CYCLES AND REALITY
PEDRO F. DOS SANTOS AND PAULO LIMA-FILHO
Abstract.In this paper we compute the equivariant homotopy type of spaces*
* of algebraic cycles
on real Brauer-Severi varieties, under the action of the Galois group Gal*
*(C=R). Appropriate sta-
bilizations of these spaces yield two equivariant spectra. The first one *
*classifies Dupont/Seymour's
quaternionic K-theory, and the other one classifies and equivariant cohom*
*ology theory Z*(-) which
is a natural recipient of characteristic classes KH*(X) ! Z*(X) for quate*
*rnionic bundles over Real
spaces X.
Contents
Introduction *
* 2
1. Preliminary results from equivariant homotopy theory *
* 6
1.1. Coefficient systems and Mackey functors *
* 6
1.2. Dold-Thom theorem *
* 7
1.3. Motivic notation *
* 7
1.4. Real bundles and equivariant Chern classes *
* 7
1.5. Thom isomorphism for real bundles *
* 8
1.6. Poincar'e duality *
* 9
1.7. Relation with group cohomology and Galois-Grothendieck cohomology *
* 10
1.8. The RO(Z=2)-graded cohomology of the Brauer-Severi curve P(H) *
* 12
2. The Z=2-homotopy type of zero-cycles *
* 13
3. Quaternionic algebraic cycles and the join pairing *
* 16
3.1. Equivariant homotopy type of algebraic cycles *
* 17
3.2. Stabilizations of cycle spaces *
* 19
4. Quaternionic K-theory *
* 22
4.1. Classifying spaces and equivariant quaternionic K-theory spectrum *
* 22
5. Characteristic Classes *
* 25
5.1. Cohomology of (Z x BU)H *
* 25
5.2. Projective bundle formula *
* 28
5.3. The quaternionic total Chern class map *
* 29
5.4. The group struture on Z0H(X) *
* 31
5.5. Remarks on the space (Z x BU)H *
* 33
References *
* 34
___________
Date: July 2001.
The first author was supported in part by FCT (Portugal) through program POCT*
*I. The second author was
partially supported by NSF..
1
2 DOS SANTOS AND LIMA-FILHO
Introduction
In [Ati66] Atiyah developed a K-theory for spaces (X, oe) with an involution *
*oe, the Real spaces
in his terminology. The construction uses the notion of a Real bundle (E, ø) o*
*ver (X, oe) which
consists of a complex vector bundle E over X, along with an anti-linear map ø :*
* E ! E covering
the involution oe and satisfying ø2 = 1. The group KR(X) is then defined as the*
* Grothendieck
group of the monoid of isomorphism classes of Real bundles over (X, oe), and th*
*e resulting theory
is called KR-theory.
In a similar fashion, J. Dupont developed in [Dup69 ] the symplectic K-theory*
* KSp(X) for Real
spaces (X, oe). His construction is similar to Atiyah's, in that KSp(X) is the *
*Grothendieck group of
the monoid of isomorphism classes of symplectic bundles over (X, oe). In this c*
*ontext, a symplectic
bundle (E, ø) over (X, oe) consists of a complex vector bundle E over X, along *
*with an anti-linear
map ø : E ! E covering the involution oe and satisfying ø2 = -1. Subsequently, *
*R. M. Seymour
reintroduced this theory in [Sey73], where he called it quaternionic K-theory a*
*nd denoted it by
KH (X). We adopt this terminology, for it avoids confusion with the non-equiva*
*riant notion of
symplectic K-theory.
A clear and conceptual reason for the existence of these two competing theori*
*es arises when one
tries to find their respective classifying spaces in the equivariant category. *
*In fact, one can extend
these theories to RO(Z=2)-graded cohomology theories KR* and KH* in the sense o*
*f [Seg68] and
[May77 ]. To this purpose, one constructs Z=2-spaces (Z x BU)C and (Z x BU)H sa*
*tisfying
[X+, (Z x BU)C]Z=2~=KR(X) and [X+, (Z x BU)H]Z=2~=KH(X);
cf. [LLFM98b ] and Proposition 4.4, respectively. These spaces are shown to h*
*ave the structure
of Z=2-equivariant infinite loop spaces in [LLFM98b , ] and Theorem 4.7, respec*
*tively, yielding the
spectra classifying the desired equivariant cohomology theories.
In order to construct such classifying spaces, we first identify Z=2 - the un*
*derlying group of the
equivariant category - with the Galois group Gal(C=R). Recall that the Brauer g*
*roup [Gro57] Br(R)
of R is also isomorphic to Z=2. This will be shown to account for the two disti*
*nct K-theories in
Section 4. The argument is roughly the following. Let P(Cn) denote the projecti*
*ve space of complex
1-dimensional subspaces of Cn. In the language os schemes, this is the set of c*
*omplex-valued points
of Pn-1, endowed with the analytic topology. The Galois group Z=2 = Gal(C=R) ac*
*ts on P(Cn)
via complex conjugation. Similarly, let H = C Cj denote the quaternions, and*
* let P(Hn) be
the projective space of complex 1-dimensional subspaces of Hn. We give P(Hn) t*
*he Z=2-action
induced by multiplication by j on the left of Hn. As a space, P(Hn) is homeomor*
*phic to P(C2n),
however the Z=2-actions on P(C2n) and P(Hn) are quite distinct. In fact, these*
* spaces are the
complex-valued points of the two inequivalent Brauer-Severi schemes of rank 2n *
*- 1 over R, under
the action of the Galois group. The aforementioned classifying spaces (Z x BU)C*
* and (Z x BU)H
are then constructed using the usual equivariant stabilization of the Grassmann*
*ians of complex
linear subspaces of P(Cn) and P(Hn), respectively.
In order to develop a theory of characteristic classes for KR* and KH*, one n*
*eeds to introduce
the appropriate equivariant cohomology theories. In [Kah87 ] B. Kahn defined ch*
*aracteristic classes
for Real bundles, taking values in Galois-Grothendieck cohomology with coeffici*
*ents in the Z=2-
modules Z(n). In the case of quaternionic bundles, Dupont poses in [Dup99 ] the*
* question of which
equivariant cohomology theory would be the natural target of characteristic cla*
*sses, but the question
was left unanswered.
QUATERNIONIC ALGEBRAIC CYCLES AND REALITY *
* 3
In this paper we provide an answer to this question by constructing a cohomol*
*ogy theory Z*Hin
which the characteristic classes for quaternionic bundles take its values. Furt*
*hermore, we extend
these characteristic classes to a natural transformation of RO(Z=2)-graded coho*
*mology theories
KH* ! Z*H. A crucial aspect of our construction is the fact that the two distin*
*ct Brauer-Severi
varieties P(Cn) and P(Hn), used in the construction of the classifying spaces f*
*or KR* and KH*,
are also used to construct classifying spaces for the corresponding cohomology *
*theories. Under this
approach, the classifying maps for the characteristic classes have a similar de*
*scription in both cases.
In order to place our constructions under the proper perspective, let us desc*
*ribe how the char-
acteristic classes for Real bundles were extended to KR* in [LLFM98b ] and [dS0*
*0]. The main
constructions go back to [LM91 ], [BLLF+93 ] and [LLFM96 ].
Let Zq(P Cn ) denote the group of algebraic cycles of codimension q in P Cn .*
* This is an
abelian topological group on which Gal(C=R) acts via topological automorphisms.*
* See [LLFM98a ]
for details and additional references. The equivariant homotopy type of Zq(P Cn*
* ) was determined
in [dS00], and it turns out to be a product of classifying spaces for equivaria*
*nt coohomology with
coefficients in the constant MacKey functor Z_. More precisely, one has a cano*
*nical equivariant
homotopy equivalence
n
Zq(P C ) ~=Z x K(Z(1), 2) x K(Z(2), 4) x . .x.K(Z(q), 2q).
The associated equivariant cohomology theory is RO(Z=2)-graded (bigraded, in th*
*is case) and the
resulting invariants arise naturally in Z=2-homotopy theory. The functor repres*
*ented by K(Z(q), 2q)
is denoted H2q,q(-; Z_). See Section 1.3 for notation.
The situation in KR-theory follows a standard, albeit non-trivial, pattern. A*
* canonical stabiliza-
tion lim-!Grq(Cn)in the Z=2-homotopy category, of Grassmannians as Gal(C=R)-spa*
*ces, produces
a classifying space BUC for KR-theory. It is easy to see that the Bredon cohomo*
*logy of BUC with
coefficients in Z_is a polynomial ring over Z on certain characteristic classes*
*, the equivariant Chern
classes for KR-bundles. Furthermore, the inclusion of Grq(Cn) ,! Zq(P Cn ) stab*
*ilizes to give a
map of equivariant infinite loop spaces
Y
c : BUC ! Z ~= K(Z(p), 2p),
p
which classifies the total Chern class.
In this paper we provide the quaternionic counterpart of the constructions in*
* KR*-theory de-
scribed above. This turns out to be a more subtle issue, and our answer was ins*
*pired by the following
observation, made in [Ati66] and [Dup69 ]. If X be a Real space, then one has a*
*n isomorphism:
(1) KR(X x P(H)) ~=KR(X) KH(X).
The first task is to determine the Z=2-homotopy type of Zq(P Hn ), under the *
*Z=2-action induced
by j, as explained above. This is the action induced by the Gal(C=R) action on *
*the complex points
P(Hn) of the Brauer-Severi variety of rank n - 1 over R. This problem was firs*
*t considered in
[LLFM98b ], where it was proved that quaternionic suspension, = H: Zq(P Hn ) ! *
*Zq(P Hn+1 ),
is a Z=2-homotopy equivalence. This reduced the problem to computing the Z=2-ho*
*motopy type
of cycle spaces of dimensions 0 and 1. Their homotopy type is quite distinct, a*
*nd this somehow
reflects the sharp difference between quaternionic bundles of even and odd comp*
*lex rank. Later, in
[LLFM98c ] the homotopy type of the space of cycles Zq(P Hn )Z=2was computed us*
*ing suspension
to a real bundle other than O(1) (which corresponds to complex suspension).
4 DOS SANTOS AND LIMA-FILHO
Based on the techniques of [LLFM98c ], and on the Z=2-equivariant perspective*
* of [dS99a], we
first establish the following splitting result.
Theorem 3.4. For k < n there are canonical equivariant homotopy equivalences:
n Yk
(2) Z2k-1(P H ) ~= F (P(H)+, K(Z(2j - 1), 4j - 2) )
j=1
and
n Yk
(3) Z2k(P H ) ~= F (P(H)+, K(Z(2j), 4j) )
j=1
where F (-, -) denotes based maps.
Note that the spaces F (P(H)+, K(Z(q), 2q))are classifying spaces for the coh*
*omology functors
H2q,q(- x P(H); Z_). This result completely determines the Z=2-homotopy type of*
* Zm (P Hn ).
We then apply a suitable stabilization procedure using the spaces Zq(P Hn ). *
* The resulting
space ZH has the property that all of its connected components are products of *
*classifying spaces
for the functors H2*,*(- x P(H); Z_), according to the splitting of Theorem 3.4.
Theorem 3.9. The space ZH is written as a disjoint union of connected spaces
1a
ZH = ZjH,
j=-1
where the equivariant homotopy type of ZjHis totally determined by
(Q 1
F (P(H)+, K(Z(2k - 1), 4k - 2),)if j is odd
(4) ZjH~= Q k=11
k=1 F (P(H)+, K(Z(2k), 4k) ) , if j is even.
Using standard results in equivariant homotopy theory (cf. [CW91 ]), we prove*
* that the complex
join pairing on algebraic cycles induces an equivariant infinite loop space str*
*ucture on ZH. We
denote by Z*Hthe resulting equivariant cohomology theory. Note that for a compa*
*ct Z=2-space X,
one has an identification
M h ji M Y
(5) Z0H(X) = [X, ZH]Z=2 = X, ZH = H4r-2ffl(j), 2r-ffl(j)(X x *
*P(H), Z_),
j2Z Z=2 j2Zr 1
where ffl(j) is 0 if j is even and 1 if j is odd. The group structure on Z0H(X*
*), coming from the
H-space structure on ZH induced by the algebraic join of cycles, has the follow*
*ing description.
Proposition 5.10. Let X be a Z=2-space, and let a.b denote the product of eleme*
*nts a, b in Z0H(X).
Consider Z0H(X) included in M Y
Hr,s(X x P(H), Z_),
j2Zr,s 1
as in (60). Then, under this inclusion we have,
a . b = a [ b + pr*(a=z) [ pr*(b=z),
where z 2 H2,1(P(H); Z_) is the fundamental class P(H), -=z denotes slant produ*
*ct with z and pr is
the projection onto the first factor in the product X x P(H).
QUATERNIONIC ALGEBRAIC CYCLES AND REALITY *
* 5
A similar stabilization procedure is then applied to the Grassmannians Gq(Hn)*
* and the result
is an equivariant infinite loop space (Z x BU)H = qj2Z BUjHwhich classifies KH**
*, as described
above. The inclusion Grq(Hn ) Zq(P Hn ) induces a total Chern class map cH : *
*(ZxBU)H ! ZH,
which turns out to be a map of equivariant infinite loop spaces; cf. (48).
In order to understand this Chern class map, we compute the equivariant cohom*
*ology group
Z0H((Z x BU)H) = [ (Z x BU)H, ZH ]Z=2. It follows from Proposition 5.10 that we*
* first need to
compute the cohomology ring H*,*(BUjHx P(H), Z_).
If E is a Real bundle over a Z=2-space X, denote by eck(E) 2 H2k,k(X; Z_) its*
* k-th equivariant
Chern class, as described in [dS00]. Let ,2nbe the universal quotient bundle ov*
*er BU2nH, and observe
that ,2n O(1) is a real bundle over BU2nHx P(H). Define classes dk 2 H2k,k(BU*
* 2nHx P(H); Z_)
using the formulas
d2n-(2i+ffi):= ec2n-(2i*ffi)(,2n O(1)) - i ec2n-(2i*ffi)-1(,2n *
*O(1)) x,
for 0 ffi 1 and 0 2i + ffi 2n, and where x 2 H2,1(P(H), Z_) ~=Z is the *
*canonical generator; cf.
Section 1.8.
Theorem 5.5. Let dk be the classes defined above. Then we have ring isomorphisms
H*,*(BU evHx P(H); Z_) ~=H*,*(BU oddHx P(H); Z_) ~=H*,*(P(H); Z_) [d1, d2,*
* . .,.dk, . .]..
The cohomology ring H*,*(P(H); Z_) is computed in Section 1.8.
The total Chern class map cH : BU H = qj2ZBUjH! ZH = qj2ZZjHsends the compone*
*nt BUjH
to the component ZjH. Its equivariant homotopy type is determined by the follow*
*ing result.
Theorem 5.9. The equivariant cohomology classes determined by total quaternioni*
*c Chern class
map cH and the splitting (41)of Theorem 3.9 are given by
(6) 1 + d2+ d4+ . .+.d2n+ . . .on BU evH
(7) d1+ d3+ . .+.d2n+1+ . . . on BU oddH
In [dSLF01 ] we work in the category of real algebraic varieties, addressing *
*the issue of replacing
continuous maps by morphisms of algebraic varieties. The K-theoretic constructi*
*ons yield a semi-
topological quaternionic K-theory for real varieties. This is related to Friedl*
*ander-Walker's semi-
topological K-theory for real varieties [FW01 ] in the same way as Seymour's KH*
** is related to
Atiyah's KR*. Furthermore, we establish relations to the algebraic K-theory of *
*real varieties and
to Quillen's computation of the K-theory of Brauer-Severi varieties in [Qui73].*
* In the level of
öm rphism spaces" into algebraic cycles, the splittings of Theorem 3.4 still ho*
*ld. Using them we
introduce the the quaternionic morphic cohomology for real varieties, and discu*
*ss Chern classes for
the quaternionic K-theory of real varieties.
This paper is organized as follows: in Section 1 we introduce the necessary b*
*ackground from Z=2-
homotopy theory needed to state our results. In Section 2 we establish a canoni*
*cal splitting for the
space of zero cycles on P Hn . In Section 3 we compute theZ=2-homotopy type of *
*Zq(P Hn ) and
define the infinite loop space of stabilized cycles ZH. In Section 4 we apply t*
*he same stabilization
procedure to the Grassmannians Gq(Hn), obtaining an equivariant infinite loop s*
*pace (Z x BU)H.
We show that (Z x BU)H classifies Dupont's quaternionic K-theory. Section 5 is*
* dedicated to
computations involving the characteristic classes for quaternionic bundles defi*
*ned in Section 4; a
6 DOS SANTOS AND LIMA-FILHO
projective bundle formula is proved and the characteristic classes for the univ*
*ersal quaternionic
bundle over (Z x BU)H are computed.
Acknowledgement. The authors would like to thank H. Blaine Lawson, Jr., for fru*
*itful conver-
sations during the elaboration of this work. The first author thanks Texas A&M *
*University and
the second author thanks the Instituto T'ecnico Superior (Lisbon) for their res*
*pective hospitality
during the ellaboration of this work.
1. Preliminary results from equivariant homotopy theory
In this section we review the definitions and results from equivariant homoto*
*py theory needed
for the purposes of this paper. Throughout this section G will be an arbitrary *
*finite group, and
later on we will specialize to the case G = Z=2.
Notation 1.1. If V is a representation of G, SV denotes the one point compactif*
*ication of V and,
for a based G-space X, VX denotes the space of based maps F (SV , X). The spac*
*e F (SV , X) is
equipped with the its standard G-space structure. The set of equivariant homoto*
*py classes [SV , X]G
is denoted by ßV(X). Given a G-space X, we denote by X+ the pointed G-space X [*
* {+}, where
+ is a point fixed by G.
1.1. Coefficient systems and Mackey functors. Let FG be category of finite G-se*
*ts and G-
maps. The coefficients for ordinary equivariant (co)homology are (contravariant*
*) covariant functors
from FG to the category Ab of abelian groups which send disjoint unions to dire*
*ct sums.
Given a contravariant coefficient system M there are Bredon cohomology groups*
* H*(-; M) with
coefficients in M. They satisfy G-homotopy invariance and the suspension axiom*
*, and they are
classical cohomology theories in the sense that they satisfy the dimension axio*
*m H0(pt; M) = M
and Hn(pt; M) = 0, for n > 0.
There are certain coefficient systems - called Mackey functors - for which Br*
*edon cohomology
can be extented to an RO(G)-graded theory. A Mackey functor M is a pair (M*, M**
*) of functors
M* : FG ! Ab and M* : FopG! Ab with the same value on objects and which transfo*
*rm each
pull-back diagram
A --f--!B
? ?
g?y ?yh
C --k--!D
in FG into a commutative diagram in Ab
M*(f)
M(A) ----! M(B)
x x
M*(g)?? ??M*(h)
M*(k)
M(C) ----! M(D)
Example 1.2. In this paper we are interested in the case where M = Z_is the Mac*
*key functor
constant at Z. This Mackey functor is uniquely determined by the following cond*
*itions; cf. [May86 ,
Prop. 9.10].
(i)Z_(G=H) = Z, for H G;
QUATERNIONIC ALGEBRAIC CYCLES AND REALITY *
* 7
(ii)if K H, the value of the contravariant functor Z_*on the projection æ : *
*G=K ! G=H is the
identity.
The RO(G)-graded cohomology groups with coefficients in a Mackey functor M ar*
*e denoted
H*(-; M) and the corresponding reduced cohomology groups are denoted eH*(-; M).*
* For each
real orthogonal representation V there is a classifying space K(M, V ) such tha*
*t, for any G-space
X,
HV (X; M) ~=[X+, K(M, V )]G.
The spaces K(M, V ) fit together to give an equivariant Eilenberg-Mac Lane spec*
*trum HM, i.e.,
given G-representations V , W , there is a G-homotopy equivalence K(M, V ) ~= W*
* K(M, V + W )
satisfying various compatibility properties; cf. [May96 ]. This implies that H**
*(-; M) satisfies the
suspension axiom in the direction of any representation:
eHV +W(SV ^ X; M) ~=eHW(X; M).
1.2. Dold-Thom theorem. Our interest on the Mackey functor Z_lies in the fact t*
*hat just as the
spaces of zero cycles (of degree zero) on the sphere Sn is a model for the non-*
*equivariant Eilenberg-
MacLane space K(Z, n), zero cycles on a representation sphere SV provide a mode*
*l for K(Z_, V ).
This is a consequence of the following equivariant version of the classical Dol*
*d-Thom theorem.
Notation 1.3. Let X be a G-space. ThePtopological group of zero cycles on X is*
* denoted by
Z0(X) . Its elements are formal sums inixi, with ni2 Z and xi2 X. There is an*
* augmentation
homomorphism deg : Z0(X) ! Z, whose kernel we denote by Z0(X) o. Note that Z0*
*(X) is
isomorphic to Z0(X+) o.
Theorem 1.4. [dS99b] Let G be a finite group, let X be a based G-CW-complex and*
* let V be a
finite dimensional G-representation, then there is a natural equivalence
ßVAG(X) ~=eHGV(X; Z_).
In particular, AG(SV ) is a K(Z_, V ) space.
1.3. Motivic notation. From now on we restrict ourselves to the case of G = Z=2*
*. We will use
motivic notation for Z=2-equivariant cohomology, for it is compatible with the *
*invariants used in
algebraic geometry.
Notation 1.5. Let s be the one dimensional real sign representation of Z=2 and *
*let 1 stand for
the one dimensional trivial representation. Then RO(Z=2) = Z . 1 + Z . s. With *
*p q, we write
(1)Rp,qfor (p - q) . 1 q . s;
(2)Sp,qfor Rp,q[ {1};p,q
(3)Hp,q(-; Z_) for HR (-; Z_);
(4)K(Z(q), p) for K(Z_, Rp,q).
1.4. Real bundles and equivariant Chern classes. Recall that a real bundle over*
* a Z=2-space
X, in the sense of [Ati66], is a complex bundle , ! X with a bundle map ø : , !*
* , which is an
anti-linear involution that covers the involution on X. Atiyah's KR-theory of X*
* is defined [Ati66]
as the Grothendieck group of isomorphism classes of real bundles over X.
It turns out that KR(-) is a Z=2-equivariant cohomology theory in the sense t*
*hat it is represented
by a Z=2-spectrum as defined above. In fact, Z x BU has a natural Z=2-action in*
*duced by complex
8 DOS SANTOS AND LIMA-FILHO
conjugation, and we denote this Z=2-space by (Z x BU)C. This is the classifying*
* space for KR-
theory, i.e., for a Z=2-space X,
KR(X) ~=[X+, (Z x BU)C]Z=2.
In [Ati66] it is proved that the usual periodicity of ZxBU is actually a (2, 1)*
* periodicity, i.e., there
is an equivariant homotopy equivalence
Z x BU ~= 2,1(Z x BU).
Hence (Z x BU)C is the zero-th space of a periodic Z=2-spectrum and there are g*
*roups KRp,q(-),
for p, q 2 Z, satisfying suspension, exact sequences and Z=2-homotopy invarianc*
*e. One can define
Chern classes for real bundles with values in H*,*(-; Z_) as usual, by pulling *
*back certain classes
from H*,*(BUC; Z_).
Theorem 1.6. There exist unique classes ecn2 H2n,n(BUC; Z_) whose image under t*
*he forgetful
map to singular cohomology is the n-Chern class cn 2 H2n(BUC; Z). Furthermore,*
* we have the
following ring isomorphism
H*,*(BUC; Z_) ~=R[ce1, . .,.ecn, . .].,
where R is the cohomology ring of a point, H*,*(pt; Z_).
Proof.This follows from the fact that BUC has an equivariant cell decomposition*
* given by the
Schubert decomposition.
Definition 1.7. For a virtual real bundle , over X, with a classifying map f : *
*X ! BUC the n-th
equivariant Chern class is
f*(cen) 2 H2n,n(X; Z_).
The equivariant Chern classes satisfy the Whitney sum and projective bundle for*
*mulas.
1.5. Thom isomorphism for real bundles. As in non-equivariant homotopy theory, *
*the exis-
tence of a Thom-isomorphism for some cohomology theory is directly related to t*
*he existence of
orientations in that theory. We will see that real bundles are HZ_-orientable a*
*nd hence there is a
corresponding Thom isomorphism theorem for real bundles.
Definition 1.8 ([May96 ]). Let G be a finite group. Let , p-!X be an n-plane G-*
*bundle over a
G-space X. An HZ_-orientation of , is an element ~ of HZ_ff(T (X)) for some ff *
*of virtual dimension
n, such that, for each inclusion i : G=H ! X the restriction i*~ to T(i*,)is a *
*generator of the free
HZ_*(S0)-module HZ_*H(T (i*,)).
Proposition 1.9. Let X be a real space and let , p-!X be a real bundle. Then , *
*is HZ_-orientable.
Proof.It suffices to consider the case where , is a real line bundle. In this c*
*ase we observe that
T (,)= P , C =P , and set ~ = c1(~) where ~ ! P , C is the tautological l*
*ine bundle
and c1 denotes the real first Chern class. Since ~|P , is trivial, then ~ desce*
*nds to a class in the
cohomology of T (,), which is also denote by ~. Consider an equivariant map i :*
* (Z=2)=H ! X.
There are two possible cases.
(i)H = Z=2: In this case T (i*,)= P C2 with the Z=2-action given by complex *
*conjugation.
Then c1(~)|P C2 is the first Chern class of the tautological bundle over P*
* C2 , which is a
generator for H*,*(P C2 , Z_) over HZ_*(S0) = H*,*(pt, Z_).
QUATERNIONIC ALGEBRAIC CYCLES AND REALITY *
* 9
(ii)H = {0}. In this case T (,)= Z=2+ ^ S2,0. We have H*,*(Z=2+ ^ S2,0, Z_) ~=*
*H*(S2; Z) and
KR(Z=2+ ^ S2,0) ~=K(S2). It is easy to see that, under these isomorphisms*
*, c1(~) is the
usual Chern class and hence it generates H*,*(Z=2+ ^ S2,0, Z_).
Definition 1.10. Given a vector bundle , ! X over X, the projection p : P , 1*
* ! X along
with the quotient map q : P , 1 ! T(,)= P , 1 =P , induce a map : T (,)*
*! T(,)^ X+
called the Thom diagonal of ,.
Proposition 1.11 (Thom isomorphism for real bundles). Let , -p!X be a real n-bu*
*ndle over a
real space X and let ~ 2 H2n,n(T (,), Z_) be an orientation for ,. Then
[~ : Hp,q(X+, Z_) -! eH2n+p,n+q(T (,), Z_)
is an isomorphism for all p, q. Furthermore, there is an equivariant homotopy e*
*quivalence
~= 2n,n
OE~ : Z0(T (,))o-!Z0 X+ ^ S o
which induces the Thom isomorphism in homology
Hep,q(T (,), Z_) ~=Hp-2n,q-n(X, Z_).
Proof.The existence of the Thom isomorphism is an immediate consequence of Prop*
*osition 1.9;
see [May96 ]. We proceed to construct an explicit map OE~ at the classifying s*
*pace level which
induces this isomorphism. Let f~ : T (,)! Z0 S2n,no be a (based) classifying m*
*ap for the
*
* f~^id
orientation class ~, hence f~(1) = 0. Consider the composition T (,)-! T (,)^ *
*X+ - --!
Z0 S2n,no ^ X+ ! Z0 S2n,n^ X+ o, where is the Thom diagonal, and the last map*
* comes
from the structure of üf nctor with smash products" (FSP) for Z0(-). This compo*
*sition induces
a function OE~ : Z0(T (,))o! Z0 S2n,n^ X+ o.
We claim that OE~ induces the Thom isomorphism in homology. Indeed, Z0(Sp,q)o*
*is a K(Z(q), p)-
space and so a model for HZ_is given by (p, q) 7! Z0(Sp,q)o. Moreover ^ induces*
* a pairing
K(Z(q), p) ^ K(Z(q0), p0) -! K(Z(q + q0), p + p0)
which gives the usual ring spectrum structure on HZ_, cf. [Dug01 ] and [dS99b].*
* It follows that the
map in homology represented by OE~ is
eHp,q(T (,), Z_) 3 a ffi~*--!p*(~ \ a) 2 Hp-2n,q-n(X, Z_),
the usual definition of the Thom isomorphism. The proof that OE~* is in fact an*
* isomorphism goes
exactly as in the non-equivariant case. Using the five lemma and the Mayer-Viet*
*oris sequence it is
possible to reduce to the case where , p-!X is the trivial bundle, in which cas*
*e T (,)~=X+ ^ S2n,n
and it is clear that OE~*is an isomorphism.
1.6. Poincar'e duality. A smooth manifold X is called a Real n-manifold if it h*
*as the structure
of a Real space (X, oe) whose tangent bundle becomes a Real n-bundle over (X, o*
*e) under the action
induced by doe.
10 DOS SANTOS AND LIMA-FILHO
Proposition 1.12. Let X be a connected Real manifold of dimension n. Then, for *
*each k 0,
there is an equivariant homotopy equivalence
i j ~ i j
(8) P : F (X+; Z0 S2(n+k),n+k ) =-!Z0 S2k,k^ X+ ,
o o
which on passage to homotopy groups induces the Poincar'e duality isomorphism
(9) H2n-p,n-q(X, Z_) ~=Hp,q(X, Z_).
Proof.By Proposition 1.11 the tangent bundle of X has an orientation, which is *
*a cohomology
class in dimension (2n, n). It follows [CW92 ] that X satisfies Poincar'e duali*
*ty as in (9)and the
duality isomorphism is given by cap product with the fundamental class z 2 H2n,*
*n(X, Z_) - which
corresponds to 1 2 H0,0(X, Z_) = H0(XZ=2; Z); cf. [CW92 ].
We now define a homotopy equivalence at the classifying space level realizing*
* the Poincar'e duality
isomorphism. Let r = n + k and let D denote the composition
2r,r ^id 2r,r
Z0(X) ^ F (X+, Z0 S o) ---! Z0(X) ^ Z0(X) ^ F (X+, Z0 S o)
-id^ffl--!Z 2r,r 2r,r
0(X) ^ Z0 S o! Z0 S ^ X+ o.
where is the diagonal map, " is the group homomorphism induced by the evalua*
*tion map
X ^ F (X+, Z0 S2r,ro)! Z0 S2r,ro, and the last arrow comes from the structure o*
*f üf nctor
with smash products" (FSP) for Z0(-). Composing D with a classifying map S2r,r*
*! Z0(X)
for the fundamental class z we obtain a map S2n,n^ F (X+, Z0 S2r,ro) ! Z0 S2r,r*
*^ X+ o,
with adjoint F (X+, Z0 S2r,ro) ! 2n,nZ0 S2r,r^ X+ o. Composing with the natura*
*l equivalence
2n,nZ0 S2r,r^ X+ o~=Z0 S2k,k^ X+ o, yields a map
i j i j
P : F (X+, Z0 S2(n+k),n+k ) ! Z0 S2k,k^ X+
o o
which induces the cap product with z. Hence P is an equivariant homotopy equiva*
*lence.
1.7. Relation with group cohomology and Galois-Grothendieck cohomology. Cohomol-
ogy theories like the one represented by HZ_ are not what algebraic geometers u*
*sually mean by
equivariant cohomology. However, it is well known that for spaces which are fre*
*e (under the action
of a finite group), the invariants given by H*,*(-; Z_) are closely related to *
*the equivariant cohomol-
ogy theories used in algebraic geometry. The goal of this section is to describ*
*e the relation between
the theory H*,*(-; Z_), and Galois-Grothendieck cohomology, in the case of Z=2-*
*actions.
Let G be a finite group. Recall that, in most geometrical contexts, the Borel*
* cohomology of a
G-space X (with coefficients in a ring R) is just the ordinary cohomology of th*
*e Borel construction
XhG := X xG EG. We denote these groups by ^H*G(X; R). Galois-Grothendieck cohom*
*ology can be
thought of as Borel cohomology with twisted coefficients: if F is a G-sheaf ove*
*r X, then Fx GEG
is sheaf over XhG and we can define H^*G(X; F) := H*(XhG; Fx GEG). These are t*
*he Galois-
Grothendieck cohomology groups of X with coefficients in F. Given any G-spectru*
*m kG there is
also a Borel type cohomology theory associated with kG, defined as the cohomolo*
*gy represented by
the G-spectrum F (EG+, kG).
We now specialize to the case G = Z=2, and kG = HZ_. Consider the Z=2-sheaves*
* Z(n), which
denote the constant sheaf Z consider as a Z=2-sheaf with the Z=2-action of mult*
*iplication by (-1)n.
The following proposition relates Galois-Grothendieck cohomology with coefficie*
*nts in the sheaves
Z(n) to the cohomology represented by F (EZ=2+, HZ_).
QUATERNIONIC ALGEBRAIC CYCLES AND REALITY *
* 11
Proposition 1.13. Let p, q be non-negative integers such that p q. There are *
*natural isomor-
phisms
(10) : Hp,q(- x EZ=2+; Z_) ! ^HpZ=2(-; Z(q)),
which assemble into a ring homomorphism
: H*,*(- x EZ=2+; Z_) ! ^H*Z=2(-; Z(*)).
In particular, if X is a free Z=2-space, H*,*(X; Z_) is periodic with period (0*
*, 2), and the periodicity
isomorphism is given by multiplication with a generator of H0,2(EZ=2; Z_) ~=Z.
Proof.The existence of periodicity in the equivariant cohomology spaces with fr*
*ee actions is a result
of Waner [Wan86 ]. Our proof uses Z0(Sp,q)oas a model for the classifying space*
* K(Z_, Rp,q). We
start by observing that the cohomology groups ^Hp(X; Z(q)) can be expressed as *
*Bredon cohomology
groups of X x EZ=2. In fact, we have ^HpZ=2(X; Z(q)) ~=Hp(X x EZ=2; Z_(q)), whe*
*re right-hand
side denotes Bredon cohomology and Z_(q) is the coefficient system determined b*
*y the Z=2-module
Z(q). Hence, ^HpZ=2(X; Z(q)) ~=[X x EZ=2+, Z0(Sp)o Z(q)]Z=2, where Z0(Sp)o Z(*
*q) is Z0(Sp)o
with the Z=2-action of multiplication by (-1)q.
A direct computation shows that there is a class ffp,qin H^pZ=2(Sp,q; Z(q)), *
*whose image under
the forgetful map to singular cohomology is the fundamental class of Sp,q. This*
* implies that there
is an equivariant map Z0(Sp,q)ox EZ=2 ! Z0(Sp)o Z(q), which is a non-equivaria*
*nt homotopy
equivalence. Composition with this map induces an equivariant homotopy equivale*
*nce
F (EZ=2+, Z0(Sp,q)ox EZ=2) ~=F (EZ=2+, Z0(Sp)o Z(q)).
Composing with the map EZ=2 ! * gives an equivariant homotopy equivalence
F (EZ=2+, Z0(Sp,q)ox EZ=2) ~=F (EZ=2+, Z0(Sp,q))o,
which induces the isomorphism in (10).
The pairing Z(q) Z(q0) ! Z(q + q0) induces a pairing
H^pZ=2(-; Z(q)) ^HpZ=2(-; Z(q0)) ! ^HpZ=2(-; Z(q + q0)),
which is easily seen to correspond, under the isomorphism (10), to the cup prod*
*uct in H*,*(- x
EZ=2+ ; Z_) (because ffp,q[ ffp0,q0= ffp+p0,q+q0). Let t be a generator of H0,2*
*(EZ=2; Z_) ~=Z and let
X be a free Z=2-space. Since X is Z=2-homotopy equivalent to X x EZ=2, the coho*
*mology of X
is a module over H*,*(EZ=2; Z_). The compatibility of with products shows tha*
*t the following
diagram commutes
Hp,q+2(X; Z_)----!^HpZ=2(X; Z(q + 2))
x x
t[-?? ??id ,
Hp,q(X; Z_)----! H^pZ=2(X; Z(q))
and hence t [ - is an isomorphism.
Notation 1.14. As observed above, if X is a free Z=2, there is a natural homomo*
*rphism
H*,*(X x EZ; Z_) ! H*,*(X; Z_)
12 DOS SANTOS AND LIMA-FILHO
making H*,*(X; Z_) into a module over H*,*(EZ=2; Z_). From now on we will use t*
* to denote both a
generator of H0,2(EZ=2; Z_) ~=Z and its image under the homomorphism above. Mul*
*tiplication by
t induces the (0, 2) periodicity in H*,*(X; Z_).
Corollary 1.15. Let X be a space with a free Z=2-action. There is an E2 spectra*
*l sequence
Hs(BZ=2; Ht(X; Z(q))) =) Hs+t,q(X; Z_)
Proof.It is shown in [Gro57] that there is an E2 term spectral sequence
Hs(BZ=2; Ht(X; Z(q))) =) ^Hs+tZ=2(X; Z_(q)).
1.8. The RO(Z=2)-graded cohomology of the Brauer-Severi curve P(H). Let P(H) de*
*note
the complex points of the Brauer-Severi variety associated with the real algebr*
*a H. Here we describe
the cohomology ring H*,*(P(H); Z_). The computations follow directly from the r*
*esults above.
We start by computing H*,q(P(H); Z_), for q = 0, 1. For q = 0, we have
H*,0(P(H); Z_) = H*(P(H)Z=2; Z) = H*(RP2 ; Z).
For q = 1, the spectral sequence (1.15)gives
8
>:
0 otherwise
Moreover, H1,1(P(H); Z_) and H2,0(P(H); Z_) are generated by the image of the h*
*omomorphism
H*,*(EZ=2; Z_) ! H*,*(P(H); Z_).
Let ffl denote the generator of H1,1(P(H); Z_) and let ffl0be its image in H1(P*
*(H); Z(1)) under the ring
homomorphism of (10). One can check that ffl026= 0 hence ffl2 is the generato*
*r of H2,2(P(H); Z_).
The group H2,1(P(H); Z_) is generated by the fundamental class of P(H), which*
* we denote by x.
Note that x2 = 0, since, by (0, 2) periodicity,
x2 2 H4,2(P(H); Z_) ~=H4(RP2 ; Z) = 0.
The same argument shows that xffl = x3 = 0.
Remark 1.16. The generator x is given by the first Chern class ~c1(O(2)) of the*
* Real bundle O(2)
over P(H).
Putting all these facts together and using that fact that t [ - is an isomorp*
*hism, we obtain a
ring isomorphism
(11) H*,*(P(H); Z_) ~=Z[ffl, x, t, t-1]=(2ffl, ffl3, xffl, x2),
where ffl, t and x have degrees (1, 1), (0, 2) and (2, 1), respectively.
QUATERNIONIC ALGEBRAIC CYCLES AND REALITY *
* 13
2. The Z=2-homotopy type of zero-cycles
A structure on a complex vector space V is a complex anti-linear map j : V ! *
*V that satisfies
either j2 = I or j2 = -I. In the first case, j is called a real structure and t*
*he pair (V, j) is a real
vector space, and in the latter case, j is called a quaternionic structure and *
*the pair (V, j) is a
quaternionic vector space. A morphism f : (V, jV) ! (W, jW ) of vector spaces w*
*ith structure is a
complex linear map from V to W commuting with the respective structures.
Any real vector space (V, j) of complex dimensin n is isomorphic as a vector *
*space with structure
to (Cn, jo), where jo denotes the usual complex conjugation on Cn. Similarly, *
*any quaternionic
vector space (V, j) of complex dimension 2n is isomorphic to (Hn, jo), where H *
*= C C_ denotes
the algebra of the quaternions and jo is induced by multiplication by _ on the *
*left of Hn.
Remark 2.1. It is clear that if (V, j) is a quaternionic vector space, then j n*
*aturally induces a
structure jd on the symmetric power Sym d(V ) which is real if d is even and qu*
*aternionic if d is
odd. In particular, one has
(12) (Sym 2k(H), j2k) ~=(C2k+1, jo) and (Sym 2k+1(H), j2k+1) ~=(Hk+1, j*
*o).
Let H_ denote the complex dual of H and let P(H) denote the projective space *
*of 1-dimensional
complex subspaces of H, which we identify with the complex subspaces of codimen*
*sion 1 in H_. In
particular, if f : H_ ! C is a non-zero linear functional then we denote its ze*
*ro locus by [f] 2 P(H).
The d-fold symmetric product SPd(P(H))inherits an anti-holomorphic involution*
* oe : SPd(P(H))!
SP d(P(H))induced by _, hence it becomes a Real space in the sense of Atiyah [A*
*ti66]. If one denotes
by [f1] . .[.fd] an element in SPd(P(H)), then the map sending [f1] . .[.fd] to*
* [f1. .f.d] induces the
classical isomorphism between SPd(P(H))and P Sym d(H) . This is an isomorphism *
*of Real spaces.
The canonical inclusion in,q: P Hn ,! P Hq , for n < q, given by setting the*
* last coordinates
zero can be described (up to linear isomorphism) in terms of symmetric products*
* as follows. Let
[fo] 2 P(H) be some point and let [fffo] denote its image under the quaternioni*
*c involution oe. Then
define
n q
(13) in,q: P H = SP2n-1(P(H))! P H = SP2q-1(P(H))
[f1] . .[.f2n-1]! ([fo][fffo])q-n. [f1] . .[.f2n-1].
Note that in,qis a morphism of Real spaces.
Following [LLFM98c ] we define, for a < b, the map
(14) rb,a: SPb(P(H))! SP(b(SP a(P(H)))
a)X
[f1] . .[.fb]7! [fi1] . .[.fia],
|I|=a
where the sum runs over all multi-indexes I = {1 i1 < . .<.ia b}. Notice th*
*at although a and
b can be even or odd, the maps rb,aare always morphisms of Real spaces.
To simplify notation, denote
n
(15) Mn = Z0(SP2n-1(P(H)))= Z0 P H ,
and let
(16) n : Mn ! Qn := Mn=Mn-1
denote the quotient map. We adopt the convention that M-1 = {0}.
14 DOS SANTOS AND LIMA-FILHO
Now, use the morphisms rb,ato construct maps
(17) Rq,n: Mq ! Mn
as follows. Given ø 2 P Hq , define for n < q,
2n-1X `q - n + j -'1
(18) Rq,n(ø) = r2q-1,2n-1(ø) + (-1)j [fo]j+ [fffo]jr2q-1,2n*
*-1-j(ø).
j=1 j
The map Rq,ncan then be extended by linearity to arbitrary 0-cycles on P Hq . F*
*inally, for n < q,
define qq,n: Mq ! Qn as the composition qq,n= n O Rq,nand let qn,n= n.
Proposition 2.2. Let {(Mn, Qn, qq,n, iq,n) | qq,n: Mq ! Qn, in,q: Mn ,! Mq, 0 *
* n q} be the
collection of groups and maps defined above. Then the following assertions hold:
a: The maps in,qand qq,nare equivariant homomorphisms for the Z=2 actions o*
*n Mn and Qn
induced by the quaternionic structure on Hn.
in-1,n qn,n
b: The sequence Mn-1 ----!Mn --! Qn is an equivariant principal fibration, *
*for all n.
c: The following diagram commutes:
Mn ----! Mq
? in,q ?
(19) qn,n?y ?yqq,n
Qn ----! Qn.
id
Proof.The first assertion is evident from the definitions.
Now, observe that the Z=2 involution on P(Hn) induced by the quaternionic str*
*ucture on Hn is a
real analytic involution. Therefore, the pair (P(Hn), P(Hn-1)) becomes a Z=2-si*
*mplicial pair after
a suitable equivariant triangulation. Therefore, the second assertion follows f*
*rom [LF97, Thm. 2.7]
In order to prove the last assertion, consider elements x1, . .,.x2q-12 P(H) *
*as free variables and
let S denote the polynomial ring S := Z[x1, . .,.x2q-1]. It is then clear that *
*r2q-1,2n-1(x1. .x.2q-1) 2
SP (2q-1(SP 2n-1(P(H)))can be seen as the coefficient of t2n-1in the polynomial
2n-1)
2q-1Y
(20) Pt2q-1(x1, . .,.x2q-1) := (1 + xit) 2 S[t].
i=1
In particular,
r2q-1,2n-1-jO in,q(x1. .x.2n-1) = r2q-1,2n-1-j(x1. .x.2n-1(fofffo)n*
*-q)
is the coefficient of t2n-1-jin the polynomial Pt2n-1(x1, . .,.x2n-1) (1 + fot)*
*q-n(1 + fffot)q-n.
QUATERNIONIC ALGEBRAIC CYCLES AND REALITY *
* 15
Using the above observations, and the definition of Rq,n, one concludes that *
*Rq,n(in,q(x1. .x.2n-1))
is the coefficient of t2n-1in
_ 2n-1 !
Y
(21) (1 + xit)(1 + fot)q-n(1 + fffot)q-n
i=1
1X `q - n + j -'1 _2n-1Y !
+ (-1)j {(fo)j+ (fffo)j} tj (1 + xit)(1 + fot)q-n(1 +*
* fffot)q-n
j=1 j i=1
_ 2n-1 ! 8
Y <
= (1 + xit). (1 + fot)q-n(1 + fffot)q-n
i=1 :
9
1X `q - n + j -'1 =
+ (-1)j {(fo)j+ (fffo)j} tj (1 + fot)q-n(1 + fffot)q-n
j=1 j ;
seen as an element in the ring Z[x1, . .,.x2n-1][[fo, fffo, t]] of formal power*
* series in the variables
fo, fffo, t with coefficients in Z[x1, . .x.2n-1]. P
We now observe that the inverse of (1 + y)N in Z[[y]] is given by 1j=0(-1)j*
* N+j-1jyj and
hence, the element
1X `q - n + j -'1
(1 + fot)q-n(1 + fffot)q-n+ (-1)j {(fo)j+ (fffo)j} tj (1 + fot*
*)q-n(1 + fffot)q-n
j=1 j
in Z[[fo, fffo, t]] is seen to be equal to
n-q ff n-q
(22) (1 + fot)q-n(1 + fffot)q-n (1 + fot) + (1 + fot) - 1
= (1 + fffot)q-n+ (1 + fot)q-n- (1 + fot)q-n(1 +*
* fffot)q-n.
If one writes (1 + yt)N = 1 + yffNt(y), then the right hand side of the formula*
* (22)above can can
be written as
(1 + fffot)q-n+ (1 + fot)q-n- (1 + fot)q-n(1 + fffot)q-n
=(1 + fffot)q-n+ (1 + fot)q-n- (1 + fffoffq-nt(fffo))(1 + fo ffq-n*
*t(fo))
=(1 + fffot)q-n+ (1 + fot)q-n
n *
* o
- 1 + fffoffq-nt(fffo) + fo ffq-nt(fo)) + fffofo ffq-nt(fffo)ff*
*q-nt(fo)
=(1 + fffot)q-n+ (1 + fot)q-n
n *
* o
- (1 + fffot)q-n+ (1 + fot)q-n- 1 + fffofo ffq-nt(fffo)ffq-nt(f*
*o)
n o
=1 - fffofo ffq-nt(fffo)ffq-nt(fo).
We now apply the latter identity to equation (21) and obtain
Rq,n(in,q(x1. .x.2n-1))= x1. .x.2n-1- fffofo(?),
16 DOS SANTOS AND LIMA-FILHO
for some element ?. Therefore,
qq,nO in,q(x1. .x.2n-1) = n (Rq,n(in,q(x1. .x.2n-1)))= n (x1. .x.2n*
*-1),
and this shows that the diagram (19) commutes, concluding the proof.
Corollary 2.3. Let X be a based Z=2-space. The maps qn,j: Z0 P Hn ! Z0 P Hj *
*=Z0 P Hj-1
induce an equivariant homotopy equivalence
i j ~qn,nx~qn,n-1x...x~qn,1nYi j
Z0 X ^ P Hn + ---------------! Z0 X ^ P Hj =P Hj-1 + .
j=1 o
i j i j
Proof.Define Mj = Z0 X ^ P Hj + and Qj = Z0 X ^ P Hj =P Hj-1 + and obse*
*rve
o
that one has an equivariant isomorphism Mj=Mj-1~=Qj. Let j : Mj ! Qj denote th*
*e projection
and let ~in,q: Mn ,! Mq be the canonical inclusion induced by the inclusion of *
*spaces when n < q.
The maps r2q-1,2n-1, described in (14), induce maps
q n
(23) ~r2q-1,2n-1: X ^ P H + ! SP(2q-1(X ^ P H )
2n-1) +
defined as the composition
q id^r2q-1,2n-1 n n o ^ n
X ^ P H + ---------!X ^ SP(2q-1(P H ) ---! SP 2q-1(X ^ P H ),
2n-1) + (2n-1) +
where the latter is the natural structural map when we see SP*(-)as a functor w*
*ith smash products.
Finally, define ~qq,n:= n O ~r2q-1,2n-1.
It is immediate from the definitions that all the assertions in Proposition 2*
*.2 hold for the new
collection (Mn, Qn, ~qq,n,~in,q) above. These assertions guarantee that the spa*
*ces and maps involved,
along with their restrictions to fixed point sets, satisfy the hypothesis of [F*
*L92, Prop. 2.13]. The
corollary then follows.
In order to fully understand the equivariant homotopy type of Z0 P Hn we ar*
*e reduced to un-
derstanding Z0 P Hj =P Hj-1 o. However, P Hj =P Hj-1 = T OP(H)(1) Hj-1is th*
*e Thom
space of the Real bundle OP(H)(1) Hj-1. It follows from Propositions 1.11 an*
*d 1.12 that the
Thom class of OP(H)(1), along with Poincar'e duality, determines a unique equiv*
*ariant homotopy
equivalence
j j-1 j-1 4j-2,2j-1
(24) Z0 P H =P H o~=Z0 T OP(H)(1) H o~=F (P(H)+, Z0 S o).
This proves the following result:
Corollary 2.4. There is a canonical equivariant homotopy equivalence
n Yn 4j-1,2j-1 Yn
Z0 P H ~= F P(H)+, Z0 S o ~= F (P(H)+, K(Z(2j - 1), 4j -.2))
j=1 j=1
The last equivalence follows from the equivariant Dold-Thom theorem proven in*
* [dS99a]; cf.
(1.4).
3. Quaternionic algebraic cycles and the join pairing
In this section we study the equivariant topology of groups of algebraic cycl*
*es under quaternionic
involution, and construct stabilizations of such objects that yield equivariant*
* Z=2-spectra.
QUATERNIONIC ALGEBRAIC CYCLES AND REALITY *
* 17
3.1. Equivariant homotopy type of algebraic cycles. Let (V, j) be a quaternioni*
*c vectorP
space. An algebraic cycle on P V of codimension q is a finite linear combinat*
*ion ki=0miAi,
where mi2 Z and Ai P V is an irreducible subvariety of codimension q in P V *
*. The following
properties hold.
Facts 3.1.
a: The collection of algebraic cycles of codimension q in P V forms an abe*
*lian topological
group Zq(P V ) under addition of cycles; cf. [LF94].
b: The quaternionic structure j on V induces a continuous involution j* :*
* Zq(P V ) !
Zq(P V ) which is also a group homomorphism; cf. [LLFM98c ]. This gives *
*an action of
Z=2 on Zq(P V ) via group automorphisms. We reserve the word equivariant*
* in the present
context to mean Z=2-equivariant under this quaternionic action.
c:PThere is a continuousPdegree homomorphism deg: Zq(P V ) ! Z which assig*
*ns to a cycle
iniAi the integer i nideg(Ai), where deg(Ai) is the degree of Ai as a*
* subvariety of
P V . For each d 2 Z, denote Zq(P V )d := deg-1(d) the subspace of cycl*
*es of degree d.
Each Zq(P V )d is a connected component of Zq(P V ).
Given two quaternionic vector spaces (V, jV) and (W, jW ), one has an equivar*
*iant external join
pairing
q0 q+q0
(25) # : Zq(P V ) x Z (P W ) ! Z (P V W )
given by the ruled join of cycles. Roughly speaking, # is the bilinear extensio*
*n of the following
operation. Given an irreducible subvariety A P V of codimension q, and an ir*
*reducible subvari-
ety B P W of codimension q0, let A#B be the irreducible subvariety of P V *
*W obtained by
taking the union of all projective lines in P V W joining points in A to poi*
*nts in B, after taking
the embeddings A P V = P V 0 P V W and B P W = P 0 W P V W*
* .
We refer the reader to [LLFM96 ] for more details.
Remark 3.2. The degree of cycles is additive with respect to addition of cycles*
* and multiplicative
with respect to the join. In other words, given cycles oe1, oe2 2 Zq(P V ) and*
* ø1 2 Zq(P W ), then
deg(oe1+ oe2)= degoe1+ degoe2 and deg(oe1#ø1)= degoe1degø1.
If one thinks of P(H) as an element in Z0(P(H)), one can use the join to defi*
*ne the quaternionic
suspension map:
q
(26) = H : Zq(P V )! Z (P V H )
c7! c#P(H).
Remark 3.3. (1) This definition parallels the construction of the complex su*
*spension map
= C : Zq(P V ) ! Zq(P V C ) for a real vector space (V, j), whose (no*
*n-equivariant)
homotopy properties were first studied in [Law89 ]. Equivariant properti*
*es for this map,
with respect to the complex conjugation involution, were studied in [Lam9*
*0 ], [LLFM98a ],
[LLFM98b ], [dS99a], [Mos98].
(2)Another useful description of the suspension map is the following. Consid*
*er a surjection
f : V ! W of quaternionic vector spaces, and let C V be a (quaternionic*
*) complement
to the kernel K := kerf, so that V is the internal direct sum K C with *
*C ~=W . The
18 DOS SANTOS AND LIMA-FILHO
=K
latter isomorphism induces a homomorphism Zq(P W ) ~=Zq(P C ) --! Zq(P C *
* K ) ~=
Zq(P V ), which is an equivariant homotopy equivalence. One can easily v*
*erify that the
resulting üp ll-back map" f* : Zq(P W ) ! Zq(P V ) is independent of the*
* choice of the
complement C.
We now proceed to determine the equivariant homotopy type of spaces of algebr*
*aic cycles
Zq(P(Hn)) of arbitrary codimension q on P(Hn). It is shown in [LLFM98a ] that, *
*given a quaternionic
vector space (V, j), the suspension homomorphism (26) = H : Zq(P V ) ! Zq(P V*
* H ) gives
an equivariant homotopy equivalence. In particular, for k < n, one obtains equi*
*variant homotopy
equivalences:
i j
(27) = n-kH: Z2k-1(P(Hk)) = Z0 P(Hk) ! Z2k-1(P(Hn))
and
(28) = n-k-1H: Z2k(P(Hk+1)) = Z1(P(Hk+1) ! Z2k(P(Hn))
Now, let (V, j) be a quaternionic vector space, and recall that Sym2(V ) has *
*a natural structure of
a real vector space; cf. Remark 2.1. It follows that the image of P V under th*
*e Veronese embedding
2 : P V ,! P Sym 2(V ) given by OP V (2) becomes a real subvariety of P Sym *
*2(V ) . Define
` '
(29) Q(V ) := T OP V (2) .
It is clear that Q(V ) can be identified with the complex suspension
= C( 2(P V )) = 2(P V )#p1 P Sym 2(V ) C ,
where p1 = P 0 C 2 P Sym 2(V ) C .
It is shown in [LLFM98c , Prop. 6.1] that the complex suspension = : Zq(P V *
*) ! Zq(Q(V ))
composed with the Veronese embedding P V ,! P Sym 2(V ) induces an equivarian*
*t homotopy
equivalence = : Zq(P V ) ! Zq(Q(V )). This fact, together with the equivalence*
* Zq(Q(Hk+1)) ~=
Zq(Q(Hk)) proven in [LLFM98c ] gives an equivalence
k+1 i k j
(30) Z2k(P H ) ~=Z0 Q(H ) .
These observations imply the following result.
Theorem 3.4. For k < n there are canonical equivariant homotopy equivalences:
n Yk
(31) Z2k-1(P H ) ~= F (P(H)+, K(Z(2j - 1), 4j - 2) )
j=1
and
n Yk
(32) Z2k(P H ) ~= F (P(H)+, K(Z(2j), 4j) )
j=1
QUATERNIONIC ALGEBRAIC CYCLES AND REALITY *
* 19
Proof.The first equivalence follows from the equivalence Z2k-1(P Hn ) ~=Z0 P Hk*
* given by (27)
and Corollary 2.4. To prove the second one, first consider the equivalences
n 2k k+1 k+1 i k j
(33) Z2k(P H ) ~= Z (P H ) = Z1(P H ) ~= Z0 Q(H )
(28) (30)
and
i j ` ` '' i j
(34) Z0 Q(Hk) = Z0 T OP Hk (2) ~=Z0 S2,1^ P Hk + ;
cf. (27), (28), (30) and Propositions 1.11 and 1.12.
Using Corollary 2.3 one obtains
i j Yn i j
Z0 S2,1^ P Hk + ~= Z0 S2,1^ P Hj =P Hj-1 +
j=1 o
Yn i j
(35) = Z0 S2,1^ T OP(H)(1) Hj-1+
j=1 o
Yn
= Z0 T OP(H)(1) Hj-1 1 o.
j=1
Finally, the canonical equivalence Z0 T OP(H)(1) Hj-1 1 o~=F P(H)+, Z0 S*
*4j,2joes-
tablished in Propositions 1.11 and 1.12, along with (31), (32) and (33), proves*
* the second assertion
of the proposition.
3.2. Stabilizations of cycle spaces. Here we use the group of algebraic cycles,*
* with the quater-
nionic Z=2 action defined above, to construct equivariant infinite loop spaces.
Consider a real vector space (V, oe), of complex dimension v, and let V *deno*
*te its complex dual.
Denote by (VH, j) the quaternionic vector space VH := V C H with quaternionic *
*structure induced
by multiplication by j on the right, and define
a2v
(36) Z(V ) := Z2v+j(VH* VH)1,
j=-2v
where Z2v+j(P VH* VH )1 denotes the spaces of algebraic cycles of codimension *
*2v + j and degree
1 in P VH* VH . Here we see Z(V ) as a Z=2-space under the quaternionic action*
* induced by j on
P VH* VH . The spaces Z(V ) have a natural basepoint 1V := P VH* {0} 2 Z2v(VH**
* VH)1 Z(V ).
Define Z(0) to be the one-point set {10}.
Given an inclusion of real vector spaces i : (V, oe) 7! (W, oe0), let i* : W *
**! V *denote the adjoint
surjection. The inclusion i induces maps
* 2w+j *
(id i)* : Z2v+j(P VH VH )1 ! Z (P VH WH )1,
given by the inclusion of cycles, and the surjection i* induces maps
* 2w+j *
(i* id)* : Z2w+j(P VH WH )1 ! Z (P WH WH )1,
20 DOS SANTOS AND LIMA-FILHO
given by the appropriate pull-back of cycles; cf. Remark 3.3. Define
a2v 2wa
(37) i] : Z2v+j(P VH* VH )1 ! Z2w+j(P WH* WH )1
j=-2v j=-2w
as the composition (i* id)*O (id i)*. This makes Z(-) into a functor from th*
*e category of finite
dimensional real vector spaces and linear monomorphisms, to the category of poi*
*nted Z=2-spaces.
The functor Z(-) comes with an additional structure, an equivariant "Whitney *
*sum" pairing,
defined as follows. Given real vector spaces (V, oe) and (W, oe0), define
(38) #V,W : Z(V ) x Z(W ) ! Z(V W )
as the map whose restriction to the components is the composition
* 2w+k * #
(39) Z2v+j(P VH VH )1x Z (P WH WH )1 -!
* *
Z2(w+v)+j+k(P VH VH WH WH )1
fi-!Z2(w+v)+j+k(P (V W )*
H (V W )*
*H )1,
where # is the join pairing (38). Here ø denotes the map on cycles induced by t*
*he composition of
linear isomorphisms VH* VH WH* WH ! (VH* WH*) (VH WH) ! (V W )*H (V*
* W )H,
where the former map is a shuffle isomorphism and the latter is the usual natur*
*al identification.
Define #0,W to be the identity map, after the natural identification Z(P 0 V *
* ) Z(P V ).
Proposition 3.5. The assignments V 7! Z(V ) along with the pairings #V,W give Z*
*(-) the struc-
ture of an equivariant (Z=2)I*-functor, in the language of [May77 ]. See also [*
*LLFM96 ].
Proof.This amounts to checking various coherence properties, and proceeds exact*
*ly as in the non-
equivariant case done in [BLLF+93 ], or in the equivariant study of real algebr*
*aic cycles, done in
[LLFM98b ].
Now, consider (C1 , oe) as a real vector space under complex conjugation oe, *
*and observe that it is
a complete Z=2-universe; cf. [May96 ]. In other words, it contains infinitely c*
*ountably many copies
of each irreducible representation of Z=2. Define ZH as the colimit
(40) ZH := lim-!Z(V ),
V C1
where V runs over all real subspaces of C1 .
Theorem 3.6. The space ZH is an equivariant Z=2-infinite loop space. In other w*
*ords, for each
real Z=2-module V there is a Z=2-space ZH,V along with coherent equivariant hom*
*otopy equivalences
ZH = ZH,0~= VZH,V.
Remark 3.7. Given any based Z=2-space X and Z=2-module V , the space VX of V -*
*fold loops in
X is the space F (SV , X) of based maps from the one-point compactification SV *
*of V into X, with
its usual topology and the usul Z=2 action on function spaces. The coherence pr*
*operties mentioned
in the statement above are the ones that define an equivariant infinite loop sp*
*ace; cf. [May96 ].
Definition 3.8. The infinite loop space structure on ZH determines an equivaria*
*nt spectrum (cf.
[May96 ]) denoted ZH, satisfying ZH(0) ~=ZH.
QUATERNIONIC ALGEBRAIC CYCLES AND REALITY *
* 21
Before proving the theorem, let us analyze the space ZH in detail. Since the *
*increasing coordinate
flag {0} C C2 . . .C1 is cofinal among the finite dimensional subspaces o*
*f C1 , one observes
that
2na
ZH = lim-!Z(Cn) = lim Z2n+j(P Hn * Hn )1.
n -!n j=-2n
Here we are making the usual identification CnH= Cn C H Hn. The maps i] in *
*the colimit
above are compositions of algebraic suspensions and coordinatewise inclusions. *
*Therefore, they are
compatible with the splittings given in Theorem 3.4.
Theorem 3.9. The space ZH is written as a disjoint union of connected spaces
1a
ZH = ZjH,
j=-1
where the equivariant homotopy type of ZjHis totally determined by
(Q 1
F (P(H)+, K(Z(2j - 1), 4j - 2),)if j is odd
(41) ZjH~= Q j=11
j=1 F (P(H)+, K(Z(2j), 4j) ) , if j is even.
Proof.It is evident that if one defines
n * n
(42) ZjH:= lim-!Zj+2n(P H H ),
n
` 1 j
then ZH = j=-1 ZH. The result now follows from the remark preceding the Propo*
*sition together
with Theorem 3.4.
Remark 3.10. Note the the equivariant homotopy type of ZjHis completely determi*
*ned by the
parity of j. Furthermore, a canonical inclusion of coordinate hyperplanes give*
*s immediately an
equivariant homotopy equivalence ZjH~=Zj+2H, for all j. For that reason we esta*
*blish the notation
ZevH:= the equivariant homotopy typeZofjH, for j even,
and
ZoddH:= the equivariant homotopy typeZofjH, for j odd.
We now proceed to prove Theorem 3.6.
Proof.The same arguments of [BLLF+93 ], [LLFM96 ], or [LLFM98b ], together with*
* Proposition
3.5, imply that the join operation induces an action of the equivariant Z=2-lin*
*ear isometries operad
on ZH. In particular, both ZH and its fixed point set ZZ=2Hhave an induced act*
*ion of the usual
linear isometries operad.
It follows from (38) that the H-space structure on ZH given by the join induc*
*es a group isomor-
*
* Z=2
phism ß0(ZH) ~=Z. Furthermore, it is easy to see (cf. [LLFM98c ]) that Zj+2n(P*
* Hn * Hn )1
is empty if j is odd, and non-empty and connected if j is even. This implies, a*
*fter passage to the
colimit, that
ß0(ZZ=2H) ~=2Z Z ~=ß0(ZH).
Therefore, the H-space ZH is Z=2-group complete, in the language of [CW91 ]. *
*It follows from
the equivariant "recognition principle" in [CW91 ] that ZH is an equivariant in*
*finite loop space; see
also [dSLF01 ].
22 DOS SANTOS AND LIMA-FILHO
Remark 3.11. We must point out that the construction of our I*-functor Z(-), in*
* Proposition
3.5, only uses embeddings V ,! W , not isometric embeddings of hermitian vector*
* spaces. As a
consequence, we can replace the linear isometries operad L by the linear embedd*
*ings operad
{ E(n) := Emb (C1 . . .C1 , C1 ) | n 2 N }.
This operad has the advantage of being algebraic in nature and is more suitable*
* for possible öm -
tivic" generalizations and various other applications to algebraic geometry.
4.Quaternionic K-theory
Given a Z=2-space X, one can use its equivariant structure to study two class*
*es of complex
bundles over X, namely the real and quaternionic bundles. This study yields two*
* distinct, albeit
related, equivariant theories: the Real K-theory studied by Atiyah in [Ati66] a*
*nd the (equivariant)
quaternionic K-theory studied by Dupont in [Dup69 ].
Let us recall the basic definitions.
Definition 4.1. Let (X, oe) be a Z=2-space and let p : E ! X be a complex vecto*
*r bundle over X.
Let ø : E ! E be a continuous map covering oe, i.e. p O ø = oe O p, and such th*
*at for any x 2 X the
resulting map ø : Ex ! Effxis anti-linear.
a: If ø2 = idthen (E, ø) is a real bundle;
b: If ø2 = - idthen (E, ø) is a quaternionic bundle.
The dimension of a real or quaternionic bundle is defined as its complex dimens*
*ion, and morphisms
between such bundles are bundle morphisms that commute with the structure maps *
*ø.
The isomorphism classes real bundles form a monoid under Whitney sum, whose G*
*rothendieck
group is called the real K-theory KR0(X) of X. Similarly, the Grothendieck grou*
*p of isomorphism
classes of quaternionic bundles gives the quaternionic K-theory groups KH 0(X) *
*of X.
In [LLFM98b ] the connective version of KR-theory is studied from the equivar*
*iant point of
view, along with a suitable theory of equivariant Chern classes. In this sectio*
*n and the next, we
provide quaternionic analogues, along with a suitable theory of equivariant Che*
*rn classes and their
equivariant deloopings. We must point out that in [Dup99 ], Dupont conjectures *
*the existence of
an appropriate theory of Chern classes for quaternionic bundles. Here we provid*
*e a quite natural
answer to his question.
4.1. Classifying spaces and equivariant quaternionic K-theory spectrum. In this*
* section
we describe a classifying space for quaternionic K-theory in the equivariant ca*
*tegory, and at the
same time we prove the existence of equivariant deloopings of this space.
Given a real vector space (V, oe) of complex dimension v, we follow (36) and *
*define
a2v
(43) Gr(V ) := Gr2v+j(VH* VH).
j=-2v
We make the assignment V 7! Gr(V ) functorial as follows. Consider an inclusi*
*on i : W ,! V of
real vector spaces of dimensions w and v, respectively. As in the previous sect*
*ion, we observe that
i induces an inclusion (id i)* : Gr2w+j(WH* WH ) ! Gr2v+j(WH* VH ) given b*
*y the inclusion
of linear subspaces, and that the surjection i* : V *! W *induces another inclu*
*sion (i* id)-1 :
QUATERNIONIC ALGEBRAIC CYCLES AND REALITY *
* 23
Gr2v+j(WH* VH ) ! Gr2v+j(VH* VH), given by taking inverse images. Let i] : Gr*
*(W ) ! Gr(V )
be the map given, on each connected component, by the composition
(44) i]:= (i* id)-1 O (id i)*.
Given any space X, and any real vector space (V, oe), we denote by V_Hthe qua*
*ternionic vector
bundle X x VH over X. Now, let ,jVdenote the universal quotient bundle over Gr2*
*v+j(VH* VH).
The proof of the following result is standard.
Lemma 4.2. The map on Grassmannians i]: Gr2w+j(WH* WH ) ! Gr2v+j(VH* VH), ind*
*uced by
an inclusion i : W ,! V , satisfies
(i])*,jV= ,jW (V=W__)H.
Definition 4.3. Define the (Z=2)-space (Z x BU)H as the colimit
(45) (Z x BU)H := lim-!Gr(V ).
V C1
Observe that, in the`same fashion as ZH (cf. Theorem 3.9), the space (Z x BU)*
*H can be written
as a disjoint union 1j=-1BU jHof (Z=2)-spaces BU jHdefined as BU jH:= lim-!G*
*r2v+j(VH* VH).
V C1
Furthermore, the coordinate-wise inclusion C1 ,! C1 , given by setting the firs*
*t coordinate zero,
induces equivariant homotopy equivalences
(46) BU jH~=BUj+2H~=. .~.=BUj+2rH~=... .
Hence all connected components of (Z x BU)H are either equivalent to BU 0Hor to*
* BU 1H, and we
denote by BU evHand BU oddHtheir respective equivariant homotopy types.
Proposition 4.4. The space (Z x BU)H classifies quaternionic K-theory. In other*
* words, given a
compact (Z=2)-space X one has a natural isomorphism
[X+, (Z x BU)H ]Z=2-'!KH 0(X).
Proof.The proof follows standard arguments, as in [Seg68], and we only outline *
*the details which
are particular to this case.
Let (X, oe) be a compact (Z=2)-space. Given an equivariant map f : X ! (Z x B*
*U)H, one can
find a real subspace W C1 , dimW = w, so that the map f factors as a composit*
*ion
X fW--!Gr(W ) iW--!q2wj=-2wBUjH (Z x BU)H,
where the iW 's are the natural maps from the directed system defining (Z x BU)*
*H.
Now, assign to f the isomorphism class of the virtual bundle f*W(,W ) - W__H,*
* where ,W
is the bundle over Gr(W ) = q2wj=-2wGr2w+j(WH* WH ) whose restriction to the c*
*omponent
Gr2w+j(WH* WH ) is the universal quotient bundle ,jW. If i : W ,! V C1 is*
* an inclusion,
then it follows from the construction of BU jHthat fV = i]O fW , and hence one *
*has equalities in
KH 0(X):
i j
f*V(,V) - V_H= f*W(i*](,V)) - V_H= f*W(,W ) (V=W_)_H- V_H
= f*W(,W ) + (V=W_)_H- V_H= f*W(,W ) - W__H,
24 DOS SANTOS AND LIMA-FILHO
where the second equality comes from Lemma 4.2. This shows that the element in *
*KH 0(X) thus
obtained is independent of the factorization through a finite dimensional Grass*
*mannian. Standard
arguments, e.g. [Seg68], show that this assignment only depends on the equivari*
*ant homotopy class
of f and that the resulting map : [X+, (Z x BU)H ]Z=2! KH 0(X) is injective.
In order to prove surjectivity, let (E, ø) be a quaternionic bundle over (X, *
*oe). One can find
sections si: X ! E,Pi = 1, . .,.k generating E, i.e., for each x 2 X the map 'x*
* : Ck ! Ex sending
(~1, . .,.~k) to ki=1~isi(x) is surjective. Now, define
OE : Hn x X ! E
P k P k
by sending (a1+ b1j, . .,.ak+ bkj; x) to i=1aisi(x) + i=1biø(si(oex)) 2 Ex.*
* It is clear that OE
is onto, by construction, and that OE is a map of quaternionic bundles for the *
*diagonal quaternionic
structure on Hn x X given by right multiplication by j on the first factor, and*
* by oe on the second
factor. It follows that the map f : X ! GrdimE(Hk ) defined as f(x) = ker(OEx)i*
*s equivariant and
satisfies f*Q ~=E, where Q is the universal quotient bundle over GrdimE(Hk ). A*
* little manipulation
with f and the directed system giving (Z x BU)H then shows that is onto.
Definition 4.5. For each j 2 Z, we denote by ,j the virtual universal quotient *
*bundle over BU jH,
of virtual dimension j, whose restriction to Gr2v+j(VH* VH) is ,jV- VH.
One must notice that the construction of Gr(V ) here parallels (in fact it pr*
*ecedes) that of Z(V ),
given in (36). Furthermore, given real vector spaces (V, oe), (W, oe0) one can *
*define
(47) V,W : Gr (V ) x Gr(W ) ! Gr(V W )
by sending L VH* VH and L0 WH* WH to ø(L L), where ø is the shuffle map *
*which switches
coordinates from (VH* VH) (WH* WH) to (VH WH)* (VH WH). The following pro*
*position
is analogous to Proposition 3.5 and is proven in a similar fashion.
Proposition 4.6. The assignments V 7! Gr(V ) along with the pairings V,W give*
* Gr(-) the
structure of an equivariant (Z=2)I*-functor, in the language of [May77 ]. See a*
*lso [LLFM96 ].
The proof of the following result is identical to the proof of Theorem 3.6.
Theorem 4.7. The direct sum operation induces an equivariant Z=2-infinite loop *
*space structure
on the space (Z x BU)H.
Definition 4.8. The infinite loop space structure on (ZxBU)H determines an equi*
*variant spectrum
(cf. [May96 ]) denoted Ksp, satisfying Ksp(0) ~=(Z x BU)H. This is the connecti*
*ve quaternionic
K-theory spectrum.
Remark 4.9. The same construction with VC replacing VH, and with the complex co*
*njugation ac-
tion on VC, would give (ZxBU)C along with the equivariant infinite loop space s*
*tructure classifying
KR ()-theory.
Let (V, oe) be a real vector space. An important feature of our constructions*
* is the fact that a
complex linear subspace L of codimension 2v + j in VH* VH is also an irreducib*
*le subvariety of
codimension 2v + j and degree 1 in P VH* VH . Furthermore, the external direct*
* sum of two such
subspaces corresponds to their algebraic join when seen as projective subvariet*
*ies. See [BLLF+93 ]
for more details. This gives a natural transformation c : Z(-) ! Gr(-) of (Z=2)*
*I*-functors.
QUATERNIONIC ALGEBRAIC CYCLES AND REALITY *
* 25
Standard arguments, such as in [BLLF+93 ], show that the resulting map of col*
*imits
(48) cH : (Z x BU)H ! ZH
is a map of equivariant infinite loop spaces. In other words, it induces a map *
*of equivariant spectra
cH : Ksp ! ZH.
Definition 4.10. The map cH : Ksp ! ZH is called the total quaternionic Chern c*
*lass map.
In order to analyze the quaternionic Chern class map in the level of classify*
*ing spaces, we first
need to understand the equivariant cohomology of (Z x BU)H x P(H). This computa*
*tion and the
subsequent analysis form the content of our next section.
5.Characteristic Classes
In this section we introduce characteristic classes for quaternionic bundles *
*and establish their
relation to the total quaternionic Chern class map cH : (ZxBU)H ! ZH, in the le*
*vel of classifying
spaces.
The characteristic classes dk(E) 2 H2k,k(X x P(H); Z_), associated to a quate*
*rnionic bundle
E ! X are defined as follows.
Definition 5.1. Let E be a rank e quaternionic bundle over a Z=2-space X. For f*
*fi = 0, 1, and i
satisfying 0 2i + ffi e, define
de-(2i+ffi)(E) := ece-(2i+ffi)(E O(1)) - (i + ffi) ece-(2i+ffi)-1*
*(E O(1)) x,
where c-1(-) = 0.
Note that, since X x P(H) is a free Z=2-space, the characteristic classes ~ck*
*(E O(1)) can also be
defined with values in Galois-Grothendieck cohomology as in [Kah87 ], according*
* to the discussion
in Section 1.7.
In order to understand the meaning of this definition, and its relation to th*
*e total Chern class
map, we first compute the equivariant cohomology of (Z x BU)H x P(H) in the nex*
*t section.
5.1. Cohomology of (Z x BU)H. We start by observing that all the components of *
*(Z x BU)H x
P(H) have the same equivariant homotopy type, hence it suffices to compute H*,**
*(BU evHxP(H); Z_).
In fact, an equivariant homotopy equivalence : BU evHxP(H) ! BUoddHxP(H) can *
*be constructed
as follows. Given a real vector space (V, oe), let
_j,j+1V: Gr 2v+j(VH* VH) x P(H) ! Gr2v+j+1((V C)*H (V C)H)
denote the composition
V,C
(49) Gr2v+j(VH* VH) x P(H) idx'---!Gr2v+j(VH* VH) x Gr1(H* H ) ---!
Gr 2v+j+1((V C)*H (V C)*
*H),
where ' : P(H) ! Gr1(H* H ) is the linear embedding sending L to L H, and V,C*
*is the Whitney
sum map (47) after one identifies H CH.
It is easy to see that
(_j,j+1V)*(,j+1V) C= ß*1(,jV) H_ ß*2(O(1)),
where the ßi's are the projections from Gr2v+j(VH* VH) x P(H) to the respectiv*
*e factors, ,jVis the
universal quotient bundle over the Grassmannian, and O(1) is the hyperplane bun*
*dle over P(H).
26 DOS SANTOS AND LIMA-FILHO
Note that O(1) is also a quaternionic bundle. Furthermore, the maps _j,j+1Vass*
*emble to give a
morphism of directed systems inducing a (Z=2)-equivariant map _j,j+1: BU jHx P(*
*H) ! BU j+1H
having the property that (_j,j+1)*(,j+1) = ,jxO(1). Here we denote ß*1(,j) ß*2(*
*O(1)) as a product
,jx O(1).
It follows that one obtains a Z=2-map _ : BUevHx P(H) ! BUoddH. Now, define
(50) := _ x id: BUevHx P(H) ! BUoddHx P(H),
and observe that is a Z=2-map which is a non-equivariant homotopy equivalence*
*. Since BU evHx
P(H) and BU oddHx P(H) are free (Z=2)-spaces, it then follows that is an equi*
*variant homotopy
equivalence.
Remark 5.2. The same procedure used to construct can be applied to produce an*
* equivariant
homotopy equivalence : ZevHx P(H) ! ZoddHx P(H). Indeed, given a real vector *
*space (V, oe), let
OEj,j+1V: Z2v+j(VH* VH) x P(H) ! Z2v+j+1((V C)*H (V C)H)
denote the composition
#V,C
(51) Z2v+j(VH* VH) x P(H) idx'---!Z2v+j(VH* VH) x Z1(H* H) ---!
Z2v+j+1((V C)*H (V C)H*
*),
As before, one can check that the maps OEj,j+1Vassemble to give a morphism of d*
*irected systems
inducing a (Z=2)-equivariant map OEj,j+1: ZjHx P(H) ! Zj+1H. Finally, define
:= OE x id: ZevHx P(H) ! ZoddHx P(H).
Non-equivariantly, one can fix a point t 2 P(H), and observe that the map (-, *
*t) is just suspension
to t. It follows that is a non-equivariant homotopy equivalence. Since ZevHxP*
*(H) and ZoddHxP(H)
are free (Z=2)-spaces, it then follows that is also an equivariant homotopy e*
*quivalence.
Before we compute the equivariant cohomology of BU evHx P(H) some notation is*
* needed.
Notation 5.3. Given a real bundle E over a Z=2-space X, its k-th equivariant Ch*
*ern class is
denoted by eck(E) 2 H2k,k(X; Z_). The generator of H2,1(P(H); Z_) ~=Z is denote*
*d by x, as before.
We denote the line bundle O(m) over P(Hn) by On(m). In the case n = 1 we write *
*O(m) instead
of O1(m). Notice that On(m) is real, if m is even, and quaternionic if m is odd*
*. Also, note that
x = ec1(O(2)). The fact that a tensor product of two quaternionic bundles is a *
*real bundle will also
be used throughout.
Remark 5.4. Let ,jVdenote the universal quotient bundle over Gr2v+j(VH* VH), c*
*f. Lemma 4.2,
and let i]: Gr2w+j(WH* WH ) ! Gr2v+j(VH* VH) be the maps defined in (44). It *
*is easy to see that
one has i*](dk(,jV)) = dk(,jW). Hence, these classes yield elements dk 2 H2k,k(*
*BU jHx P(H); Z_) that
are compatible with the equivalences BU jH~=BUj+2H, described in (46). It follo*
*ws that we obtain
well-defined classes in H2k,k(BU evHxP(H); Z_) . Using the homotopy equivalence*
* : BUevHxP(H) !
BU oddHx P(H), we define the corresponding classes in BU evHx P(H), and also de*
*note these classes
by dk. The following facts are easily verified:
a: Let f : X ! (Z x BU)H be a classifying map for a quaternionic bundle E o*
*ver X. Then:
dk(E) := (f x id)*(dk).
QUATERNIONIC ALGEBRAIC CYCLES AND REALITY *
* 27
b: Let E ! X be a quaternionic bundle of rank e, and let æ denote the forge*
*tful functor
from equivariant cohomology to singular cohomology. Then, denoting the f*
*undamental
cohomology class of P(H) by fi, we have
(
ck(E) x 1 + ck-1(E) x fi, k e mod 2
æ(dk(E)) =
ck(E) x 1 , k 6 e mod 2.
In particular, considering the universal bundles ,jkover Gr2k+j(Hk * Hk)*
*, we see that the
classes æ(dk) generate the singular cohomology of (Z x BU)H x P(H) over H*
**(P(H); Z).
We now compute the equivariant cohomology of (Z x BU)H.
Theorem 5.5. Let dk be the classes defined in (5.1). Then we have the following*
* ring isomorphism
H*,*(BU evHx P(H); Z_) ~=H*,*(P(H); Z_) [d1, d2, . .,.dk, . .].
Proof.Since BUevHxP(H) is free, it suffices to compute the Galois-Grothendieck *
*cohomology groups
with Z(n) coefficients. For this we use the spectral sequence (cf. Section 1.7)
Er,s2(n) := Hr(BZ=2; H(BU evHx P(H); Z(n))) ) ^Hr+sZ=2(BU evHx P(H); Z*
*(n)),
0,q0 p+p0,q+q0
and the pairing of spectral sequences Ep,qr(n) Epr (n0) ! Er (n + n0), i*
*nduced by the cup
product. This pairing makes E*,*r(*) into a spectral sequence of Z2x Z=2-graded*
* rings.
Set
F2p,q(n) := Hr(BZ=2; H(P(H); Z(n))),
and note that E*,*2(*) is a module over F2*,*(*). Corresponding to each of the *
*classes dk there are
elements in E0,2k2(k) which are universal cycles. These cycles are denoted edk.*
* The correspondence
dk 7! edkdefines a ring homomorphism : F2*,*(*) [d1, d2, . .,.dn, . .].! E*,**
*2(*).
Observe that the action of j* in H2q(BU evHx P(H); Z) is multiplication by (-*
*1)q, hence
(
Hq(BU evHx P(H); Z), q = 2q0, n = q0 mod 2
E0,q(n) =
0 , otherwise
As noted in remark 5.4 , the image of the dk's, under the forgetful functor t*
*o singular cohomology,
generates the cohomology of BU evHx P(H) over H*(P(H); Z). This implies that *
*0,2q(q) is an
isomorphism. Since, Ep,0(n) = Hp(BZ=2; Z(n)), we see that p,0(n) is also an is*
*omorphism. By
Zeeman's comparison theorem, it follows that
E*,*1(*) ~=F1*,*(*)[d1, d2, . .,.dn, . .]..
A standard argument can be used to show that there is actually a ring isomorp*
*hism
^H*Z=2(BU evHx P(H); Z(*)) ~=^H*Z=2(P(H); Z(*)) [d1, d2, . .,.dk,*
* . .]..
By Proposition 1.13, we conclude that
H*,*(BU evHx P(H); Z_) ~=H*,*(P(H); Z_) [d1, d2, . .,.dk, . .]..
The following result will be used subsequently.
28 DOS SANTOS AND LIMA-FILHO
Lemma 5.6. Let E ! X be a quaternionic bundle of dimension e, and let z 2 H2,1(*
*P(H); Z) be
the fundamental homology class. Then we have
e - 2
pr*(de-2k(E)=z) = de-2k-1(E),k = 0, . .,.b_____c,
2
where -=z denotes slant product with z and pr is projection to the first factor*
* of X x P(H).
Proof.We will prove the case where e is even. The other case is proven similar*
*ly. Recall from
Theorem 5.5 that the equivariant cohomology of BU evHx P(H) is generated over H*
**,*(P(H); Z_) by
the classes dk, k 1. The result will follow once we prove the identity
pr*(d2k=z) = d2k-1,
where pr : BU evHx P(H) is the projection onto the first factor. Observe that,*
* by construction,
the restriction of d2k to Gr2k-2(Hr* Hs)is zero. Hence, by Theorem 5.5, pr*(d*
*2k=z) must be a
multiple of d2k-1. Since
æ(pr*(d2k=z)) = æ(d2k-1) = c2k-1x 1,
it follows that pr*(d2k=z) = d2k-1.
5.2. Projective bundle formula. In the previous section we defined characterist*
*ic classes, dk(E),
for a quaternionic bundle E over X, with values in the cohomology theory H*,*(X*
* x P(H); Z_). By
analogy with Chern classes, it is natural to look for a relation between the cl*
*asses dk(E) and the
structure of H*,*(P(E) x P(H); Z_), as module over H*,*(X x P(H); Z_). The next*
* result addresses
this question.
p *
* p
Proposition 5.7. Let P E -!X be the projectivization of a rank e quaternionic *
*bundle E -!X.
Let , in KH (P E ) be the universal quotient bundle over P E . Then H*,*(P E xP*
*(H); Z_) is a free
H*,*(XxP(H); Z_)-module generated by dk(,), k = 0, . .,.e-1. Moreover, the clas*
*ses d0(,), . .,.de(,)
satisfy the following relation
Xe
(52) (-1)kdk(,)de-k(p*E) = 0.
k=0
Proof.Let p1 : X x P(H) ! X and p2 : X x P(H) ! P(H) be the projections. Then *
*G :=
p*1(E) p*2(OP(H)(1)) is a real bundle, and there is an equivariant homeomorph*
*ism
* * *
P G = P p1(E) p2(OP(H)(1)) ~=P p1(E) ~=P E x P(H).
It follows from the projective bundle formula for real bundles that, if OG(1) i*
*s the dual of the
tautological line bundle over P E and t = ~c1(OG(1)), then H*,*(P E xP(H); Z_)*
* is a free H*,*(X x
P(H); Z_)-module generated by 1, t, . .,.te-1. Moreover, the classes ~c0(p*G), *
*. .,.~ce(p*G) satisfy the
relation
X
(53) (-1)2i+ffit2i+ffi~ce-2i+ffi(q*G) = 0,
0 ffi 1
0 2i+ffi e
where q : P G ! X x P(H) is the bundle projection.
A simple computation shows that the classes tk can be expressed in terms of t*
*he classes dk(,).
In fact, denoting the first Chern class of the real bundle O(2) by x as before,*
* we have
(54) t2i+ffi= d2i+ffi(,) + i d2i+ffi-1(,)x,
QUATERNIONIC ALGEBRAIC CYCLES AND REALITY *
* 29
for 0 ffi 1 and 0 2i + ffi e.
Hence H*,*(P E xP(H); Z_) is also generated by 1, d1(,), . .,.de(,). It remai*
*ns to show that (52)
holds. Now,
de-(2i-ffi)(p*E) = ce-(2i-ffi)(q*G) - i ce-(2i-ffi)-1(q*G)x,
for 0 ffi 1 and 0 2i + ffi e (where we've set c-1(G) := 0). Hence from *
*(53), (54)and the
equality x2 = 0, we get
Xe X
(55) (-1)k dk (,) de-k(p*E) = (-1)2i+ffid2i+ffi(,) de-(2i+ffi)(p*E)
k=0 0 ffi 1
0 2i+ffi e
X i j
= (-1)2i+ffit2i+ffi- i t2i+ffi-1xce-(2i+ffi)(q*G) - i ce-(2i+ffi)*
*-1(q*G) x
0 ffi 1
0 2i+ffi e
X i *
* j
= (-1)2i+ffi-1i t2i+ffice-(2i+ffi)-1(q*G) - t2i+ffi-1ce-(2i*
*+ffi)(q*G)=x0.
0 ffi 1
0 2i+ffi e
5.3. The quaternionic total Chern class map. Recall from Theorem 3.9 that the s*
*pace ZH
splits equivariantly as a product of classifying spaces for the functors H*,*(-*
* x P(H); Z_). Given
any equivariant map, X ! ZH, such a splitting determines a set of classes in H**
*,*(X x P(H); Z_).
We have seen in Definition 4.10 that the inclusion of linear spaces in the sp*
*ace of all algebraic
cycles induces an equivariant map cH : (Z x BU)H ! ZH. In this section we will*
* compute the
cohomology classes determined by the map cH and the splitting (41)of Theorem 3.*
*9.
Proposition 5.8. Let ,n 2 KH (P Hn ) be the universal quotient bundle over P H *
*, and let
n Yn
(56) _n : Z0 P H ! F (P(H)+, K(Z(2i - 1), 4i - 2))
i=1
be the equivariant homotopy equivalence of corollary 2.4, and let jn : P Hn ,!*
* Z0 P Hn denote
the natural inclusion. Then the composition
n Yn
OEn := _n O jn : P H ! F (P(H)+, K(Z(2i - 1), 4i - 2))
i=1
classifies d1(,n), d3(,n), . .,.d2n-1(,n).
Proof.For n = 1, we need to identify the element ff of H2,1(P(H) x P(H); Z_) cl*
*assified by the
composition P(H) j-!Z0(P(H))-_!F (P(H)+, K(Z(1), 2). Since _ is the map that re*
*alizes Poincar'e
duality, ff is the Poincar'e dual of the diagonal P(H) x P(H). From the pr*
*ojective bundle
formula (Proposition 5.7) it is easy to see that
ff = P( ) = c1(pr*1O(1) pr*2O(1))= d1(,1).
Assume that the proposition holds for k < n, and note that Proposition 2.2(c)*
* implies that OEn
restricts to OEk on P Hk , for k < n. The projective bundle formula shows that *
*d2i-1(,n) is the only
30 DOS SANTOS AND LIMA-FILHO
class whose restriction to P Hk is d2i-1(,k), for all i = 1, . .,.k. Hence it *
*suffices to show that
pn O OEn classifies dn(,n), where pn denotes the projection onto the last facto*
*r in (56).
Observe that pn O OEn is given by the composition
n i n n-1 j n n-1 `
P H + -!P H =P H -! Z0 P H =P H o -!F (P(H)+, K(Z(2n - 1), 4n - 2)*
*),
where ß is the projection, j is the natural inclusion and ` is the equivalence *
*given in (8). The
P Hn xP(H)
adjoint map (` O j)_: ___________! K(Z(2n - 1), 4n - 2), can be described as t*
*he composition
P Hn-1 xP(H)
n-1
(57) T O(1) H ^ P(H)+
-^id--!TO(1) Hn-1^ P(H) n-1
+ ^ P(H)+ = T O(1) H ^ (P(H) ^ P(H))+
f~^d1 i 4(n-1),2(n-1)j 2,1 4n-2,2n-1
----! Z0 S ^ Z0 S ! Z0 S = K(Z(2n - 1), 4n -*
* 2),
o o o
where f~ classifies the Thom class of O(1) Hn-1. One can easily check that t*
*he pull-back of
[f~] under the projection ß : P Hn ! T O(1) Hn-1 coincides with d2n-2(,n). *
* Therefore,
[pn O OEn] = d2n-2(,n)d1(,n) = d2n-1(,n), and this completes the proof of the P*
*roposition.
We can now compute the classes determined by total quaternionic Chern class m*
*ap cH
Theorem 5.9. The equivariant cohomology classes determined by total quaternioni*
*c Chern class
map cH : (Z x BU)H ! ZH and the splitting (41)of Theorem 3.9 are
(58) 1 + d2+ d4+ . .+.d2n+ . . .on BU evH
(59) d1+ d3+ . .+.d2n+1+ . . . on BU oddH
Proof.Recall that there are natural equivalences BU jH~=BUj+2Hand ZjH~=Zj+2Hand*
*, moreover, the
map cH : (Z x BU)H ! ZH is compatible with these equivalences. Thus cH induces *
*maps
cevH: BUevH! ZevH and coddH: BUoddH! ZoddH
and it suffices to compute the equivariant cohomology classes they classify.
The map coddH: BUoddH! ZoddHclassifies an element D1+ D3+ . .+.D2n-1+ . .w.ith
D2i-12 H4i-2,2i-1(BU oddHx P(H); Z_) , i 1.
Note that by construction, we have cH(Gr2q-3(Hk * Hk)) Z2q-3H(Hk * Hk) hence
(D2q-1)|Gr2q-3(Hk* Hk)= 0,k 0.
It follows from Theorem 5.5 that there are constants ~1, ~2 such that
D2q-1= ~1d2q-1+ ~2 x . d2q,
where x 2 H2,1(P(H); Z_) is the fundamental class of P(H). To compute ~1, ~2 we*
* observe that, by
Proposition 5.8, the restriction of D2q-1to P Hq is d2q-1(,q), where ,q is the*
* universal quotient
bundle over P Hq . Since the inclusion
q 2q-1 q* 2q-1 q* 2q-1 q* q 2q-1
P H Gr (H )Gr (H ) Gr (H H ) BUH
classifies ,q, it follows that D2q-1|P Hq= d2q-1(,q). Thus ~1 = 1, ~2 = 0 and D*
*2q-1= d2q-1.
QUATERNIONIC ALGEBRAIC CYCLES AND REALITY *
* 31
Now, consider the map cevH: BUevH! ZevH. It classifies an element 1 + D2+ D4+*
* . .+.D2n+ . . .
with D2i2 H4i,2i(BU evHx P(H); Z_) i 1. Once again, we observe that cH(Gr2q-*
*2(Hk * Hk))
Z2q-2H(Hk * Hk), implying
(D2q-2)|Gr2q-2(Hk* Hk)= 0,k 0.
As before, we conclude that there are constants ~1, ~2 such that D2q = ~1d2q+ ~*
*2 x . d2q-1.
To compute ~1, ~2 it suffices to compute the image of D2q under the forgetful m*
*ap to singular
cohomology. In [LM88 ] it is shown that, non-equivariantly, Z1(P Hn 1~=K(Z, 2) *
*x . .x.K(Z, 4n -
2), and that under this equivalence cH classifies the total Chern class. One c*
*an show that the
decomposition (34)is compatible with this non-equivariant splitting. It follow*
*s that æ(D2k) =
ck + ck-1fi, hence ~1 = 1 and ~2 = 0.
5.4. The group struture on Z0H(X). In this section we compute the group structu*
*re induced by
the algebraic join of cycles on ZH. Recall from Proposition 3.5 that the algebr*
*aic join # induces a
pairing
# : ZH x ZH ! ZH
satisfying
0
# : ZjHx ZjH! Zj+jH.
For a Z=2-space X, one has an identification
M h ji M Y
(60) Z0H(X) = [X, ZH]Z=2 = X, ZH = H4r-2ffl(j), 2r-ffl(j)(X x*
* P(H), Z_),
j2Z Z=2 j2Zr 1
where ffl(j) is 0 if j is even and 1 if J is odd. Given the splitting above one*
* might conjecture that
the group structure induced by # is induced by the cup on H*,*(X x P(H); Z_), h*
*owever this is not
the case as we will show. From (60)it follows that the group structure on Z0H(X*
*) is completely
determined by the cohomology class represented by the map # under the equivalen*
*ces
0 j+j0 Y 4r-2ffl(j+j0), 2r-ffl(j+j0)jj0
[ZjHx ZjH, ZH ]Z=2~= H (ZH x ZH x P(H), Z_).
r 1
Also recall that BU jHmaps to ZjHby cH and that the following diagram commutes
0 j+j0
BUjHx BUjH____//_BUH
cHxcH|| cH||
fflffl|0# fflffl|0
ZjHx ZjH______//_Zj+jH
We claim that the maps cH above induce injective maps
i j
H*,*(ZjHx P(H); Z_) ! H*,*(Z0 BU jH x P(H); Z_)
1
and hence the pairing on Z0H(-) is completely determined by the formula for the*
* quaternionic Chern
class of a Whitney sum.
Let us start with the case j = 1. Let ' : P(H1 ) ! BU 1Hbe the map induced by*
* the inclusions
'n : P(Hn) ! Gr1(Hn * Hn) that send L to L Hn. Composing with cH gives a ma*
*p from
P(H1 ) to Z1H. The linear extension Z0(P(Hn))! Z1Hof this composition is, by th*
*e quaternionic
32 DOS SANTOS AND LIMA-FILHO
suspension theorem, an equivariant homotopy equivalence when restricted to the *
*component of 1.
fcH
This map factors as Z0(P(Hn))1-e'!Z0 BU 1H1-! Z1H, where e'and fcHdenote the li*
*near extension
of ' and cH, respectively. It follows that, for any cohomology theory H*, the m*
*ap H(fcH) : H*(Z1H) !
H*(Z0 BU 1H1) is injective. For 1 6= j odd it suffices to observe that fcHis co*
*mpatible the canonical
equivariant homotopy equivalence ZjH~=Zj+2H, given by inclusion of hyperplanes.
Suppose now that j is even. We need to show that fcHinduces an injective map
i j
(idxfcH)* : H*,*(ZjHx P(H); Z_) ! H*,*(Z0 BU jH ) x P(H); Z_).
1
The same argument as before shows that we may replace BU jHby BU evH. Recall fr*
*om Remark 5.2
that there are equivariant homotopy equivalences : BU evHx P(H) ! BU oddHx P(*
*H) and :
ZevHx P(H) ! ZoddHx P(H). By construction and are compatible with cH so tha*
*t the following
diagram commutes
(idxcH)* *,* ev
H*,*(ZevHxOP(H);OZ_)__//_H (Z0(BUOHO)1x P(H); Z_)
*|| *||
| (idxcH)* *,* |odd
H*,*(ZoddHx P(H); Z_)_//_H (Z0 BU H 1x P(H); Z_).
It follows that the map on top is injective, as desired.
Proposition 5.10. Let X be a Z=2-space, and let a.b denote the product of eleme*
*nts a, b in Z0H(X).
Consider Z0H(X) included in
M Y
Hr,s(X x P(H), Z_),
j2Zr,s 1
as in (60). Then, under this inclusion we have,
a . b = a [ b + pr*(a=z) [ pr*(b=z),
where z 2 H2,1(P(H); Z_) is the fundamental class P(H), -=z denotes slant produ*
*ct with z and pr is
the projection onto the first factor in the product X x P(H).
Proof.By the preceeding remarks it suffices to show the following formula for t*
*he quaternionic
Chern class of a Whitney sum holds
cH(E F ) = cH(E) [ cH(F ) + pr*(cH(E)=z) [ pr*(cH(F )=z).
Recall that cH(E) is defined as a combination of Chern classes of the real bund*
*le E O(1), where
O(1) is the hyperplane bundle over P(H). To simplify notation we will use ~Ean*
*d ~Fto denote
E O(1) and F O(1), respectively. Let e , f be the dimensions of E and F , r*
*espectively. We
QUATERNIONIC ALGEBRAIC CYCLES AND REALITY *
* 33
have,
be_2cXh i bf_2cXh *
* i
(61) cH(E) [ cH(F ) = ece-2i(E~) - iece-2i-1(E~)x [ ecf-2j(F~) - jecf-2j*
*-1(F~)x
i=0 j=0
be+f_2cXrX
= ece-2s(E~) [ ecf-2(r-s)(F~)
h r=0 s=0 *
* i
- sece-2s-1(E~) [ ecf-2(r-s)(F~) + (r - s)ece-2s(E~) [ *
*ecf-2(r-s)-1(F~) x,
where we've set eck(- ) equal to 1 if k is zero and eck(- ) equal to zero if k *
*< 0. By Lemma 5.6 we
have
be_2cX bf_2cX
(62) pr*(cH(E)=z) [ pr*(cH(F )=z) = de-2i-1(E) [ df-2j-1(F )
i=0 j=0
be+f_2cXrX
= ece-2s-1(E~) [ ecf-2(r-s)-1(F~)
h r=0 s=0 *
* i
- sece-2s-2(E~) [ ecf-2(r-s)-1(F~) + (r - s)ece-2s-1(E~) [ *
*ecf-2(r-s)-2(F~) x.
Thus, we get
(63) cH(E) [ cH(F ) + pr*(cH(E)=z) [ pr*(cH(F )=z)
be+f_2cX2rX h *
* i
= ece-s(E~) [ ecf-2r+s(F~) - rece-2s-1(E~) [ ecf-2r+s(F~) + rece-2s(*
*E~) [ ecf-2r+s-1(F~) x
r=0 s=0
be+f_2cX
= ece+f-2r(E~ ~F) - rece+f-2r-1(E~ ~F)x = cH*
*(E F ).
r=0
The Proposition follows.
5.5. Remarks on the space (Z x BU)H. Here are two facts about (Z x BU)H which s*
*eem to be
quite interesting. Both of them are particular cases of results of Karoubi [Kar*
*00].
Remark 5.11. In [Kar00] , Karoubi observes that there is an involution on BU su*
*ch that BU hZ=2=
BSp . We claim that the involution on BU evHsatisfies this. It is clear that *
*BU evHis homotopy
equivalent to BU and that {BU evH}Z=2 = BSp . We now proceed to show that {BU*
* evH}Z=2 =
{BU evH}hZ=2.
The proof mimics one of the proofs of the well known fact BO = BU hZ=2, usin*
*g Dupont's
quaternionic K-theory instead of KR-theory. Let X be a Z=2-space. From [Dup69 ]*
*, we know that
there is a natural splitting
KH (X x P(H)) ~=KH (X) KR(X).
34 DOS SANTOS AND LIMA-FILHO
This implies that there is a natural equivariant homotopy equivalence
F (X x P(H)+, (Z x BU)H) ~=F (X+, Z x BU) _ F (X+, (Z x BU)H).
Applying this equivalence to XxEZ=2 instead of X and using the equivariant homo*
*topy equivalence
X x P(H) x EZ=2 ~=X x P(H),
we obtain an equivariant homotopy equivalence
F (X x EZ=2+, (Z x BU)H) ~=F (X+, (Z x BU)H).
Remark 5.12. Another interesting fact about (Z x BU)H is that
(Z x BU)H ~= 4,0(Z x BU).
To see this, consider a map f : S4 ! BSp which represents a generator of ß4(BSp*
* ). The tensor
product induces an equivariant map
S4,0^ (Z x BU)H ! Z x BU.
Its adjoint (Z x BU)H ! 4,0(Z x BU) is an equivariant map which is a non-equiv*
*ariant homotopy
equivalence, and whose restriction to the fixed points induces the homotopy equ*
*ivalence
Z x BSp ~= 4BO ;
see [Bot59].
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Department of Mathematics, Texas A&M University, USA
Departamento de Matem'atica, Instituto Superior T'ecnico, Portugal
Department of Mathematics, Texas A&M University, USA