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]. References [Ati66] M. F. Atiyah, K-theory and reality, Quart. J. 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