Algebraic cycles and infinite loop spaces - Springer Link

Invent. math. 113,373-388 (1993)

Inventiones mathematicae 9 Springer-Verlag 1993

Algebraic cycles and infinite loop spaces Charles P. Boyer I, H. Blaine Lawson 2, Jr., Paulo Lima-Filho 3, Benjamin M. Mann I and Marie-Louise Michelsohn 2"* 1 University of New Mexico, Albuquerque, NM 87106, USA 2 SUNY at Stony Brook, Department of Mathematics, Stony Brook, NY 11794, USA 3 Department of Mathematics, The University of Chicago, Chicago IL 60637, USA Oblatum XII-199l & 4-II-1993

Summary. In this p a p e r we use recent results a b o u t the t o p o l o g y of C h o w varieties to a n s w e r a n o p e n q u e s t i o n in infinite loop space theory. T h a t is, we c o n s t r u c t a n infinite loop space structure o n a certain p r o d u c t of E i l e n b e r g - M a c L a n e spaces so that the total C h e r n m a p is a n infinite loop map. A n a n a l o g o u s result for the total Stiefel-Whitney m a p is also proved. F u r t h e r results o n the structure of stabilized spaces of alebraic cycles are o b t a i n e d a n d c o m p u t a t i o n a l consequences are also outlined.

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Chow varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 The complex join . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 l,-functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 The Chow monoid functor . . . . . . . . . . . . . . . . . . . . . . . . . . 6 The main theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Real cycles and Stiefel-Whitney universal map . . . . . . . . . . . . . . . . . . 8 Computational consequences . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

373 377 378 380 381 383 385 386 387

I Introduction Recently some progress h a s been m a d e in the s t u d y of the t o p o l o g y of the C h o w varieties of algebraic cycles in IP" [ L a w ] . In t h a t analysis certain m e t h o d s of h o m o t o p y t h e o r y played a n i m p o r t a n t role. In t u r n it has also b e e n s h o w n t h a t the C h o w varieties yield in a n a t u r a l way some of the f u n d a m e n t a l c o n s t r u c t i o n s of t o p o l o g y [ L M 1 ] . In this p a p e r we shall use these n a t u r a l c o n s t r u c t i o n s to answer * All authors were partially supported by the NSF

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some outstanding questions in homotopy theory. The results will also have direct application to the study of the spaces of algebraic cycles. The main problem we solve concerns the universal Chern class. If BUq denotes the classifying space for complex vector bundles of rank q and K(77, 2j) denotes the Eilenberg-MacLane space, then the total Chern class for the universal bundle over BUq determines a map q

c:BUq --+ ~I K(71, 2j).

(1.0.1)

j=0

By taking the limit as q ~ oo one obtains a map c:BU ~ f i K(7/, 2j) = K(71, even).

(1.0.2)

j=o

It is well known that BU is an H-space with multiplication induced by the Whitney sum of bundles and, in fact, this structure may be enriched to make (BU, ~ ) an infinite loop space. (Since a space may be an infinite loop space in several inequivalent ways we keep track of the structure map in the notation). The space K(77, even)also has a natural infinite loop space structure given by the direct product of the standard structures on each factor K(77, 2j). However, with this structure on K(Z, even) the map c is not even a map of H-spaces. This is easily seen as follows. Considering homotopy classes of maps of a finite complex X into each of these spaces yields the K-theory K ( X ) = [X, BU] and the cohomology HeY(X;71) = IX, K(7I, even)] of even degrees. The infinite loop space structures above yield the standard addition on these groups. However, as Grothendieck [ G r o ] pointed out in 1958, the total Chern class is additive only if one redefines the additive structure on H~ 71) by using the cup product. Specifically, we set

G(X,z)d~=f H ~

I-I H2i(X;71)) i>=l

with additive structure given in H o (X; 71) as usual, and in (l • lqi _>_1H 2i(x; ~)) by the formula (l+xl+x2+

...)(l+x'~+x'2+

. . . )= l + (xl + x2) + (xz + xlx'~ + x'2) . . . .

An analogous formula defines multiplication, and with respect to this formula the map c: K ( X ) ~ G(X, 2~) is a ring homomorphism. In [Segl] Segal asked whether the map c extends to a stable natural transformation between cohomology theories. Equivalently, does K(71, even) admit an infinite loop space structure 9 such that c:(BU, ~ ) -* (K(7Z, even), , ) is an infinite loop space map? Such an infinite loop space structure must, of course, enhance the multiplication given in the formula above. In fact, in [Segl] Segal introduced a new infinite loop space structure on K(:E, even) which enhanced this multiplication. In [Ste2] and [Stel] Steiner explored Segal's constructions and explained it in greater detail. However, it was shown by Snaith [Sna] that this structure does not provide the desired result.

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Segal's question, and its ramifications, continued to attract attention over the years. There has been, for example, very interesting work by Kozlowski [Kozl, Koz2] and also by Kraines and Lada [KL2]. Nevertheless, the question has, to this time, remained unanswered. In this paper we shall resolve the question by using constructions with Chow varieties that are compatible with well-known constructions in infinite loop space theory. We recall that results of [Law] and [LM1] showed that: (1) Certain Chow varieties build both BU and K(7/, even) in a particularly regular and compatible way. (2) There is a natural multiplication of cycles, the complex join ~r162which extends to an H-space multipliction on both BU and K(7/, even). (3) (BU, 0) ~ (BU, ~r as H-spaces. We use the complex join to construct actions of the operad of linear isometric embeddings by using May's formalism of •,-functors. Then applying May's infinte loop space machine leads to our first main result: Theorem A. The multiplication on K(7I, even) induced by the complex join :~r enriches to an infinite loop space structure with respect to which the total Chern class map c:(BU, :~r --* (K(Z, even), :~r is an infinite loop map. Remarks. (1) Our main constructions take place at the finite-dimensional level and extend a well-known construction of the space BU as an infinite loop space [May 2, pp. 16 18]. We use the complex join to extend the Whitney sum of linear subspaces of ~'" to a pairings of general algebraic cycles. We then construct an d,-functor which associates to an n-dimensional hermitian inner product space V the monoid of effective algebraic cycles of codimension-n on IP(V @ V). There is a compatible functor which associates to V the set of degree-I cycles, i.e., the Grassmannian of linear subspaces of codimension-n on ~'(V@ V). The formalism of d,-functors then takes over to prove Theorem A. (2) It is not surprising that the Chow varieties, which are holomorphic objects, model an interesting iterated-loop space structure on K(Z, even). It is, in fact, a recurring theme in the recent study of moduli spaces that certain finite dimen r sional complex manifolds provide good models for building loop spaces. Recent examples of this phenomenon include instantons and monopoles, linear control systems, and various holomorphic mapping spaces (see, for example, [AJ, BHMM, BM1, BM2, CCMM, Don, MM, Seg2, Tau]). Segal [Segl] also asked the analogous question in the real case. That is, is there an infinite loop space structure on

K(7Z/2,,) ae~ I~ K(2E/2, i)

(1.0.3)

i>_o

so that the total Stiefel-Whitney map is an infinite loop map? Here again Segal introduced a new infinite loop space structure on 1.0.3 enhancing the real analogue of the multiplication explained above but Snaith [-Sna] proved this structure does not have the required property. Using the results of Lam [Lam] on real algebraic cycles we prove our second main result.

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Theorem B. The multiplication on K(7//2, ,) induced by Lam's algebraic join :~r enriches to an infinite loop space structure with respect to which the total StiefelWhitney map w:(BO, •) ~ (K(7Z/2,,), $r is an infinite loop map.

Thus Theorems A and B answer Segal's questions in the affirmative. We conjecture that the analysis in this paper extends, with appropriate modifications, to produce an analogous theorem in the group actions cases studied in [LM2]. For readers unfamiliar with infinite loop space theory we point out that when a space S carries an infinite loop space structure, it determines a spectrum (i.e. a sequence of spaces S = So, S1, $ 2 , . . . , ) such that Si = f2Si+l and thereby generalized homology and cohomology theories. Our main results imply the following. Corollary C. The functor G ~ defined to be HeY(", 7l) with the "Grothendieck addition" above, enhances to a full 9eneralized cohomology theory G* with the property that the total Chern class becomes a natural transformation c:k* ~ G* from connective K-theory. Similarly, the functor GO ~ def H*(', 7//2) with the analogous addition, enhances to a generalized cohomology theory GO* so that the total StiefeI-Whitney class becomes a natural transformation w:ko* ~ GO* from connective K O-theory.

This result has a number of important consequences. For example, given a finite covering map f: X --* Y, there is a transfer homomorphism f defined in each theory. The main results here imply that the total Chern class map c and the total Stiefel-Whitney class map w commute with these transfer homomorphisms. In [Koz2] Kozlowski directly constructed such a transfer homomorphism for the groups GO~ which appears to coincide with ours. Our main results also imply that the homomorphisms in homology induced by c and w are module homomorphisms over the Dyer-Lashof algebra. This has strong implications for calculation of the Dyer-Lashof operations on the homology of (K(7/, even), :r and (K(7//2, *), $). The remainder of this paper is organized as follows: Sections 2 through 4 review the basic facts about Chow monoids, the complex join and J,-functors. Section 5 gives the central construction of the Chow monoid functor. In Sect. 6 we prove Theorem A. In Sect. 7 we prove Theorem B and comment on the work of Snaith. Section 8 concludes with a brief summary of some computational consequences of our main results. We anticipate that these computations should be useful, in the spirit of IBM1], in obtaining unstable homological information about individual Chow varieties and intend to return to this question in a future paper. The paper of Totaro [Tot] should be helpful in this regard.

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2 Chow varieties In the following two sections we summarize selected results of [Law] and [LM1] necessary to state and prove our main result. We begin by recalling Definition 2.1 Let p + q = n. By the set of all finite formal sums c = ~ n , (1) F o r each ~, n, is a positive subvariety of complex dimension p n-space IP", and where (2)

Chow variety cg~,d(~" ) ~f C~p,d we mean the W~ where integer, and W, is an irreducible algebraic (and codimension-q) in complex projective

deg(c) def ~ n~ degree(W~) = d.

Each cg~,dcan be realized in a canonical way as a projective algebraic variety. In particular, each ~q,d is canonically a compact Hausdorff space. Under addition of cycles, the disjoint union def

~ , , = II ~,~, d>O

is an abelian topological monoid and thus has a homotopy-theoretic group completion ~2B~,.. Here we set ~ , o = {0}. There is a natural model for this space. Fix a p-dimensional linear subspace : o e l P " , and consider the analytic embeddings ~ , d = C ~ , d + 1 defined by c ~-+c + {o- Set

cgqp= lira ~ , d d~oo

with the compactly generated topology. The standard mapping telescope argument gives a homotopy equivalence 7/• cg~,= f2BC~,,. The first main result of [Law] is the following. Theorem 2.2 For each q and p there is a homotopy equivalence 7/x egg = f i K(7/, 2j). (2.2.1) j=o We focus now on cycles of degree 1 and observe that ~ , 1 is precisely the complex Grassmann manifold of codimension-q planes in projective p + q space. The natural inclusion cg~,1 m cgq of the degree-1 cycles into the space of all cycles, stabilizes to an inclusion map q

c: BUq~ IF] K(7/, 2j)

(2.2.2)

j=0

by fixing q and letting p --* m , [-Law]. The image of this map c lies in the connected component determined by 1 in K(7/, 0) = 7/. Letting q + m in 2.4, (cf. [Law, LM1]) gives a map c: BU ~ f i K(7/, 2j) def K(7/, even). j=o The first main result of [LM1] is the following.

(2.2.3)

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Theorem 2.3 The natural inclusion c of(2.2.2) represents the total Chern class of the universal complex q-plane bundle over BUq. Hence, the natural inclusion c of(2.2.3) represents the universal total Chern class on the classifying space BU for K-theory. The space BU is the first of a natural family of spaces a~=f f i ~ ( d )

(2.3.1)

d=O

which were introduced in [LM1] and are defined by stabilizing with respect to fixed degree. Specifically, we have

~(d) = lim ~ , d

(2.3.2)

p,q~c~

for d > 0 and ~(0) = {0}. Note that ~(1) = BU.

3 The complex join Both the domain and the range of the map (2.2.3) are naturally H-spaces. The structure on BU is given by the map BU x BU ~ BU classifying the Whitney sum of vector bundles. The structure on K(Z, even) is given by the cup product. The Cartan formula for the total Chern class of a sum of bundles amounts to the assertion that (2.2.3) is a map of H-spaces. A main assertion of this paper is that these H-space structures enrich to infinite loop space structures (the standard one on BU and a new one on K(TI, even) so that (2.2.3) becomes a map of infinite loop spaces. Our first step is to recall the complex join pairing which was introduced in [Law] and shown in [LM1] to give the cup product pairing on K(7/, even). To define the complex join we fix disjoint linear embeddings IP"1 LI~?"2 ~ lP"' +,2+1 Definition 3.1 Given two closed subsets W1 c l P " ' and W2 c IP"2, the complex join of W1 and W2 is the subspace W1 ~r W2 c IPTM +"2 +1 consisting of the union of all lines joining W1 and W2. The complex join has a useful alternative description. Note that the disjoint subspaces IP"' and ~,2 determine a splitting (12"~+"2+2=(IY~+lx(I? "2+1 for IP"~+"~+~. F o r a closed subset W c ~'", let us set C ( W ) = n - ~ ( W ) w {0} ~ 112"+1, where n: II~"+~ - {0} ~ ~" is the canonical projection. Then given W1 and W2 as above, we have

C(Wl~C~W~) =

C(Wl)

x

C(W2).

From this description it is clear that gr is strictly associative. It is also commutative up to the natural equivalence induced by the automorphism of IP"' +,2+ 1 corresponding to the interchange of factors in the homogeneous coordinates: I12"1+I x 112"2+1 --* II;"2+1 x II~"'+1. Remark 3.2 It is interesting to note that, as pointed out in [Law], W1 #r W2 can be described as ai- 1 W1 n a f 1 I4:2 where

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is the vector bundle obtained by projection in homogeneous coordinates (a2 is defined similarly). Combined with the Complex Suspension Theorem of [Law], this allows us to interpret ;ge, up to homotopy, as the intersection pairing on cycles. The complex join of irreducible varieties W1 and Wz is an irreducible variety whose codimension is the sum of the codimensions of W1 and W2, and whose degree is the product of the degrees (cf. [Law]). The complex join extends bilinearly to algebraic cycles and induces algebraic maps q' q+q' :~,r:~,,lx ~dl,,,,~, ~ C~+p,+,,,~,~,.

(3.2.1)

The next step is to show that this pairing can be extended to the stabilized spaces. The most natural extension is obtained by replacing the space cg~ with the naive group completion 2~ q. Algebraically, ~ q is the free abelian group generated by the p-dimensional subvarieties of PP+q, i.e., the set of all finite sums c = y'n~ W,, where the n, are now arbitrary integers. The group ~eq can be written as a union of quotients of the compact spaces c~,.d x ~qp;.~, under the maps (c, c') ~-* c - c'. The group ~eq is given the weak topology for this family of compact subsets. With this topology, the components are precisely determined by the degree, where deg(c) = ~n~ deg(W~). Theorem 3.3 [LM1, LF]

The map 7Zx~

i

~

~

defined for (n, c) 9 7/x c~,e by i(n, c) = c + (n - d)lo, is a homotopy equivalence. Furthermore, passin 9 to the homotopy-theoretic 9roup completion gives a factorization of i:

by homotopy equivalences which determine homotopy equivalences 7Z x ~

--+

f2 B~g, --*

of the spaces obtained by takin 9 limits over p and q.

The complex join defines a continuous biadditive mapping

y,/,/, ~.~,, ~,r

~+,r ) ~p+p'+

1

and if one sets %r lira ~q, p,q~oo

then the join can be extended (cf. [Law]) to a continuous biadditive pairing /x ~q -~ ~e.

(3.3.1)

Now consider the family of spaces ~ = LI~=o ~(d) given in 2.3.1 and 2.3.2, with the canonical identification 9(1) = BU.

(3.3.2)

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C.P. Boyer et al.

For each d > 0 there is a natural inclusion ja: ~(d) ~ ~ ( d ) into the component ~ ( d ) of cycles of degree d in ~ . Taken together these give a topological embedding = fi @(d)~ d=O

fi

~(d)=~.

(3.3.3)

d:--:o

One verifies directly that the following diagram commutes. ~(1)x~(l)

~

9(1)

@x@

-~

@

~A~

~c -~

~.

(3.3.4)

Observe now that the map cg~,l x ~q;,1 ~ (t~p+p,+ , q + q ' 1,1 which is induced by the complex join, corresponds to the direct sum of subspaces. Theorem 3.4 Via 3.3.2 and 3.3.4, one obtains a commutative diagram BU x BU

~

~,

BU ~

(3.41)

where ~ is the map classifying the Whitney sum of bundles.

Theorem 3.5 Under the canonical identification ~ ~- K(71, even), the complex join ^ ~ ~ ~ is, up to homotopy, exactly the map given by the cup product. Remark 3.6 It is shown in [LM1] that translating back to the identity component via a fixed flag of linear subspaces, gives a sequence of embeddings il

12

13

BU = @(1) c @(2) c 9(3) = . . . 9 ( ~ ) ~

K(77, even)

and hence an intriguing filtration on the space K(7/, even) which begins with BU.

4 J,-functors One efficient method of generating infinite loop spaces and infinite loop maps is via the work of May [May 1, May2] using the linear isometrics operad 5a of Boardman and Vogt [BV] and J,-functors. This theory has a direct application to our context. Following FMay2] we introduce the following notation. Let J , be the category whose objects are finite dimensional hermitian inner product spaces and whose morphisms are linear isometric embeddings. Let Y- denote the category of

Algebraic cycles and infinite loop spaces

381

compactly generated, Hausdorff topological spaces with base point. The morphisms in @ (i.e., the continuous maps) are given the compact-open topology. Definition 4.1 An J,-functor (T, co), is a continuous functor T: J , ~ Y together with a commutative, associative, and continuous natural transformation co: T x T ~ To @ such that (a) If x ~ T V and if l ~ T{0} is the basepoint, then co(x, 1) = x e T(V@ {0}) = TV. (b) If V = V'@ V", then the map TV' -+ TVgiven by x ~ co(x, 1) is a homeomorphism onto a closed subset. By an 50-space we mean a space on which there is an action of the linear isometries operad 50 (cf. [May2, pp. 10 ff]). In particular an 50-space is an H-space. A map of 50-spaces is one which commutes with the action of 5 ~ The fundamental property of these objects is that any connected 50-space has a canonically determined infinite loop space structure, and any map of connected 50-spaces is an infinite loop map [May 2]. Theorem 4.2 [May2] I f T is an J.-functor, then T(C ~ ) =

lira T(V),

where the limit is taken over finite dimensional subspaces of IE ~, is an 50-space. Any natural transformation 4: T ~ T' of J,-functors induces a mapping cb: T(~ ~) ~ T'(C ~) of 50-spaces. Example 4.3 For each hermitian inner product space V of dimension n, let To V = Grass,( V(~ V) denote the Grassmannian of n-dimensional linear subspaces of V@ Vwith distinguished point 1 = V(~ {0}. Given a linear isometric embedding f : V ~ W, define Tof" T o V ~ ToW on a plane U ~ V @ V by T o f ( U ) = ((fV)• There is a natural transformation coo: T o x T o To o @ given by the external direct sum, i.e., for U ~ To V and U' ~ To V', coo(U, U') = ~(U|

U')

where ~: V @ V ~ V ' ( ~ V ' ~ V ~ V ' ( ~ V G V ' is the obvious shuffle isometry interchanging the middle two factors. One easily checks that To is an J,-functor, and it follows from the discussion in [May 2, p. 16], that To(r ~) = BU

(4.3.1)

with its standard loop space structure.

5 The Chow monoid functor We shall now construct an J , - f u n c t o r T, parallel to the one given in 4.3 above, but with values in the Chow monoid of all cycles on projective space. To each hermitian inner product space V of dimension n, let

T(V) = ~ ( ~ ( V @

V))

(5.0.2)

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C.P. Boyer et al.

be the Chow monoid of effective algebraic cycles of codimension n on P ( V ~ V), with distinguished element 1 = ~ ( V ~ {0}). In dimension 0 we set T{0} = ]N with distinguished element 1. For a linear isometric embedding f: V --* W, we define T ( f ) : T(V) ~ T(W) on a codimension n cycle c by

T ( f ) c = F ( f (V)" ~ {O} )~,e(f(~ f),(c)

(5.0.3)

where ~;r denotes the complex join, which is defined on any pair of cycles which live in disjoint linear subspaces of ~'(V(~ V). Recall that the complex join is given in homogeneous coordinates by taking the direct product of the homogeneous (conical) varieties. In particular, it is a strictly associative pairing. Note that T(f)(1) = 1. Using the complex join we define a pairing co: TV• TV' --* T ( V G V') by setting

co(c, c') = z , (c$r

(5.0.4)

where, as above, T: VG V ~ V ' ~ V' --, VG V ' ~ V ~ V' is the map which interchanges the middle two factors. The pairing co is associative, commutative and continuous. Theorem 5.1 ( T, co) is an J,-functor.

Proof We first show that T is a functor. For this we must verify that T(f~176 for morphisms f: V ~ W and 9: U ~ V. Fix a cycle c ~ (g~,(~(U ~ U)), and note that T(f) T(g)c = T ( f ) {~(g(U)• (~ {0} ) :~r (~ g),(c) } = IP(f(V) • (~ {0}) :~r ( f O f ) , {F(g(U)• (~ {0}) gr = ~(T(V) • ~ {0}) $r {~'(T(o(U)•

{0}) $r

~ g), (c)}

(~fg),(c)}

= F ( f ( V ) • (~f(o(U)z) ~ {0} $r (f9 (~fg),(e)

= ~((fg( U))• ~ {0} ) $r (fg ~fg),(c) =

T(f9)c.

We now show that co is a natural transformation. For this we must check that for morphismsf: V --* W a n d f ' : V' ~ W', we have ~ o ( T f x Tf') = T ( f ~ f ' ) o c o . Fix cycles c ~ T V and c' ~ TV'. Then

r

• Tf')(c, c') =

%((Tf)(c)~r

= % ( { F ( ( f V ) " ~ {Ov})~r

= T, { ~ ((fV) ' ~ {0v })$r = ~(((f~f')(V~

[]

V') • ~ {0v' } ) $ r

V'))" ~ { 0 v e v , } ) ~ , r 1 6 2

= T(f~f')(o~(c, c')) as desired.

$r

V') • ~ {0v'})~r ),(c):~(f'

~ f ' ) , ( c ' ) })

~f'),(c') }

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383

6 The main theorems Let T be the Chow monoid functor constructed in the previous section. We see by direct inspection that T(~E~)= 9 = ] j 9(d), d=>0

(6.0.1)

where 9 is defined in w Theorem 6.1 The stabilized cycle space 9 is an ~f'-space, where the structure maps are defined via the join pairing f~r : ~(k) x ~ ( f ) - . @(kE). Furthermore, the infinite loop space structure thereby induced on ~(1) ~ BU agrees with the standard one. Proof The first assertion follows from the theory of J . - f u n c t o r s [May2, Chap I]. For the second assertion one notes that the restriction of T to degree-1 cycles (i.e., to the Grassmannians of n-planes in r 2.) coincides with May's construction in [ M a y l , pp. 16-18], which, as he proves there, gives the standard infinite loop space structure on BU. [] As noted in [LM1], each map ~r @(1) x ~(d) ~ ~ ( d ) makes ~(d) a geometric module for (@(1), :~r -_ (BU, q~). Theorem 6.1 asserts that these maps assemble together to give a map of ~L~a-spaces:~r ~(1) x 9 ~ 9 . We now observe that Tis actually an abelian monoid-valued J . - f u n c t o r whose additive structure is compatible with the join pairing, in other words, in May's terminology, T is an J . - m o n o i d . Hence, we have the following. Theorem 6.2 9 is an Eo~-rin9 space. It follows immediately that there is associated to ~ an Eo~-ring spectrum and a natural map j: @ ~ F ~ of @ into the 0'th space of the spectrum. F@ is an ~-space, and j is an ~r map. In particular, the connected component of 1, (F~)(1) is an infinite loop space and j: 9 0 ) -~ (F9)(1) is an infinite loop space map. By its construction the space F@ is an additive group completion of 9 , i.e., F ~ -~ s --- ~2BC~, ~ Z • ~. Actually, it is elementary to verify (cf. [May 1]) that the homotopy-theoretic group completion Y~BX of an ~q~-space X is again an ~-space, and the canonical map X ~ O B X is an ~ - s p a c e map. Either way we have proved the following result. Theorem 6.3 The natural map ~ ~ OB~s of ~ into its additive group completion, is a map of.L~-spaces. In particular, there is an infinite loop space structure on the 1-component Y2B~,(l) so that the map 9 0 ) -~ ~2B~,(1) is an infinite loop space map. This already settles Segal's question. Remark 6.4 The connective cohomology theory G * ( X ) associated to the spectrum we have defined has G(X, Z) (defined at w as the abelian group G~ and as a consequence of our results, the Chern map c: K ( X ) ~ G(X, 7/) is extended to a map of cohomology theories c: k*(X) ~ G*(X). In [Segl], Segal also asked if G(X, Z ) could be extended to a multiplicative cohomology theory and one might in addition ask if the Chern map c, in the case of a positive answer to this question, is a map of multiplicative theories. The answer to this question is actually negative,

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as shown recently by Totaro [Tot]. Notice that this does not conflict with the E~-ring structure of our spectrum, for the additive structure on G(X, 77), as well as on BU, is provided by the multiplicative structure of the E~-ring space ~. On the geometric side, one would like to have a construction involving algebraic cycles which extends the tensor product in the level of BU. However, as shown by Totaro, this structure, if it exists, is not sufficiently compatible with the complex join (now viewed as an additive operation) to generate another ring spectrum. As the reader has probably noticed, there is a second, more elementary way of putting an &a-structure on the group completion of @. The construction of Sect. 5 can simply be repeated with the Chow monoids replaced by their nai've topological group completions. In particular, if one defines Z(V) = ~r~(F(V|

V)),

where ~ , ( ~ ' ( V ~ V)) is the naive group completion of c g , ( ~ ( V ~ V)) (cf. w above), then the entire discussion of Sect. 5 carries over without change to show that 77 is an J,-functor and the universal map Uv: T(V) --* Z(V) is a morphism of J,-functors. Consequently, the limit Z(C~) = ~ =

I~I ~((d) d=

-

(6.4.1)

~3

is canonically an &a-space. Clearly the natural mapping ~ ~ ~e is an &a-space map. This has the following immediate consequence, parallel to 6.3 above. Theorem 6.5 The natural map ~ ~ ~ of ~ into its naive additive 9roup completion, is a map of &a-spaces. Corollary 6.6 There is an infinite loop space structure induced by the complex join on ~Y(1) such that the total Chern class map ~(1) ~ ~e(1) is an infinite loop map. This establishes Theorem A of the introduction. We note that Theorem 6.5 can be interpreted in terms of the filtraction o f ~ ( ~ ) given in 3.6. Translation to the identity component of ~ is compatible with the join pairing, and in particular, the map gr respects this filtration, i.e., we have ~:c(N(d) x ~(d')) c @(dd') for each d, d'. Theorem 6.5 implies that this filtration is actually compatible with an infinite loop space structure on @( ~ ). It seems likely that the assertions of 6.3 and 6.5 are equivalent. This is indeed the case.

Proposition 6.7

There is a homotopy equivalence 12B~, & ;,~ of &a-spaces so that f2BC~,

N is a commutative diaoram of &a-spaces. Proof Observe that the J , - m o n o i d T acts on T x T by diagonal translations and that the homotopy quotient B ( T x T, T, *) obtained via the triple bar construction

still is an J , - m o n o i d , cf. [May 2].

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385

Since T is an abelian J , - m o n o i d , the natural map T(V) ~ B(T(V)• T(V), T(V), *) is a group completion, i.e., a "model" for (2BT(V), see [LF]. Furthermore, there is a commutative triangle of monoid morphisms

B(T(V)x T(V), T(V), *)

~(v) z(v) whose vertical arrow is a homotopy equivalence, cf. [LF]. The triangle naturally induces a triangle of J , - m o n o i d morphisms and the theorem follows by taking direct limits. []

7 Real cycles and Stiefel-Whitney universal map Here we combine the techniques and constructions above with the results obtained by Lam [Lam] in his Ph.D. thesis to study the Stiefel-Whitney universal map and prove Theorem B. For a real vector space Vlet V~ = V ~ , C denote its complexification and, for a linear map f: V ~ Wbetween real vector spaces, letfr = f | 1: Vr ~ We denote the induced complex linear map. Note that complex conjugation of Vr induces an involution )oo:IP(V~)~ ~'(Vr and similarly an involution of n'(Vcq~Vr We denote by 2 the induced involution on the spaces of effective cycles

~,(~'(v~@ v~)). Definition 7.1 F o r a real inner product vector space (V, ( , ) ) define the monoid of real algebraic cycles of codimension n in P(Vr 9 Vr to be the space

RT(V) = IRC#~,(IP(Vr ~ Vr

deZ {a ~ c#~,(~(Vr ~ Vc))12(a) = a}.

Similarly we define the submonoid of 2-averages (or Galois sums) of codimension n cycles to be the space

DT(V) = D~,(IP(Vr ~ Vr

de~ {a + 2(a)la6c#~,(P(Vr

Vr

_ RT(V).

F o r a linear isometric embedding f: V ~ W define RT(f): RT(V) ~ RT(W) by

RT(f)a = lP(f(V)~ G {0}) ~r162 ~fr Next define DT(f) by restriction, observing that 2 commutes with (fr @fr finally define the Whitney sum

co:RT(V)x RT(W) ~ RT(VO W) by ~o(~, ~') = ~ ( ~ G ' ) , where, as before, T is the shuffle isometry

~: v~.| v~@ v~:@ v~ ~ v~| v~@ v~| v~:.

and

386

C.P. Boyer et al.

Since 2(a$ctr') = 2(tr)~r and 2 2 = 1, we conclude that if either tr ~ DT(V) or a' ~ DT(W), then ~o(a, tr')e DT(V~ W). Proposition 7.2 With the operations above defined, both (RT, ~o) and (OT, co) are

J,-monoids and DT is an J,-submonoid of RT. Proof The arguments here are completely analogous to those given for Theorem 5.4 and so we omit the details.

[]

Corollary 7.3 The homotopy quotient B(RT, DT, *) is an J,-monoid. We see, as before, that the direct limit of B(RT(V), OT(V), ,) over the finite dimensional subspaces of R ~, denoted g, is an ~-space, where 5 ~ is the real linear isometries operad. The same applies to ~ T a n d to ~T, the corresponding limits of RT(V) and DT(V), respectively. The component of cycles of degree 1 in RT(V) can be canonically identified with the real Grassmannians Gr(n, V ~ V). Lam [Lam] has shown that the (additive) group completion g + of g is homotopy equivalent to lqi K(Z/2, i) and that the composition, induced by passage to limits, BO ~ ~ T + -'* g +

(7.3.1)

represents the total Stiefel-Whitney class. Combining Lam's results with our previous observations and May's machinery, one sees that the above composition is equivalent to a map of infinite loop spaces BO ~ Fd ~, where Fo~ is the zero-th space of the ~-spectrum associated to

B(RT, DT, *). This proves Theorem B of the introduction and thus completes the affirmative solution to Segal's questions.

Remark 7.4 In [Sna], Victor Snaith shows that the infinite loop space structures constructed by Segal on the spaces K(Z, 2) and K(7//2,,) are not compatible with the total Chern and Stiefel-Whitney maps. The key to this argument is his Lemma 2.1 which establishes a certain property of the operad maps on the Eilenberg-MacLane spaces. Our operad maps definitely do not have this property. It is for this reason that his arguments do not apply to our case.

8 Computational consequences

It is well known that the homotopy theory of infinite loop spaces is extremely rich and that, for every infinite loop space X and every prime p, H , ( X ; Z/p) admits an infinite family of homology operations, called Dyer-Lashof operations [DL], distinct from the duals of the Steenrod algebra. These homology operations are defined using the map ~ of 4.7. We refer the reader to [CLM] for an extensive summary of these operations and their algebraic properties. Next we highlight the following facts: (1) The homology and cohomology of Eilenberg-MacLane spaces, and hence of 9 ( ~ ), are completely described in terms of the Steenrod algebra. (2) Kochman [Koc-I completely described H,(@(1) = BU) as a module over the Dyer-Lashof algebra.

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387

(3) Theorem A, Theorem 2.6, and K o c h m a n ' s results completely determine the action of the Dyer-Lashof algebra on the generators, over the Steenrod algebra, of

H , ( ~ ( ~ ); Z/p).

(4) The Nishida relations (see, for example [ C L M ] ) calculate Dyer-Lashof operations on elements of the form a(x) where a is an element of the Steenrod algebra and x a H . ( X ; Z/p}. We emphasize that, taken together, these facts give complete calculational control of H , ( ~ ( oo ); Z/p) as a module over the D y e r - L a s h o f algebra. One can use these calculations to obtain new information about H . ( ~ , a ; Z/p) for various values of p, q, and d. Analogous computations can also be made in the real case corresponding to the spaces of real algebraic cycles.

Acknowledgements. Finally,

we would like to thank D. Kraines, J. Stasheff, and B. Totaro for calling our attention to the history surrounding the Chern map problem.

References [AJ] [BHMM] [BM 1] [BM 2]

[BV] [CCMM] [CLM] [DL] [Don] [Gro] [KL] [Koc] [Koz 1] [Koz 2] [Lam] [Law] [LF]

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C.P. Boyer et al. Lawson, H.B., Jr., Michelsohn, M.-L.: Algebraic cycles, Bott periodicity, and the Chern characteristic map. In: Wells, R.O., Jr. (ed.) The mathematical heritage of H. Weyl. (Proc. Symp. Pure Math., vol. 48, pp. 241 264) Providence, RI: Am. Math. Soc. 1988 Lawson, H.B., Jr., Michelsohn, M.L.: Algebraic cycles, and group actions. In: Differential Geometry, Harlow: Longman 1991, pp. 261 278 May, J.P.: The geometry of iterated loop spaces. (Lect. Notes Math., vol. 271 Berlin Heidelberg New York: Springer 1972 May, J.P.: E -ring spaces and E~-ring spectra. (Lect. Notes Math., vol. 577) Berlin Heidelberg New York: Springer 1977 Mann, B.M., Milgram, R.J.: Some spaces of holomorphic maps to complex Grassmann manifolds. J. Differ. Geom. 33, 301-324 (199l) Segal, G.: The multiplicative group of classical cohomology. Q. J. Math. Oxf. II. Ser. 26, 289-293 (1975) Segal, G.: The topology of rational functions. Acta Math. 143, 39-72 (1979) Snaith, V.P.: The total Chern and Stiefel-Whitney classes are not infinite loop maps. Ill. J. Math. 21, 300-304 (1977) Steiner, R.: Decompositions of groups of units in ordinary cohomology. Q. J. Math., Oxf. 90 (no. 120), 483 494 (1979) Steiner, R.: Infinite loop structures on the algebraic k-theory of spaces. Math. Proc. Camb. Philos. Soc. 1 (no. 90), 85 111 (1981) Taubes, C.H.: The stable topology of self-dual moduli spaces. J. Differ. Geom. 29, 163 230 (1989) Totaro, B.: The total Chern class is not a map of multiplicative cohomology theories. Math. Z. (to appear) Totaro, B.: The map from the Chow variety of cycles of degree 2 to the space of all cycles. Math. Sci. Res. Inst. (Preprint, 1990)