- Email: [email protected]

An H-space is a topological space X with basepoint e and a multiplication map m: X2 = XXX—>X such that e is a homotopy identity element, (We take all maps and homotopies in the based sense. We use k-topologies throughout in order to avoid spurious topological difficulties. This gives function spaces a canonical topology.) We call X a monoid if m is associative and e is a strict identity. In the literature there are many kinds of ü-space: homotopyassociative, homotopy-commutative, ^««-spaces [3], etc. In the last case part of the structure consists of higher coherence homotopies. In this note we introduce the concept of homotopy-everything H-space {E-space for short), in which all possible coherence conditions hold. We can also define £-maps (see §4). Our two main theorems are Theorem A, which classifies E-spaces, and Theorem C, which provides familiar examples such as BPL. Many of the results are folk theorems. Full details will appear elsewhere. A space X is called an infinite loop space if there is a sequence of spaces Xn and homotopy equivalences Xnc^tiXn+i for n^O, such that X = Xo. T H E O R E M A. A CW-complex X admits an E-space structure with TQ(X) a group if and only if it is an infinite loop space. Every E-space X has a (i'classifying space" BX, which is again an E-space.

1. The machine» This constructs numerous J3-spaces. Consider the category é of real inner-product spaces of countable (algebraic) dimension and linear isometric maps between them. As examples we have JR00 with orthonormal base [eu e2, e8, • • • }, and its subspace Rn with base {ei, £2, • • • , en], which is all there are up to isomorphism. We topologize ó(Af B), the set of all isometric linear maps from A to B, by first giving A and B the finite topology, which makes each the topological direct limit of its finite-dimensional subspaces. LEMMA.

The space é(A% R°°) is contractible.

This is a consequence of two easily constructed homotopies: (a) ii~i2:A-+A®A, (b) i\C^u : R^—ïR00 ® -R00, for some isomorphism u. 1117

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Suppose we have a functor T defined on the category $, taking topological spaces as values, and a continuous natural transformation co: TAXTB->T(A ®B) called Whitney sum, such that: (a) Tf is a continuous function of ƒ Çzé(A, B) ; (b) TR° consists of one point; (c) o) preserves associativity, commutativity and units; (d) TR00 is the direct limit of the spaces TRn. B. TR00 is an E-space. If T happens to be monoid-valued$ the classifying space [2] BTR00 agrees with that from Theorem A. THEOREM

As a (noncanonical) multiplication on TR00 we take TR™ X TR00 - » T(R°° © R°°) —> TR00, co Tf where/: .R00©/?00—»2?°° is any linear isometric embedding. The Lemma provides homotopy-associativity, since ƒ o ( / © 1 ) ~ / o (1©/), homotopy-commutativity, and all higher coherence homotopies. In the examples below we define TA explicitly only for finitedimensional A, and note that axiom (d) extends the definition to the whole of 6. In each case the maps Tf and the Whitney sum co are obvious, in view of the inner products. 1. TA = 0(A), the orthogonal group of A. Then TR00 = 0. 2. TA = U(A ® C), the unitary group of A ® C. Then TR00 = U. 3. TA ~BO(A), a suitable classifying space for 0(A), for example that given by [2]. Then TR» = BO. 4. TA = F(A), the space of based homotopy equivalences of the sphere SA, which is the one-point compactification i W o o of A, with 00 as basepoint. Then TR00 = F. There is also a semisimplicial analogue, in which T takes semisimplicial values and ó(A,B) is replaced by its singular complex. 5. TA =TopC<4). A ^-simplex of Top(^4) is a fibre-preserving homeomorphism of A XAk over Ak, where Ak is the standard ^-simplex. Then rjR°°=Top. 6. The semisimplicial analogues of Examples 1-4. 7. The orientation-preserving versions of Examples 1-6. 8. TA=PL(A), defined as Top(^l) except taking only piecewise linear homeomorphisms of A XA*. This fails. To cure this we need a new machine. Suffice it to say that as a fe-simplex of (P(A, B) we take a pair (£, ƒ), where £ is a p. 1. subbundle of the product bundle B XA* over Ak, a n d / : £© (A XAk)^B XAk is a p. 1. fibrewise homeomorphism that extends the inclusion of £. EXAMPLES.

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THEOREM C. We have the Espaces O, SO, F, U, PL, Top, etc., their coset spaces F/PL, T — "PL/0" etc., and all their iterated classifying spaces. The natural maps between these are all E-maps, including 0-+PL and PL->T.

2. Categories of operators. There are two variants: with or without permutations. D E F I N I T I O N . In a category of operators (B (a) the objects are 0, 1, 2, • • • ; (b) the morphisms from m to n form a topological space (B(m, n), and composition is continuous; (c) we are given a strictly associative continuous functor © : (B X(B —KB such that m®n = m+n; (d) if (B has permutations, we are also given for each n a homomorphism Sn—»(B(w, n), Sn the symmetric group on n letters. (We omit any symbol for this homomorphism.) In the case with permutations we impose two further axioms: (i) if irCzSm and p&Sn then [email protected] lies in Sm+n and is the usual sum permutation; (ii) given any r morphisms a*-: mc^ni and 7r£S f , we have 7r(n) o (a\ © a 2 © • • • © a r ) = fl"(ai © ai © • • • © ar) o 7r(m), where m = 2 r a ; , n = '2ni, w permutes the factors of ai©aj2ffi • • • ®ar, and the permutation 7r(n) £ S n is obtained from ir by replacing i by a block of ni elements. We require functors to preserve all this structure. EXAMPLES. 1. Endx, for a based space X. Endx(m, n) is the space of all (based) maps Xm—>Xn, where Xn is the nth power of X. The functor © is just X . This example has permutations. D E F I N I T I O N . The category (B of operators acts on X, or X is a (&-space, if we are given a functor (B—»End*. 2. Ci. &(m, n) is the set of all order-preserving maps { l , 2 , • • • ,m\ —>{l, 2, • • • , n). Then an Ct-space is a monoid. 3. $. S(m, n) is the set of all maps {l, 2, • • • , rn}—±{\, 2, • • • , n\, including permutations. Then an S-space is an abelian monoid. D E F I N I T I O N . We call X an Espace if we are given a category (B of operators with permutations acting on X, for which (B(w, 1) is contractible for all n. (We do not single out any canonical (B.) 4. #. Define ê(rn, n) =$((R°°)m, (R°°)n) as in §1. By the Lemma any 4-space, for instance TR», is an .E-space. 5. Qn, a category of operators on the ^th loop space X = Q n F, the space of all maps (In, d/ n )—»(F, o), where In is the standard w-cube, dln its boundary, and o the basepoint of F. A point aGQn(kt 1) is a

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collection of k w-cubes 7? linearly embedded in In1 with disjoint interiors, and with axes parallel to those of In. It acts on X as follows: given (fuf2, • • • , fk)&Xk, the map a(fu f2, • • • , ƒ*) : I n -> Y is given by ƒ» on each little cube if, and zero elsewhere. Similarly for Qw(&, r). We topologize Q„(&, 1) as a subspace of JR2Aîn. We observe that Qn(&, 1) is (^ — 2)-connected, so that as n tends to oo, Theorem A becomes plausible. We say the category (B of operators, without permutations, is in standard form if every morphism aim-^n has uniquely the form [email protected]@ • • • ®an, where a«:w<—>1. For categories with permutations the definition is more complicated. Of our examples, 2, 3 and 5 are in standard form, but 1 and 4 are not. 3. The bar construction. Suppose given a category (B of operators, in standard form. We consider words [ceo|«i| • • • |«A]> where k*z0 and each a»- is a morphism in (B and a0 o ai o • • • o a& exists. D E F I N I T I O N . The category W°(& has as morphisms from m to n those words [ce0| «i| • • • | ak] for which the composite exists and is in (B(m, n), subject to the relations and their consequences: [ a © j 8 ] = [ a © l | l © j 8 ] = [ l © | 8 | a © l ] for appropriate identities 1; [l] is an identity in W°(&; [a\ T] = [a o w] and [w\ /3] = [ir o /3] if (B has permutations w. Composition in W°(& is by juxtaposition. To form the category PF(B, we take for each morphism x in PT°(B a cube C(x) of suitable dimension, having x as vertex, and identify the faces not containing x with certain cubes C(xi) of lower dimension, where Xi runs through the words formed from x by one "amalgamation." The categories W°(& and W(& inherit obvious identification topologies. For composition we have C(x) o C(y)C.C(x o y) as a face containing xoy, and © : C(x) XC(y)=C(x®y). The augmentation e: W(&—»(B is defined by e[ce 0 |«i| • • • \oik] =ce0 o ai o • • • oak and eC(x) = ex. A PFCt-space, with a as in §2, is approximately an ^ - s p a c e [3]. In particular, the familiar pentagon in W(X(4ti 1) is now subdivided into five squares. For the following theorems we need a slightly different category (B' augmented over (B, in which (B'(l, 1) differs from (B(l, 1) by a whisker. However, we may replace (B' by (B in the theorems whenever the identity 1 in (B(l, 1) is an isolated point. We call an augmentation functor 0: <3—»(B ftbre-homotopically trivial if for each n there exists a section x- ®(^, l)-*<5(», 1) such that xod is fibrewise homotopic to the identity map of Q(n, 1), 5 n -equivariantly if (B and 6 have permutations.

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D. (a) e: W(&'—>($>,-*(&isfibre-homotopically trivial. (b) Given any category of operators 6 augmented over (B by a fib?ehomotopically trivial functor, there exists a functor W(&'—>(2> that lifts W(B'-->(B. (It is not unique.) THEOREM

T H E O R E M E. Suppose X and Y have the same homotopy type, and W(&' acts on X. Then we can make W($>' act on Y.

4. Maps between //-spaces. Suppose X and Y axe fiRB-spaces and ƒ : X—» Y a map. We call ƒ a W(Sb-homomorphism if it commutes with the action of W(&. We need a weaker homotopy notion. Let <£n be the "linear" category with objects 0, 1, 2, • • • , n and one morphism i—*j whenever i^j. We can generalize the bar construction in §3 to form TF((BX £ n ), a category which we make act on (w+l)-tuples of spaces. (Note that in (BX «Cn © is not quite a functor, because it is not everywhere defined.) We define a homotopy (R-map from X to F as an action of W((BX£i) on the pair (X, F), that induces the given PRB-structures on X and F. Similarly, an action on £-spaces X and F of a suitable category Q such that G(X n , F) is contractible for all n is called an E-map. T H E O R E M F. Let X and Y be W&-spaces, and ƒ : X—» F a homotopy &-map that is also a homotopy equivalence. Then any homotopy inverse g: Y—+X admits the structure of homotopy ($>'-map.

We cannot form the category of PRB-spaces and homotopy (B-maps, because unless one of them is a PF(B-homomorphism, the composite of two homotopy (B-maps is defined only up to a homotopy, which is defined only up to a homotopy, which is . . . . Instead we form a semisimplicial complex K, whose w-simplexes are actions of W((BX £n) on (n + 1)-tuples of spaces; in particular its vertices are TF(B-spaces and its edges are homotopy (B-maps. G. This complex K satisfies the ^restricted Kan extension in which the omitted f ace is not allowed to be the first or the

THEOREM

condition" last.

This result provides everything we need for composition up to homotopy etc., and allows the formation of the category of W(&spaces and homotopy classes of homotopy (B-maps. 5. Structure theory. The following theorem is essentially due to Adams. T H E O R E M H. Given a Wd-space X, there is a universal monoid MX equipped with a homotopy d-map i: X—>MX> such that any homotopy

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(H-mapf: X—>G to a monoid G factors uniquely as goi with g a monoid homomorphism. Moreover, if X is a CW-complex the map i is a homotopy equivalence. Our main technical result for proving Theorem A is: THEOREM

Wa-structure.

J. Let X be an E-space, so that in particular it admits a Then the classifying space [l\ BMX is also an E-space.

For Theorem A we then define BX = BMX. 6. Cohomology theories. Given an £-space Y such that F is a CW-complex and w0(Y) is a group, we define [ l ] a graded additive cohomology theory on CPF-pairs by setting tn(X, A) = [X/A, BnY], t~n(X, A) = [X/A, ÛWF]

for w ^ O ,

whose coefficient groups vanish for n>0. Let us call a cohomology theory with this property connective. T H E O R E M K. Every connective graded additive cohomology theory on CW-pairs arises from some such Espace F, which is uniquely determined up to homotopy equivalence of Espaces.

In particular the £-space ZXBU gives rise to the connective Ktheory cK. This is more usually obtained by appealing to Bott periodicity and killing off the unwanted coefficient groups. In other cases we cannot appeal to Bott periodicity: D E F I N I T I O N . We define connective piecewise linear K-theory cKPL by using the .E-space ZXBPL: for n>0 we set cKnPL(X, A) = [X/A, Bn(Z X BPL)]. REFERENCES

1. E. H. Brown, Cohomology theories, Ann. of Math. (2) 75 (1962), 467-484. 2. R. J. Milgram, The bar construction and abelian H-spaces, Illinois J. Math. 11 (1967), 242-250. 3. J. D. Stasheff, Homotopy associativity of H-spaces. I, Trans. Amer. Math. Soc. 108 (1963), 275-292. UNIVERSITY OF WARWICK, ENGLAND

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