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SynopsisThe special unitary groupSU(n)has the stable homotopy type of a wedge ofn− 1 finite complexes. The ‘first’ of these complexes is ΣℂPn–1, which is well known to be indecomposable at the prime 2 whethernis finite or infinite. We show that the ‘second’ finite complex is again indecomposable at the prime 2 whennis finite, but splits into a wedge of two pieces whennis infinite.

Let V be an elementary Abelian 2-group of rank n, which it will usually be more useful to consider as an n-dimensional vector space over \({\Bbb F}\) 2, the field with 2 elements. We write H*(V) and H *(V) for the cohomology and homology of BV with \({\Bbb F}\) 2-coefficients. Of course, BV is homotopy equivalent to the cartesian product of n copies of ℝP ∞. The importance of H*(V) as a module or algebra over the mod 2 Steenrod algebra, A, in unstable homotopy is now well established. It is therefore to be expected that H*(V) itself has much inner subtlety.

1. Few mod 2 finite simply-connected spaces are known to support an H-structure. These examples arise from products of Lie groups and seven spheres. It has been shown in (12) and (14) that the space of a mod 2 finite H-space of given rank can belong to only a finite number of different homotopy types. The situation at odd primes is very different; many examples of mod p finite H-spaces are known, such as p-localized odd-dimensional spheres. Therefore there are infinitely many homotopy types of prescribed rank. If one insists that the mod pH-space has a strictly associative multiplication, or equivalently has the homotopy type of a topological group, the picture is less clear. Examples have been constructed which do not arise from Lie groups and spheres but there are strong restrictions on just what can occur by these techniques, see (5). There are only finitely many known examples in each rank. We establish that there can in fact only be a finite number of possible homotopy types for the spaces.

SynopsisThe H-spaces considered have no homology p-torsion and are rationally equivalent as H-spaces to products of even dimensional Eilenberg–Maclane spaces. We obtain conditions which ensure that if the cohomology with coefficients in the ring of integers localized at the prime p is a polynomial algebra, then the Pontrjagin ring with these same coefficients is polynomial. A topological consequence is that BSUP has just one homotopy associative, homotopy commutative H-structure.

We prove that B U p B{U_p} has only two standard H H -structures.

Hans Scheerer proved that if two simply connected compact Lie groups are homotopically equivalent, then the groups are isomorphic. We give a conceptually simpler proof which shows that the result depends only on the 2 and 3 primary homotopy of the Lie groups.

It is shown that a compact, simply connected, Lie group G is p-regular if and only if K o ( Ω G , Z / p Z ) {K_o}(\Omega G,Z/pZ) is a primitively generated polynomial algebra. It is conjectured that this remains true for G a finite, simply connected, associative, H-space.

The third homotopy group of a finite H-space is shown to have no torsion.

Let X be a CW-complex with finite skeletons and with H*(X, Z) free of p-torsion, where p is some odd prime. We shall prove two theorems on the action of the Steenrod powers

of the Steenrod squares for the Q2-cohomology rings of finite complexes whose integral homology is free of 2-torsion, where Q2 is the ring of rationals with odd denominators. These are not cohomology operations as they only satisfy generalized naturality properties, but under favourable conditions they provide information which it does not seem possible to obtain using either primary or secondary cohomology operations. The treatment of these generalized cohomology operations given in this note is not complete; only those properties are considered which are needed for the immediate applications. In later papers the author hopes to extend both the theory and the applications of these generalized operations as indicated in [13]. All the additive and several of the multiplicative results given in this paper have quite precise analogues for odd primes. These are not considered here, as the most elegant treatment of the corresponding generalized reduced power operations for odd primes involves the use of theorems of J. F. Adams which have not yet been published. I would like to thank Professor J. F. Adams and Dr. I. M. James for the considerable assistance they have given in the preparation of this paper, which essentially formed a part of the author's doctoral thesis.

Let X be a finite connected CW-complex which is a topological monoid. It is shown that if the rank of X is less than five, then the rational type of X is that of a Lie group. 0. The classification of the homotopy types of complexes with few cells which support H-space multiplications now seems significantly nearer completion than was the case just a short time ago, [8] and [17]. If one can show that the only possible rational cohomology rings of these H-spaces are those which are known to occur, then one might hope to generate all possible homotopy types from the known examples using Zabrodsky's technique of mixing rationally equivalent H-spaces [16]. Now to determine the rational cohornology rings of H-spaces with few cells is not an easy problem. In fact without information on torsion in the integral homology and/or some geometric or Hopf algebraic associativity assumption connected with the Hspace multiplication, very little is known. In this note we consider the rational cohomology rings of such H-spaces where we make no torsion assumptions but where the H-space multiplication is strictly associative. 1. Let X be a finite connected CW-complex whichi supports an H-space multiplication. Thle rational cohomology ring H*(X, Q), is isomorphic to that of a product of odd dimensional spheres [9]. The number of spheres, r(X), is called the rank of X. We define t(X), the type of X, to be the set of integers { q } where { 2q -1 } is the set of dimensions of these spheres. This is not quite the usual definition of type but it is convenient here. THEOREM 1.1. Let X be a finite connected CW-complex which supports a strictly associative H-space multiplication. If r(X) < 4; then t(X) = t(G) where G is a Lie group. Representatives for all the Lie groups which occur in Theorem 1 .1 can be obtained from products of S'; SU(n), 2 _n_5; Sp(n) 2 < n

In this thesis, we establish a pair of systems of higher order cohomology operations that act on the Ζp-ordinary cohomology of spaces that are free of p-torsion. These "pyramids" of operations are generated by the p-divisibility of certain sums of "pseudo" primary cohomology operations that operate on the p-local cohomology of p-torsion-free spaces. The properties of these higher order operations allow us to prove theorems that either generalise or improve (in the sense of decreasing indeterminacy) several results in the literature.

We prove that BUp has only two standard H-structures. The two most familiar H-multiplications on BU arise from the direct sum of complex vector bundles and from the tensor product of virtual line bundles; BU with these multiplications will be denoted by BU and BUe respectively [2]. Both multiplications are standard, define a standard H-space (X, e, ,u) to be an H-space whose rational Pontrjagin ring H*(X, Q) induced by ,u is both associative and graded commutative. We consider H-spaces localized at a fixed prime p. THEOREM 1.1. Let m be a standard H-multiplication on BU. Then BUpm is H-equivalent toU or U . We recall that two H-spaces (X, e, ,u) and (Y, f, v) are H-equivalent if there exists a homotopy equivalence k: X -* Y such that v(k X k) k,u. The two classes of multiplications on BUp in Theorem 1.1 can easily be distinguished for the Frobenius map (: H2(BUM, Z/pZ) -H2p(BUm, Z/pZ) defined by ((x) = xP is nontrivial when m = ED but is the zero homomorphism if m = ?. Theorem 1.1 completes results about standard multiplications on BUp proved in [4]. The strongest result in this context was given in ?4 of [4]. X is assumed to have the homotopy type of a connected CW-complex with finite skeleta. THEOREM 1.2. Let X be a standard H-space andp an odd prime. Assume that (a) H*(X, Z/pZ) is a polynomial algebra, (b) Dim QH2'(X, Z/pZ) S 1 for all i and the equality holds at least for 1 < i p 1, (c) t: H2(X, Z/pZ) -H2p(X, Z/pZ) is nontrivial. Then Xp R BUp' as an H-space. We can now add a companion result for B Up. THEOREM 1.3. Let X be a standard H-space andp an odd prime. Assume that (a) H*(X, Z/pZ) is a polynomial algebra, (b)' Dim QH2i(X, Z/pZ) S 1 for all i and the equality holds at least for 1 < i < P, Received by the editors January 23, 1984. 1980 Mathematics Subject Classification. Primary 55R35; Secondary 55N15. 'Author supported by Kuwait University Research grant SM102. ?1985 American Mathematical Society 0002-9939/85 $1.00 + $.25 per page