Stewart Priddy

Stable splittings derived from the Steinberg module

inaccessible. Examples include Mahowald’s maps ~,[5] based on Snaith’s splittings and, more recently, certain maps used in Kuhn’s proof of the Whitehead conjecture[6, 71. These latter maps are based on our splitting of B(Z/2)k. Our main result shows that the suspension spectrum of a product of lens spaces B(E/P)~ can be split using the Steinberg idempotent of F,,[GL,(ff,,)]. Let Sp”(S”) denote the n-fold symmetric product of the sphere spectrum. We recall Sp”(S”) = K(Z) by the Dold-Thorn theorem. Let D(k) be the cofiber of the diagonal map d: SpP”-‘(S’)+SpJ’“(S”). Then D(co) = K(Z/p). Let M(k) = X-‘-o(k)/D(k 1). In mod-p cohomology H*(M(k)) has a basis consisting of admissible Steenrod operations of length exactly k.

Homology of classical groups over finite fields and their associated infinite loop spaces

Infinite loop spaces associated with ImJ.- Permutative categories of classical groups over finite fields.- K-theory of finite fields and the ImJ spaces.- Calculations at the prime 2.- Calculations at odd primes.- The homology of certain finite groups.- Detection theorems at the prime 2.- Detection theorems at odd primes.- Homology operations associated with the classical groups.

The complete stable splitting for the classifying space of a finite group

for the classifying spaces of finite groups. In a previous paper [ 171, one of us determined a necessary and sufficient condition for a spectrum to be an indecomposable summand in BG, for G a p-group. In this paper we give the multiplicity of each indecomposablc summand thus giving complete splittings for all finite groups G. The result is given in terms of the rank of a certain matrix which depends only on the subgroups of G and on the modular representation theory of their outer automorphism groups. Splittings of BG arc equivalent to idcmpotcnt decompositions of the identity in the ring of homotopy classes of stable maps [ UG. ,!I(; ). Such decompositions, which are usually diflicult to obtain, can be partially studied via the ring homomorphism

The transfer and Whitehead's conjecture

In this paper we present a proof of G. W. Whiteheads conjecture about symmetric products of the sphere spectrum S. Our methods are based on the transfer, the Steinberg module, and the structure of the Hecke algebra. Our results are valid for all primes and extend those of the first author for p = 2 [7]. As originally stated, the conjecture asserts that is zero on p -primary components in positive degrees [11]. By considering the quotients L(k) = Σ -k SP pk S/SP k-1- S , this is seen to be equivalent to the exactness of on homotopy groups, where ∂ k is the connecting map and e is the inclusion of the bottom cell. Here and throughout all spaces and spectra are localized at p .

Classification of BG for groups with dihedral or quarternion Sylow 2-subgroups

Abstract A classification is given of the stable homotopy type of BG for all groups cited in the title. This includes a complete stable decomposition of BG into indecomposable summands and a description of the corresponding mod-2 cohomology rings. Examples are given which show that all the resulting possibilities do, in fact, occur. Corresponding results are also given for the semidihedral groups.

Transferred Chern classes in Morava K-theory

Let η be a complex n-plane bundle over the total space of a cyclic covering of prime index p. We show that for k E {1,2,...,np} \ {p,2p,..., np} the k-th Chern class of the transferred bundle differs from a certain transferred class w k of η by a polynomial in the Chern classes c p ,..., c np of the transferred bundle. The polynomials are defined by the formal group law and certain equalities in K(s)*B(Z/p x U(n)).

Transfer and complex oriented cohomology rings

For nite coverings we elucidate the interaction between trans- ferred Chern classes and Chern classes of transferred bundles. This involves computing the ring structure for the complex oriented cohomology of vari- ous homotopy orbit spaces. In turn these results provide universal examples for computing the stable Euler classes (i.e. Tr(1)) and transferred Chern classes for p-fold covers. Applications to the classifying spaces of p-groups are given. AMS Classication 55N22; 55R12

A classification of the stable type of BG

We give a classification of the

ANOTHER SYSTEMATIC PHENOMENON IN THE COHOMOLOGY OF THE STEENROD ALGEBRA

p

Cohomology of groups with metacyclic Sylow -subgroups

--local stable homotopy type of