Victor Snaith

Localized stable homotopy of some classifying spaces

In this note I will give new, simplified proofs of some of the results announced in (18) and proved in ((19), I, § 4, II, §§ 2, 9).These results mostly date from 1975/6 at which time they were proved using my crude stable decomposition of ω n σ n X (20) and differential-geometric techniques with the Becker-Gottlieb transfer. Since then more systematic approaches have been developed towards the stable decompositions ((4); (5); (6); (10); (12)) and towards the transfer ((7), I and II; (8)). Actually, the system-atization of the stable decompositions was already latent in (15) if only I had realized it!

A conjecture of Barratt-Jones-Mahowald concerning framed manifolds having Kervaire invariant one

To SETTLE the question of the existence or non-existence of a framed manifold having a nontrivial Kervaire invariant (or Arf invariant) is one of the main long-standing problems in algebraic topology. The Kervaire invariant is a Z/Zvalued invariant which may be formulated in many contexts. Originally it occurred as an invariant in framed surgery theory and for this approach the reader may consult Cl23 for example. It is reformulated in [7] in terms of the Adams spectral sequence for the stable homotopy of spheres. In particular the only open cases were reduced to determining whether ht E Exts ‘*+‘(Z/2,2/2) is an infinite cycle, producing a non-trivial element 8, in the 2’[+’ 2 stem of the stable homotopy of spheres. More recently the Kahn-Priddy theorem [8] and the algebraic Kahn-Priddy theorem [lo] have been used to convert it to a problem in the stable homotopy of infinite dimensional real projective space [WP”. The Kahn-Priddy theorem gives a stable map T: [WF’aSO which is a split surjection of stable homotopy groups (localized at the prime 2)

The Second Chinburg Conjecture for Quaternion Fields

Introduction Class-groups of group-rings The evaluation of

The 2-primary J -homomorphism

[X]

K 3 of truncated polynomial rings over fields of characteristic two

Quaternion fields over

Algebraic K-Theory Eventually Surjects onto Topological K-Theory

\mathbf{Q}_2

On the K-Theory of the Kahn–Priddy Map

The invariant in Cases A, B and C The evaluation of

Higher Algebraic K-Theory: An Overview

[M]

Unitary K-homology and the lichtenbaum-quillen conjecture on the algebraic K-theory of schemes

The conjecture in Cases A, B and C Epilogue Bibliography Index.

On the stable homotopy of symplectic classifying and thom spaces

In this paper every space will be 2-local, for example BO will mean the 2-localization of the space usually denoted BO .