A K-theoretic refinement of topological realization of unstable algebras

Details A K-theoretic refinement of topological realization of unstable algebras A K-theoretic refinement of topological realization of unstable algebras

2002-09-04

arxiv.org/abs/math/0209032v2

Mathematics

Algebraic Topology

22 pages

Scientific paper

In this paper we propose and partially carry out a program to use $K$-theory to refine the topological realization problem of unstable algebras over the Steenrod algebra. In particular, we establish a suitable form of algebraic models for $K$-theory of spaces, called $\psi^p$-algebras, which give rise to unstable algebras by taking associated graded algebras mod $p$. The aforementioned problem is then split into (i) the \emph{algebraic} problem of realizing unstable algebras as mod $p$ associated graded of $\psi^p$-algebras and (ii) the \emph{topological} problem of realizing $\psi^p$-algebras as $K$-theory of spaces. Regarding the algebraic problem, a theorem shows that every connected and even unstable algebra can be realized. We tackle the topological problem by obtaining a $K$-theoretic analogue of a theorem of Kuhn and Schwartz on the so-called Realization Conjecture.

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