$K$-Bad Spheres

TL;DR

This paper investigates the $K$-completion of spheres in algebraic topology, revealing that for certain dimensions and primes, the $K$-homology differs from that of the original sphere, indicating 'bad' behavior.

Contribution

It identifies specific spheres that are '$K$-bad', demonstrating non-isomorphism in $K$-homology after $K$-completion for certain cases.

Findings

$K$-homology of some spheres differs after $K$-completion

Identification of '$K$-bad' spheres for specific $n$ and $p$

Highlights limitations of $K$-completion in topological analysis

Abstract

In this paper we look at the $E$-completion of topological spaces where $E$ is a $p$-local ring spectrum. After a brief review of the concept of $E$-completion, we specialize to the case where $E=K$, $p$-local complex periodic $K$-theory, and consider the $K$-theory of the unstable sphere $S^{2n+1}$. We show that for certain values of $n$ and an odd prime $p$, the $K$-homology of the $K$-completion is not isomorphic to the $K$-homology of the sphere itself, thus in the terminology of Bousfield and Kan, these spheres are '$K$-bad'.