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Commuting varieties and the rank filtration of topological K-theory

Simon Gritschacher

Abstract

We consider the space of $n$-tuples of pairwise commuting elements in the Lie algebra of $U(m)$. We relate its one-point compactification to the subquotients of certain rank filtrations of connective complex $K$-theory. We also describe the variant for connective real $K$-theory.

Commuting varieties and the rank filtration of topological K-theory

Abstract

We consider the space of -tuples of pairwise commuting elements in the Lie algebra of . We relate its one-point compactification to the subquotients of certain rank filtrations of connective complex -theory. We also describe the variant for connective real -theory.

Paper Structure

This paper contains 6 sections, 16 theorems, 103 equations.

Table of Contents

  1. Introduction
  2. Connective $K$-theory via commuting isometries
  3. The unstable rank filtration of $ku_n$
  4. The stable rank filtration of $ku$
  5. The homotopy type of the subquotients
  6. Statements for real $K$-theory

Key Result

Theorem 1.1

For $n\geq 1$, the subquotients of the rank filtration of $ku_n$ are

Theorems & Definitions (31)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • proof
  • ...and 21 more