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Proper actions and decompositions in equivariant K-theory

Andrés Angel, Edward Becerra, Mario Velásquez

Abstract

In this paper we study a natural decomposition of $G$-equivariant $K$-theory of a proper $G$-space, when $G$ is a Lie group with a compact normal subgroup $A$ acting trivially. Our decomposition could be understood as a generalization of the theory known as Mackey machine under suitable hypotheses, since it decomposes $G$-equivariant K-theory in terms of twisted equivariant K-theory groups respect to some subgroups of $G/A$. Similar decompositions were known for the case of a compact Lie group acting on a space, but our main result applies to discrete, linear and almost connected groups. We also apply this decomposition to study equivariant $K$-theory of spaces with only one isotropy type. We provide a rich class of examples in order to expose the strength and generality of our results. We also study the decomposition for equivariant connective $K$-homology for actions of compact Lie groups using a suitable configuration space model, based on previous papers published by the third author.

Proper actions and decompositions in equivariant K-theory

Abstract

In this paper we study a natural decomposition of -equivariant -theory of a proper -space, when is a Lie group with a compact normal subgroup acting trivially. Our decomposition could be understood as a generalization of the theory known as Mackey machine under suitable hypotheses, since it decomposes -equivariant K-theory in terms of twisted equivariant K-theory groups respect to some subgroups of . Similar decompositions were known for the case of a compact Lie group acting on a space, but our main result applies to discrete, linear and almost connected groups. We also apply this decomposition to study equivariant -theory of spaces with only one isotropy type. We provide a rich class of examples in order to expose the strength and generality of our results. We also study the decomposition for equivariant connective -homology for actions of compact Lie groups using a suitable configuration space model, based on previous papers published by the third author.

Paper Structure

This paper contains 10 sections, 26 theorems, 173 equations.

Table of Contents

  1. Introduction
  2. Mackey machine for equivariant vector bundles and compact normal subgroup
  3. Semidirect products with abelian normal subgroup
  4. Decomposition of $G$-vector bundles
  5. Induction
  6. Decomposition in equivariant K-theory (proper case)
  7. Structure of actions with only one isotropy type
  8. K-theory of actions with only one isotropy type
  9. Examples
  10. Additional proofs regarding the assumption (K)

Key Result

Theorem 1.1

Let $G$ be a Lie group satisfying assumption (K), and $X$ be a proper $G$-space on which the normal subgroup $A$ acts trivially. There exists a natural isomorphism This isomorphism is functorial on $G$-maps $X\to Y$ of proper $G$-spaces on which $A$ acts trivially.

Theorems & Definitions (55)

  • Theorem 1.1
  • Remark 2.1
  • Lemma 2.2
  • Remark 2.3
  • Proposition 2.4
  • proof
  • Definition 2.5
  • Proposition 2.6
  • proof
  • Proposition 2.7
  • ...and 45 more