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Algebraic kk-theory and the KH-isomorphism conjecture

Eugenia Ellis, Emanuel Rodríguez Cirone

Abstract

We relate the Davis-Lück homology with coefficients in Weibel's homotopy K-theory to the equivariant algebraic kk-theory using homotopy theory and adjointness theorems. We express the left hand side of the assembly map for the KH-isomorphism conjecture in terms of equivariant algebraic kk-groups.

Algebraic kk-theory and the KH-isomorphism conjecture

Abstract

We relate the Davis-Lück homology with coefficients in Weibel's homotopy K-theory to the equivariant algebraic kk-theory using homotopy theory and adjointness theorems. We express the left hand side of the assembly map for the KH-isomorphism conjecture in terms of equivariant algebraic kk-groups.
Paper Structure (36 sections, 42 theorems, 143 equations)

This paper contains 36 sections, 42 theorems, 143 equations.

Table of Contents

  1. Introduction
  2. Notation, conventions and preliminaries
  3. Algebras of polynomial functions on a simplicial set
  4. Isomorphism conjectures
  5. Model structure on G-simplicial sets determined by a family of subgroups
  6. Equivariant homology and assembly map
  7. The Baum-Connes conjecture with coefficients
  8. The $\mathrm{KH}$-isomorphism conjecture
  9. Algebraic kk-theory
  10. Homotopy invariance
  11. Matrix stability
  12. Excision
  13. Bivariant K-theory categories
  14. Adjoint theorems revisited
  15. Crossed product with G//H
  16. ...and 21 more sections

Key Result

Lemma 2.1

Let $B$ be an algebra.

Theorems & Definitions (104)

  • Lemma 2.1: cortho*Lemma 3.1.2 and Proposition 3.1.3
  • Remark 2.2
  • Remark 2.3
  • Example 2.4
  • Definition 2.5
  • Remark 2.8
  • Remark 2.9
  • Lemma 2.12: friendly*Proposition 2.2.6
  • proof
  • Remark 2.14
  • ...and 94 more