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The rational HK-conjecture: transformation groupoids and a revised version

Robin J. Deeley, Rufus Willett

Abstract

We prove the rational HK-conjecture for a large class of transformation groupoids in the case when the relevant action has torsion-free stabilizers. A revised version of the rational HK-conjecture in the case of (possibly) torsion stabilizers is introduced and proved for a large class of transformation groupoids. In particular, this revised version holds for Scarparo's counterexamples to the original rational HK-conjecture. The key tools used are the Baum-Connes conjecture and a Chern character defined by Raven.

The rational HK-conjecture: transformation groupoids and a revised version

Abstract

We prove the rational HK-conjecture for a large class of transformation groupoids in the case when the relevant action has torsion-free stabilizers. A revised version of the rational HK-conjecture in the case of (possibly) torsion stabilizers is introduced and proved for a large class of transformation groupoids. In particular, this revised version holds for Scarparo's counterexamples to the original rational HK-conjecture. The key tools used are the Baum-Connes conjecture and a Chern character defined by Raven.
Paper Structure (14 sections, 24 theorems, 131 equations)

This paper contains 14 sections, 24 theorems, 131 equations.

Table of Contents

  1. Preliminaries
  2. Baum-Schneider homology and Raven's Chern character
  3. Derived categories and Baum-Schneider bivariant homology
  4. Isomorphism between homologies
  5. Raven's Chern character
  6. Main results
  7. An HK conjecture that allows torsion stabilizers
  8. Computational tools and examples
  9. Baum-Schneider homology and group hyperhomology
  10. Step 1: computation of $R\Gamma_ck_{\widehat{Y}}$ for $Y$ an appropriate simplicial complex
  11. Step 2: computation of $R\Gamma_ck_{\widehat{X}}$ for $X$ finite-dimensional
  12. Step 3: computation of $\text{Ext}_{G\text{-}k[G_{tor}]}(R\Gamma_ck_{\widehat{Y}},R\Gamma_ck_{\widehat{X}})$
  13. Step 4: computation of $H^*_{G,!}(Z,X)$
  14. Step 5: computation of $H_*(G,\Gamma_c(I^{-\bullet}))$

Key Result

Theorem 2

Conjecture hk revised intro above holds if $\mathcal{G}$ is a transformation groupoid.

Theorems & Definitions (79)

  • Conjecture 1
  • Theorem 2
  • Conjecture 1.1
  • Conjecture 1.2
  • Definition 1.3
  • Definition 2.1
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • ...and 69 more