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Bordisms and unbounded $KK$-theory

Robin J. Deeley, Magnus Goffeng, Bram Mesland

Abstract

This monograph studies $KK$-theory in its unbounded model. The central object is the $KK$-bordism group obtained by imposing the $KK$-bordism relation on unbounded $KK$-cycles. In the paradigm of noncommutative geometry, an unbounded $KK$-cycle is a noncommutative geometry in its own right and our approach allow for the study of mildly noncommutative geometries (orbifolds, foliations et cetera) as if they were closed manifolds. The techniques we introduce enable us to directly import manifold techniques and arguments into the important yet technical field of unbounded $KK$-theory. Recent decades has seen a tremendous progress in the study of the unbounded model for $KK$ as well as secondary invariants, the first motivated by refining computational tools in Kasparov's $KK$-theory and the second by applications to geometry and topology. The aim of this work is to provide a common framework for these two areas: equipping unbounded $KK$-cycles with a geometrically motivated relation recovering Kasparov's $KK$-theory that is computationally tractable for working with secondary invariants.

Bordisms and unbounded $KK$-theory

Abstract

This monograph studies -theory in its unbounded model. The central object is the -bordism group obtained by imposing the -bordism relation on unbounded -cycles. In the paradigm of noncommutative geometry, an unbounded -cycle is a noncommutative geometry in its own right and our approach allow for the study of mildly noncommutative geometries (orbifolds, foliations et cetera) as if they were closed manifolds. The techniques we introduce enable us to directly import manifold techniques and arguments into the important yet technical field of unbounded -theory. Recent decades has seen a tremendous progress in the study of the unbounded model for as well as secondary invariants, the first motivated by refining computational tools in Kasparov's -theory and the second by applications to geometry and topology. The aim of this work is to provide a common framework for these two areas: equipping unbounded -cycles with a geometrically motivated relation recovering Kasparov's -theory that is computationally tractable for working with secondary invariants.

Paper Structure

This paper contains 85 sections, 156 theorems, 493 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Analytic versus geometric models for $K$-homology
  3. Main results and overview of the monograph
  4. Acknowledgements
  5. Preliminaries
  6. Elements of unbounded $KK$-theory
  7. Examples of cycles and chains
  8. Suspension operations on chains
  9. Suspension by $[0,1]$
  10. Suspension by $[0,\infty)$
  11. Suspension by $\mathbb{R}$
  12. Shubin-suspension by $\mathbb{R}$
  13. Shubin-suspension by $[0,\infty)$
  14. Dirac operators on $C^*$-bundles
  15. Group actions on cycles and chains
  16. ...and 70 more sections

Key Result

Theorem 1

There are canonical equivariant $KK$-bordisms associated with There is also well defined operations for descent of equivariant cycles, that for any second countable locally compact group $G$ produces a map Here $\mathcal{A}\rtimes_{ c} G:=C_c(G,\mathcal{A})$ with the twisted convolution product. And finally, for a proper, cocompact action of $G$ on a metric space $X$ we have an assembly map

Figures (7)

  • Figure 1: A $C^*$-bundle $\mathcal{E}_B$ over a compact manifold with boundary as in Example \ref{['diraconcstarbundleex']}.
  • Figure 2: A graphical representation of a $KK$-bordism, with the shaded part on the right representing the collar neighbourhood of the boundary.
  • Figure 3: Gluing of two $KK$-bordisms $\mathfrak{X}$ and $\mathfrak{X}'$ along a common boundary $\mathfrak{Y}_2$.
  • Figure 4: Straightening the angles on the unit square.
  • Figure 5: Picture of $W\times [0,1]$, for $W$ the grey surface with boundary, representing $\Psi(\mathpzc{N},T)$ as appearing in Lemma \ref{['restsups']}.
  • ...and 2 more figures

Theorems & Definitions (332)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 3.1
  • Proposition 3.2
  • ...and 322 more