A categorical perspective on the Atiyah–Segal completion theorem in KK-theory

For G an almost-connected Lie group, we study G-equivariant index theory for proper co-compact actions with various applications, including obstructions to and existence of G-invariant Riemannian metrics of positive scalar curvature. We prove a rigidity result for almost-complex manifolds, generalising Hattori's results, and an analogue of Petrie's conjecture. When G is an almost-connected Lie group or a discrete group, we establish Poincaré duality between G-equivariant K-homology and K-theory, observing that Poincaré duality does not necessarily hold for general G.

show abstract

“…(A, B) in the sense of [4]. In [2,Theorem 3.19], a generalisation of the Atiyah-Segal completion theorem is proved:…”

Section: Poincaré Dualitymentioning

confidence: 99%

Positive scalar curvature and Poincaré duality for proper actions

Guo

Mathai

Wang

2019

J. Noncommut. Geom.

View full text Add to dashboard Cite

show abstract

“…In this case, the K-group K * (BΓ, BΛ) is defined as the K-group of the σ-C*-algebra of continuous function on BΓ whose restriction to BΛ is zero. The Kasparov product also works in the category of σ-C*-algebras (for the K-theory and KK-theory of σ-C*-algebras, see [Phi89,Bon02,AK17]). Then we have the Milnor lim…”

Section: Rational Injectivity and Nondegeneracymentioning

confidence: 99%

The relative Mishchenko--Fomenko higher index and almost flat bundles I: The relative Mishchenko--Fomenko index

Kubota

2018

Preprint

Self Cite

View full text Add to dashboard Cite

In this paper, we introduce an alternative definition of the Chang-Weinberger-Yu relative higher index as a relative analogue of the Mishchenko-Fomenko index pairing. We use this definition to construct the dual relative higher index map, which is related to the index pairing with almost flat bundles. Contents * 3.2. The proof of µ CWY * = µ MF * 3.3. The proof of µ DG * = µ MF * 4. Rational injectivity and nondegeneracy 5. Almost flat vector bundles on manifolds with boundary 5.1. Almost flatness for relative vector bundles 5.2. Relative quasi-representations and almost monodromy correspondence 6. Relative index pairing with coefficient in a C*-algebra 6.1. Index pairing with stably h-relative representations 6.2. Relative Hanke-Schick obstruction 6.3. The Hanke-Pape-Schick codimension 2 obstruction 7. Relative quantitative index pairing 7.1. Quantitative K-theory and almost * -homomorphism 7.2. Quantitative index pairing 7.3. Relative quantitative index pairing 8. Dual assembly map and almost flat bundles 8.1. K-homology group of mapping cone C*-algebras 8.2. Range of the dual assembly map 9. Partitioning and the K-theoretic van Kampen theorem Appendix A. A note on Real Kasparov theory References

show abstract

“…In [AK15], the authors study the relative homological algebra of compact group equivariant KK-theory in connection with the Atiyah-Segal completion theorem. Here, we introduce a categorical counterpart of freeness of group actions on C * -algebras, J n G -injectivity.…”

Section: Introductionmentioning

confidence: 99%