The Novikov conjecture on Cheeger spaces

Pierre Albin, Eric Leichtnam, Rafe Mazzeo, Paolo Piazza

Research output: Contribution to journalArticlepeer-review

Abstract

We prove the Novikov conjecture on oriented Cheeger spaces whose fundamental group satisfies the strong Novikov conjecture. A Cheeger space is a stratified pseudomanifold admitting, through a choice of ideal boundary conditions, an L2-de Rham cohomology theory satisfying Poincaré duality. We prove that this cohomology theory is invariant under stratified homotopy equivalences and that its signature is invariant under Cheeger space cobordism. Analogous results, after coupling with a Mischenko bundle associated to any Galois covering, allow us to carry out the analytic approach to the Novikov conjecture: we define higher analytic signatures of a Cheeger space and prove that they are stratified homotopy invariants whenever the assembly map is rationally injective. Finally we show that the analytic signature of a Cheeger space coincides with its topological signature as defined by Banagl.

Original languageEnglish (US)
Pages (from-to)451-506
Number of pages56
JournalJournal of Noncommutative Geometry
Volume11
Issue number2
DOIs
StatePublished - 2017

Keywords

  • Cheeger spaces
  • Higher index theory
  • Higher signatures
  • Ideal boundary conditions
  • K-theory
  • L2-cohomology
  • Stratified homotopy invariance
  • Stratified spaces

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Mathematical Physics
  • Geometry and Topology

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