The Novikov conjecture on Cheeger spaces

  • Pierre Albin

    University of Illinois at Urbana-Champaign, USA
  • Eric Leichtnam

    Institut de Mathématiques de Jussieu-Paris Rive Gauche, France
  • Rafe Mazzeo

    Stanford University, USA
  • Paolo Piazza

    Università di Roma La Sapienza, Italy

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 -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.

Cite this article

Pierre Albin, Eric Leichtnam, Rafe Mazzeo, Paolo Piazza, The Novikov conjecture on Cheeger spaces. J. Noncommut. Geom. 11 (2017), no. 2, pp. 451–506

DOI 10.4171/JNCG/11-2-2