Journal of Noncommutative Geometry

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Volume 11, Issue 2, 2017, pp. 451–506
DOI: 10.4171/JNCG/11-2-2

Published online: 2017-06-29

The Novikov conjecture on Cheeger spaces

Pierre Albin[1], Eric Leichtnam[2], Rafe Mazzeo[3] and Paolo Piazza[4]

(1) University of Illinois at Urbana-Champaign, USA
(2) Institut de Mathématiques de Jussieu-Paris Rive Gauche, France
(3) Stanford University, USA
(4) Università di Roma La Sapienza, Italy

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 $L^2$-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.

Keywords: Stratified spaces, $L^2$-cohomology, ideal boundary conditions, Cheeger spaces, higher signatures, stratified homotopy invariance, K-theory, higher index theory

Albin Pierre, Leichtnam Eric, Mazzeo Rafe, Piazza Paolo: The Novikov conjecture on Cheeger spaces. J. Noncommut. Geom. 11 (2017), 451-506. doi: 10.4171/JNCG/11-2-2