The L-Homology Fundamental Class for IP-Spaces and the Stratified Novikov Conjecture

TL;DR

This paper introduces a new fundamental L-homology class for IP-spaces, a class of stratified spaces satisfying Poincaré duality, and proves the stratified Novikov conjecture for these spaces using a novel spectrum map.

Contribution

It constructs a fundamental L-homology class for IP-spaces and proves the stratified Novikov conjecture by relating IP-bordism to symmetric L-theory spectra.

Findings

Constructed a fundamental L-homology class for IP-spaces.

Proved the stratified Novikov conjecture for IP-spaces.

Established a spectrum map from IP-bordism to symmetric L-theory.

Abstract

An IP-space is a pseudomanifold whose defining local properties imply that its middle perversity global intersection homology groups satisfy Poincar\'e duality integrally. We show that the symmetric signature induces a map of Quinn spectra from IP bordism to the symmetric $L$-spectrum of $\Z$, which is, up to weak equivalence, an $E_\infty$ ring map. Using this map, we construct a fundamental $L$-homology class for IP-spaces, and as a consequence we prove the stratified Novikov conjecture for IP-spaces.