Poincar\'e duality, Hilbert complexes and geometric applications

TL;DR

This paper explores properties of $L^2$ cohomology groups on open Riemannian manifolds, establishing duality results, defining an $L^2$ signature, and applying findings to stratified pseudomanifolds and Friedrichs extensions.

Contribution

It introduces a new Hilbert complex framework for $L^2$ cohomology sequences and establishes Poincaré duality and signature definitions under specific conditions.

Findings

The first $L^2$ cohomology sequence is a Hilbert complex containing the minimal and contained in the maximal.

Poincaré duality holds for the finite-dimensional second sequence.

Applications include results on intersection cohomology and Friedrichs extensions.

Abstract

Let $(M,g)$ an open and oriented riemannian manifold. The aim of this paper is to study some properties of the two following sequences of $L^2$ cohomology groups: $H^i_{2,m\rightarrow M}(M,g)$ defined as the image $\im(H^i_{2,min}(M,g)\rightarrow H^i_{2,max}(M,g))$ and $\bar{H}^i_{2,m\rightarrow M}(M,g)$ defined as $\im(\bar{H}^i_{2,min}(M,g)\rightarrow \bar{H}^i_{2,max}(M,g))$. We show, under certain hypothesis, that the first sequence is the cohomology of a suitable Hilbert complex which contains the minimal one and is contained in the maximal one. We also show that when the second sequence is finite dimensional then Poincar\'e duality holds for it and that, in the same assumptions, when $dim(M)\equiv0\ mod\ 4$ we can use it to define a $L^2$ signature on $M$. Moreover we show several applications to the intersection cohomology of compact smoothly stratified pseudomanifolds and we get some results about the Friedrichs extension $\Delta_{i}^\mathcal{F}$ of $\Delta_{i}$.