An exact sequence in \(L^2\)-cohomology
type: Item
instance of: scholarly article
title: An exact sequence in \(L^2\)-cohomology
zbMATH Open document ID: 0951.58024
DOI: 10.1215/S0012-7094-98-09510-2
author: Gilles Carron
published in: Duke Mathematical Journal
publication date: 2000-03-27T00:00:00Z
review text: The author studies the (reduced) \(L^2\)-cohomology groups of a complete Riemannian manifold \(M\). These can be defined as vector spaces of harmonic and square integrable differential forms. Carron restricts his attention to such \(M\) which fulfill a certain type of Sobolev inequality plus an integral curvature bound. Then in particular the \(L^2\)-cohomology spaces are finite dimensional. The main result is that for such manifolds we get an exact sequence in \(L^2\)-cohomology of the pair \((M,M-D)\), where \(D\) is an open bounded submanifold with smooth boundary. Consequences of this are absolute and relative Gauss-Bonnet theorems for the \(L^2\)-Euler characteristic. The main ingredient in the proof is the construction of a Green operator for the Laplacian (on differential forms) under the assumptions of the main result. The geometric conditions are rather strong. Examples are convex \(n\)-dimensional Riemannian submanifolds of Euclidean space with infinite volume and with \(L^n\)-integrable second fundamental form; or \(n\)-manifolds which are flat at infinity where the volume of balls grows like \(r^n\) (in both cases \(n>4\)).
Mathematics Subject Classification ID: 58J10
Mathematics Subject Classification ID: 58A14
Mathematics Subject Classification ID: 53C21
zbMATH DE Number: 1425165
zbMATH Keywords: reduced \(L^2\)-cohomology
zbMATH Keywords: Mayer-Vietoris cohomology sequence
zbMATH Keywords: Sobolev inequalities
zbMATH Keywords: Gauss-Bonnet formula
zbMATH Keywords: Green operator
MaRDI profile type: Publication
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full work available at URL: s0012-7094-98-09510-2
OpenAlex ID: W2052390781
description: scientific article; zbMATH DE number 1425165