Euler characteristic of Fredholm quasicomplexes
type: Item
instance of: scholarly article
title: Euler characteristic of Fredholm quasicomplexes
zbMATH Open document ID: 1154.58010
publication date: 2008-09-02T00:00:00Z
full work available at URL: 2006 11.pdf
review text: The aim of this article is to define the class of so--called Fredholm quasicomplexes, which generalize complexes of Fredholm operators between Hilbert spaces, and to extend the notion of Euler characteristic to this setting. The author defines a quasicomplex to be a sequence of operators \(d_i\) between Hilbert spaces such that \(d_{i+1}\circ d_i\) is a compact operator for all \(i\). Then he defines the appropriate notions of chain mappings and of homotopy of chain mappings in this setting. Generalizing the fact that Fredholm operators are the ones whose image in the Calkin algebra is invertible, the author next introduces a notion of Fredholm quasicomplexes, which is stable under compact perturbations. For bounded complexes, the Fredholm property is proved to be equivalent to the identity mapping being homotopic to zero. The generalization of the Euler characteristic is obtained by proving that there are compact perturbations of the operators in a Fredholm quasicomplex which form a complex, and that any two such perturbations are homotopic.
Mathematics Subject Classification ID: 58J10
Mathematics Subject Classification ID: 47A53
Mathematics Subject Classification ID: 19K56
zbMATH DE Number: 5319693
zbMATH Keywords: Fredholm complex
zbMATH Keywords: quasicomplex
zbMATH Keywords: Euler characteristic
author: N. N. Tarkhanov
reviewed by: Andreas Cap
MaRDI profile type: Publication
OpenAlex ID: W1977732576
cites work: Q4256630
cites work: Q4859903
DOI: 10.1007/S10688-007-0032-9
published in: Functional Analysis and Its Applications
description: scientific article; zbMATH DE number 5319693