An index theorem for families of Fredholm operators parametrized by a group
type: Item
instance of: scholarly article
title: An index theorem for families of Fredholm operators parametrized by a group
zbMATH Open document ID: 0673.58004
DOI: 10.1016/0040-9383(88)90010-9
author: Daniel S. Freed
published in: Topology
publication date: 1988-01-01T00:00:00Z
review text: The author studies the problem of realizing real cohomology of some infinite dimensional Lie groups by left invariant differential forms. The stable complex general linear group GL(\(\infty)\) is considered as a subgroup of GL, the group of invertible operators in a separable complex Hilbert space \({\mathcal H}\). For all \(p\geq 1\) the Schatten ideals \(L^ p\) of compact operators in \({\mathcal H}\) are used to construct Banach Lie groups \(GL^ p\). By a theorem of R. Palais all these groups are homotopy equivalent to GL(\(\infty)\). It is proved that every singular cohomology class of GL(\(\infty)\) and \(GL^ 1\) is representable by an invariant form, and that the real cohomology of \(G^ p\), \(p>1\), is represented by invariant forms in sufficiently large degrees depending on p. For smooth principal \(GL^ p\)-bundles \({\mathcal P}\to {\mathcal M}^ a \)Chern-Weil theorem is proved, expressing the Chern character classes \(ch_{\ell}({\mathcal P})\) by a differential form on \({\mathcal M}.\) To study the cohomology of the classifying space BGL(\(\infty)\) explicitly the author constructs a tower of groups \({\mathcal G}^ p=GL/GL^ p\). In absence of a natural good topology \({\mathcal G}^ p\) is considered as an abstract group only. There exists a projection \(\pi\) : \(Fred_ 0\to {\mathcal G}^ p\) from the space of Fredholm operators with index zero to \({\mathcal G}^ p\). Although GL\(\to {\mathcal G}^ p\) is not a bundle, the pullback of GL to \(Fred_ 0\) is a principal \(GL^ p\)-bundle and a model of the universal fibration EGL(\(\infty)\to BGL(\infty)\). The author regards smooth maps T: \(G\to Fred_ 0\) from a Banach Lie group G to \(Fred_ 0\) such that the composition \(G\to^{T}Fred_ 0\to^{\pi}{\mathcal G}^ p\) is a homomorphism of abstract groups. In this case he constructs a left invariant \(L^ p\)-valued 2-form \(\Omega\) on G, which is used to represent \(T^*ch_{\ell}\) as a left invariant form on G for \(\ell \geq p\). This is the index theorem of th…
Mathematics Subject Classification ID: 58A12
Mathematics Subject Classification ID: 55R35
Mathematics Subject Classification ID: 47A53
Mathematics Subject Classification ID: 58B15
Mathematics Subject Classification ID: 58B25
Mathematics Subject Classification ID: 47L30
zbMATH DE Number: 4102110
zbMATH Keywords: classifying space of K-theory
zbMATH Keywords: Chern-Weil theorem
zbMATH Keywords: infinite dimensional Lie groups
reviewed by: K. H. Mayer
MaRDI profile type: Publication
full work available at URL: 0040-9383(88)90010-9
OpenAlex ID: W2093239908
Wikidata QID: Q126422573
description: scientific article; zbMATH DE number 4102110