Equivariant moduli problems, branched manifolds, and the Euler class
type: Item
instance of: scholarly article
title: Equivariant moduli problems, branched manifolds, and the Euler class
zbMATH Open document ID: 1026.58010
DOI: 10.1016/S0040-9383(02)00022-8
author: K. Cieliebak
author: Ignasi Mundet i Riera
author: Dietmar A. Salamon
published in: Topology
publication date: 2003-06-17T00:00:00Z
review text: The authors give the following definitions. A \(G\)-moduli problem \((G\) is a compact oriented Lie group) is a triple (\textbf{B,E,S}) such that \({ \mathbf B}\) is a Hilbert manifold without boundary equipped with a smooth \(G\)-action; \textbf{E} is a Hilbert space bundle over \textbf{B}, also equipped with a smooth \(G\)-action, such that \(G\) acts by isometries on the fibres of \textbf{E} and the projection \({\mathbf E}\to{\mathbf B}\) is \(G\)-equivariant; \({\mathbf S}:{\mathbf B} \to{\mathbf E}\) is a smooth \(G\)-equivariant Fredholm section of constant Fredholm index such that the determinant bundle \(\det({\mathbf S})\to{\mathbf B}\) is oriented, \(G\) acts by orientation preserving isomorphisms on the determinant bundle, and the zero set \({\mathbf M}=\{x\in {\mathbf B}/{\mathbf S}(x)=0\}\) is compact. A finite dimensional \(G\)-moduli problem \((B,E,S)\) is called oriented if \(B\) and \(E\) are oriented and \(G\) acts on \(B\) and \(E\) by orientation preserving diffeomorphisms. A \(G\)-moduli problem (\textbf{B,E,S}) is called regular if the isotropy subgroup \(G_x=\{x\in G\mid g*x=x\}\) is finite for every \(x\in{\mathbf M}\). A morphism from a \(G\)-moduli problem (\textbf{B,E,S}) to another one \(\mathbf B',\mathbf E',\mathbf S'\)) is a pair \((\psi,\Psi)\) such that \(\psi\) is a smooth \(G\)-equivariant embedding of a neighbourhood \({\mathbf B}_0 \subset{\mathbf B}\) of \({\mathbf M}\) into \({\mathbf B} \), \(\Psi\) is a smooth injective bundle homomorphism of \({\mathbf E}_0={\mathbf E}_{{ \mathbf B}_0}\) into \({\mathbf E}'\) and a lift of \(\psi\), and the sections \textbf{S} and \textbf{S}' satisfy \({\mathbf S}'\circ\psi=\Psi\circ{\mathbf S}\), \({\mathbf M}'= \psi ({\mathbf M})\). Moreover, the linear operators \(d_x\psi:T_x{\mathbf B}\to T_{\psi (x)}{\mathbf B}'\) and \(\Psi_x:{\mathbf E}_x\to {\mathbf E}_{\psi(x)}'\) induce isomorphisms \(d_x\psi: \ker{\mathbf D}_x\to\ker {\mathbf D}_{\psi(x)}'\) and \(\Psi_x: \text{coker} {\mathbf D}…
Mathematics Subject Classification ID: 58D29
Mathematics Subject Classification ID: 57N20
Mathematics Subject Classification ID: 58B15
Mathematics Subject Classification ID: 57R91
Mathematics Subject Classification ID: 57R20
zbMATH DE Number: 1929320
zbMATH Keywords: equivariant cohomology
zbMATH Keywords: Euler class
reviewed by: Enrique Outerelo
MaRDI profile type: Publication
full work available at URL: s0040-9383(02)00022-8
OpenAlex ID: W2051389858
description: scientific article; zbMATH DE number 1929320