Equivariant surgery theories and their periodicity properties
type: Item
instance of: scholarly article
title: Equivariant surgery theories and their periodicity properties
zbMATH Open document ID: 0712.57001
author: Karl Heinz Dovermann
author: Reinhard Schultz
published in: Lecture Notes in Mathematics
publication date: 1992-09-17T00:00:00Z
review text: The book under review is an exposition and survey of certain formal properties of equivariant surgery theories. Before giving more details about the contents of the book, it might be useful to briefly describe this subject. Equivariant surgery is concerned with the study of manifolds with group action up to variation by bordisms constructed from ``elementary surgeries''. If N is a compact n-dimensional smooth G-manifold, let \(N^ H\) denote the fixed-point set of a subgroup \(H\subseteq G\). For convenience, I will assume that G is a finite group. Let \(f_ 1: S^ k\times D^{n-k}\to N\) be a framed embedding such that (i) \(f_ 1\) is an H-equivariant smooth map, (ii) \(f_ 1(S^ k\times 0)\subset N^ H\), and (iii) the image \(f_ 1(S^ k\times D^{n-k})\) is a smooth H- tubular neighbourhood of \(f_ 1(S^ k\times 0)\) in N. These properties fix the H-action on \(S^ k\times D^{n-k}\) as the product of the trivial action on the first factor and a linear H-representation on the second factor. If we also assume that the image of \(f_ 1\) is disjoint from all its translates by elements of G-H, then \(f_ 1\) extends to a G- embedding \[ f:\coprod_{\alpha \in G/H}(S^ k\times D^{n- k})_{\alpha}\to N \] induced by the action of G. The basic operation in equivariant surgery is to construct the bordism \[ W=(N\times I)\cup_ f\coprod_{\alpha \in G/H}(D^{k+1}\times D^{n-k})_{\alpha}, \] referred to as the trace of an elementary surgery on the class \(f_ 1(S^ k\times 0)\). The manifold \(N'\) appearing as the ``top'' boundary of W is the result of performing equivariant surgery on the given class in N. There is a map F: \(W\to N\) extending the identity on \(N\times 0\subset W\) which maps the equivariant handle to a G-orbit of points in N. More generally, if we start with a G-map h: \(N\to M\), where M is another G-manifold of the same dimension, then this construction provides a bordism between the maps h and \(h': N'\to M.\) There are various natural questions connected with this cons…
Mathematics Subject Classification ID: 57-02
Mathematics Subject Classification ID: 57R67
Mathematics Subject Classification ID: 57R91
Mathematics Subject Classification ID: 57S17
Mathematics Subject Classification ID: 57R65
Mathematics Subject Classification ID: 57S15
zbMATH DE Number: 47878
zbMATH Keywords: stratification by fixed sets of subgroups
zbMATH Keywords: Equivariant surgery
zbMATH Keywords: manifolds with group action
zbMATH Keywords: elementary surgeries
zbMATH Keywords: equivariant handle
zbMATH Keywords: normal maps
zbMATH Keywords: equivariant homotopy equivalences
zbMATH Keywords: equivariant maps which are homotopy equivalences
zbMATH Keywords: pseudo-equivalences
zbMATH Keywords: structure set
zbMATH Keywords: equivariant surgery obstruction groups
zbMATH Keywords: surgery exact sequence
zbMATH Keywords: Gap Hypothesis
zbMATH Keywords: periodicity properties of the surgery obstruction groups
zbMATH Keywords: equivariant surgery with coefficients
reviewed by: Ian Hambleton
MaRDI profile type: Publication
DOI: 10.1007/BFB0092354
description: scientific article; zbMATH DE number 47878