Surgery of involutions with middle-dimensional fixed point set
type: Item
instance of: scholarly article
title: Surgery of involutions with middle-dimensional fixed point set
zbMATH Open document ID: 0656.57021
author: Karl Heinz Dovermann
author: Reinhard Schultz
published in: Pacific Journal of Mathematics
publication date: 1987-01-01T00:00:00Z
review text: Let X and Y be simply-connected closed manifolds with smooth \({\mathbb{Z}}_ 2\)-actions and let f: \(X\to Y\) be a degree one equivariant map. The authors study the problem of finding a \({\mathbb{Z}}_ 2\)-bordism of f to f': X'\(\to Y\) which is a (not necessarily equivariant) homotopy equivalence. There are several technical assumptions made in order to make \({\mathbb{Z}}_ 2\)-equivariant surgery possible. The special hypothesis emphasized in this paper in addition to others is that ``the dimension of the fixed point set is at most (1/2)dim X and there is exactly one component of the fixed set whose dimension is equal to (1/2)dim X''. The paper is very much in the spirit of \textit{K. H. Dovermann}'s [Mich. Math. J. 28, 267-287 (1981; Zbl 0491.57014)]. The main results of the paper give necessary and sufficient conditions for the vanishing of the surgery obstructions in terms of signatures and Arf invariants. The techniques are adapted from standard surgery theory.
Mathematics Subject Classification ID: 57R85
Mathematics Subject Classification ID: 57R67
Mathematics Subject Classification ID: 57S25
zbMATH DE Number: 4071941
zbMATH Keywords: smooth \({\mathbb{Z}}_ 2\)-actions
zbMATH Keywords: degree one equivariant map
zbMATH Keywords: \({\mathbb{Z}}_ 2\)-bordism
zbMATH Keywords: homotopy equivalence
zbMATH Keywords: \({\mathbb{Z}}_ 2\)-equivariant surgery
zbMATH Keywords: dimension of the fixed point set
zbMATH Keywords: surgery obstructions
zbMATH Keywords: signatures
zbMATH Keywords: Arf invariants
reviewed by: Amir H. Assadi
MaRDI profile type: Publication
full work available at URL: pjm.1987.130.275
OpenAlex ID: W1977927361
DOI: 10.2140/PJM.1987.130.275
description: scientific article; zbMATH DE number 4071941