Equivariant controllable cutting-pasting and cobordism with vector fields
type: Item
instance of: scholarly article
title: Equivariant controllable cutting-pasting and cobordism with vector fields
zbMATH Open document ID: 0687.57016
author: Katsuhiro Komiya
published in: Publications of the Research Institute for Mathematical Sciences, Kyoto University
publication date: 1987-01-01T00:00:00Z
review text: Let G be a finite group of odd order, \({\mathcal M}_ n(G)\) the set of n- dimensional closed G-manifolds (smooth), \(N_ n(G)\) the cobordism group of n-dimensional closed G-manifolds, \(N^ R_ n(G)\) the Grothendieck group of the semigroup constructed by the Reinhart cobordism classes in \(N_ n(G)\), and \(SKK_ n(G)\) the Grothendieck group of the semigroup \({\mathcal M}_ n(G)/\sim\) with \(\sim\) the SKK-equivalence. In this paper, the author obtains two short exact sequences which contain the groups \(N_ n(G)\), \(N^ R_ n(G)\) and \(SKK_ n(G)\). And then he shows that for \(M,N\in {\mathcal M}_ n(G)\) with \(M^{H,1}=\emptyset\), \(N^{H,1}=\emptyset\) for any subgroup H of G, \([M]=[N]\) in \(N^ R_ n(G)\) iff \([M]=[N]\) in \(SKK_ n(G)\), where \(M^{H,1}\) denotes the 1- dimensional components of the H-fixed point set of the G-manifold M.
Mathematics Subject Classification ID: 57S17
Mathematics Subject Classification ID: 57R85
Mathematics Subject Classification ID: 57R65
Mathematics Subject Classification ID: 57R91
zbMATH DE Number: 4126332
zbMATH Keywords: finite group actions
zbMATH Keywords: equivariant cutting and pasting
zbMATH Keywords: cobordism group of n-dimensional closed G-manifolds
zbMATH Keywords: fixed point set
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2977/prims/1195176849
OpenAlex ID: W1966302598
cites work: Handlebody decompositions for G-manifolds
cites work: Äquivariantes kontrolliertes Schneiden und Kleben
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cites work: Q3924849
cites work: Cutting and pasting involutions
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cites work: Fixed point free equivariant homotopy classes
DOI: 10.2977/PRIMS/1195176849
description: scientific article; zbMATH DE number 4126332