Orientation reversing involutions on closed 3-manifolds
type: Item
instance of: scholarly article
title: Orientation reversing involutions on closed 3-manifolds
zbMATH Open document ID: 0709.57006
author: Masako Kobayashi
published in: Osaka Journal of Mathematics
publication date: 1989-01-01T00:00:00Z
review text: Let M be an arbitrary connected, closed, orientable 3-manifold admitting an orientation reversing PL involution \(\tau\). Various algebraic and geometric consequences (concerning M and the fixed point set Fix(\(\tau\))) have been derived from these hypotheses. In the present paper the author rounds off results of this kind. She observes that Fix(\(\tau\)) is either a closed 2-manifold separating M into two pieces which are interchanged by \(\tau\) or a nonseparating disjoint union of closed surfaces and isolated points. For either case she lists a complete set of properties of the group \(G=H_ 1(M)\) and the Betti numbers of \(F=Fix(\tau)\) which follow from the stated hypotheses. Completeness is to be understood in the following sense: if G is any abelian group and F is any orientable closed 2-manifold (or a disjoint union of a closed 2-manifold and a finite discrete set) such that G and the Betti numbers of F possess the listed properties, then G and F can be realized as \(H_ 1(M)\) and Fix(\(\tau\)) for some connected, closed, orientable 3-manifold M and an orientation reversing involution \(\tau\) of M.
Mathematics Subject Classification ID: 57M60
Mathematics Subject Classification ID: 57N10
Mathematics Subject Classification ID: 57S17
zbMATH DE Number: 4166070
zbMATH Keywords: 3-manifold
zbMATH Keywords: orientation reversing PL involution
zbMATH Keywords: fixed point set
reviewed by: Joze Vrabec
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4166070