Stiefel-Whitney classes and toral actions
type: Item
instance of: scholarly article
title: Stiefel-Whitney classes and toral actions
zbMATH Open document ID: 0578.57009
DOI: 10.1016/0166-8641(85)90054-9
published in: Topology and its Applications
publication date: 1985-01-01T00:00:00Z
full work available at URL: http://hdl.handle.net/2027.42/25539
review text: The Stiefel-Whitney classes of a manifold were originally defined as primary obstructions to finding frame fields on the manifold. A generic collection of vector fields gives rise to a possibly degenerate frame field and the points of degeneracy are Poincaré dual to some Stiefel- Whitney class. A smooth action of a Lie group on a manifold gives rise to a collection of vector fields, hence a frame field whose points of degeneracy are just the fixed point sets of the actions. One expects, therefore, a strong connection between first point sets of smooth actions and Stiefel-Whitney classes. The purpose of this paper is to show that this is so for toral actions. For \(H\subset T^ n\) we denote by F(M,H) the fixed point set of H on M and by D(F,(M,H)) the Poincaré dual of F(M,H). Then we have: Theorem A. If \(T^ n\) acts effectively on \(M^{n+k}\) then \(\omega_ i(M)=0\) for \(i>2k\) and \(\omega_{2k}(M)=\sum_{H\approx T^ k}D(F(M,H)).\) Note that the above sum is taken over all subgroups of \(T^ n\) isomorphic to \(T^ k\). This sum may not be finite, but is locally finite, and hence is a well-defined cohomology class. The action of \(T^ n\) gives rise to n vector fields on M. One can find a subcollection of \(n-k+1\) fields which is a generic collection, and then the proof of theorem A follows quickly. A similar result for Pontryagin classes of \(T^ n\) vector bundles and fixed point sets of isotropy subgroups was obtained by Gomez using very different methods. Theorem B. If \(T^ n\) acts smoothly and effectively on \(M^{n+k}\) then \[ \omega_{2k-1}(M)=\sum_{H\approx T^{k-1}\times {\mathbb{Z}}_ 2}D(E(M,H)). \] Here E(M,H) denotes the components of F(M,H) of codimension 2k-1. The proof of theorem B is considerably more difficult, since we cannot obtain a generic collection of \(n-k+2\) vector fields from the action. - As an application of theorem B we compute the toral degrees of symmetry of the Dold manifolds \(P(4r+1,2)\).
Mathematics Subject Classification ID: 57S15
Mathematics Subject Classification ID: 57R20
Mathematics Subject Classification ID: 57R25
zbMATH DE Number: 3925811
zbMATH Keywords: finding frame fields on a manifold
zbMATH Keywords: first point sets of smooth actions
zbMATH Keywords: Stiefel-Whitney classes
zbMATH Keywords: toral actions
zbMATH Keywords: Poincaré dual
zbMATH Keywords: Pontryagin classes
zbMATH Keywords: toral degrees of symmetry of the Dold manifolds
author: Janey Daccach
author: Arthur G. Wasserman
MaRDI profile type: Publication
OpenAlex ID: W1994561651
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description: scientific article; zbMATH DE number 3925811