Stiefel-Whitney classes and toral actions
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 \(P(4r+1,2)\).
toral degrees of symmetry of the Dold manifolds
scientific article; zbMATH DE number 3925811