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Cohomology Theory of Topological Transformation Groups

  • Wu Yi Hsiang
Book

Part of the Ergebnisse der Mathematik und ihrer Grenzgebiete book series (MATHE2, volume 85)

About this book

Introduction

Historically, applications of algebraic topology to the study of topological transformation groups were originated in the work of L. E. 1. Brouwer on periodic transformations and, a little later, in the beautiful fixed point theorem ofP. A. Smith for prime periodic maps on homology spheres. Upon comparing the fixed point theorem of Smith with its predecessors, the fixed point theorems of Brouwer and Lefschetz, one finds that it is possible, at least for the case of homology spheres, to upgrade the conclusion of mere existence (or non-existence) to the actual determination of the homology type of the fixed point set, if the map is assumed to be prime periodic. The pioneer result of P. A. Smith clearly suggests a fruitful general direction of studying topological transformation groups in the framework of algebraic topology. Naturally, the immediate problems following the Smith fixed point theorem are to generalize it both in the direction of replacing the homology spheres by spaces of more general topological types and in the direction of replacing the group tl by more general compact groups.

Keywords

Cohomology Kohomologie Topologische Transformationsgruppe Topology Transformation Groups algebra theorem

Authors and affiliations

  • Wu Yi Hsiang
    • 1
  1. 1.Department of MathematicsUniversity of CaliforniaBerkeleyUSA

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-642-66052-8
  • Copyright Information Springer-Verlag Berlin Heidelberg 1975
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Springer Book Archive
  • Print ISBN 978-3-642-66054-2
  • Online ISBN 978-3-642-66052-8
  • Series Print ISSN 0071-1136
  • Buy this book on publisher's site