English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

A fixed point theorem for actions of toral and finite p-groups is obtained for finite simplicial complexes satisfying certain conditions on their fundamental groups and their universal covering spaces. Examples are constructed relating this result to other well-known results. 1. Introduction. Fixed point theorems have long been regarded as a very important part of the theory of group actions. Most fixed point theorems for general spaces derive from the Smith theory which is homological in nature. In this note we shall show, for finite simplicial complexes, how a fundamental group which is a nontrivial free product forces the existence of fixed points for toral and p-group actions. The role of the fundamental group in the study of group actions on manifolds and nice topological spaces has become increasingly important. Of course, it is crucial for aspherical manifolds because their homotopy type is determined solely by the fundamental group. For example, if M is a closed aspherical manifold and G is a compact Lie group of homeomorphisms, then: 1. G is a A:-torus where k < r = rank of center trx(M), if G is connected (C-R-2). 2. In particular, if r — 0, G is finite (C-R-2). 3. If r — 0, the torsion of the outer automorphism group of <nx(M) is trivial, then G itself must be trivial (C-R-l). These results have been generalized and extended in many directions by various authors: R. Schultz, S. Schwartzman, M. C. and H. T. Ku, Asadi-Burghelea, W. Browder and W. C. Hsiang, R. Schoen and S. T. Yau, and K. B. Lee to name just several. The fundamental groups of closed aspherical manifolds are indecomposable while those of connected sums are free products. Connected sums usually destroy aspheri- city and the fundamental group no longer dictates the topology of the sum. However, connected sums and free products still play a significant role, for example, in showing that in each arbitrary bordism class one may find closed manifolds without circle actions (C-R-3, 3.7), in E. Bloomberg's construction of closed non- aspherical 4-manifolds which admit no finite group actions (B), and in R. Schultz' study of hypertoral actions (S).

A fixed point theorem for simplicial complexes whose fundamental groups are free products