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Group actions and higher signatures.

S Weinberger1

  • 1Department of Mathematics, Princeton University, Princeton, NJ 08544.

Proceedings of the National Academy of Sciences of the United States of America
|March 1, 1985
PubMed
Summary

This study investigates if a manifold (M) can have the same R-homology type as one with a free group (pi) action, focusing on homologically trivial actions. New results are presented, even for simply connected cases, with ties to the Novikov higher signature conjecture.

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Area of Science:

  • Topology
  • Algebraic Topology
  • Geometric Topology

Background:

  • Manifolds admitting free group actions are fundamental objects in topology.
  • The R-homology type classifies manifolds up to certain homological equivalences.
  • Understanding group actions on manifolds is a central theme in geometric topology.

Purpose of the Study:

  • To investigate whether a closed manifold (M) possesses the R-homology type of a manifold admitting a free action of a nontrivial finite group (pi).
  • To analyze this problem specifically for group actions that are 'homologically trivial'.
  • To explore connections to the Novikov higher signature conjecture, particularly when the fundamental group of M is nontrivial.

Main Methods:

  • The study focuses on the properties of manifolds under 'homologically trivial' group actions.
  • It leverages concepts from algebraic topology, including R-homology and group actions.
  • The research examines the relationship between the manifold's fundamental group and the existence of free group actions.

Main Results:

  • The paper presents new findings regarding the R-homology type of manifolds admitting specific types of finite group actions.
  • Results are established even for the case of simply connected manifolds.
  • The study highlights the intricate relationship between these manifold properties and the Novikov higher signature conjecture.

Conclusions:

  • The research contributes novel insights into the classification of manifolds admitting group actions.
  • The findings extend existing knowledge in algebraic and geometric topology.
  • The work demonstrates the relevance of studying homologically trivial actions for understanding manifold structures and related conjectures.

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