A Family Version of Lefschetz-Nielsen Fixed Point Theory

Doctoral Dissertation

Abstract

In classical Lefschetz-Nielsen theory, one defines the Lefschetz invariant L(f) of an endomorphism f of a manifold M. The definition depends on the fundamental group of M, and hence on choosing a base point * in M and a base path from * to f(*). Our goal is to develop a family version of Lefschetz-Nielsen theory, i.e., for a smooth fiber bundle p:E–> B and a fiber bundle endomorphism f:E–> E. A family version of the classical approach involves choosing a section s:B–> E of p and a path of sections from s to fs. Not only is this artificial, but such a path does not always exist.

To avoid this difficulty, we replace the fundamental group with the fundamental groupoid. This gives us a base point free version of the Lefschetz invariant. In the family setting, we define the Lefschetz invariant using a bordism theoretic construction, and prove a Hopf-Lefschetz theorem. We then describe our ideas for extending the algebraic base point free invariant to get an algebraic version of the Lefschetz invariant in the family setting.

Attributes

Attribute NameValues
URN
  • etd-07082004-165859

Author Vesta Mai Coufal
Advisor Bruce Williams
Contributor Bruce Williams, Committee Member
Degree Level Doctoral Dissertation
Degree Discipline Mathematics
Degree Name PhD
Defense Date
  • 2004-06-22

Submission Date 2004-07-08
Country
  • United States of America

Subject
  • topology

Publisher
  • University of Notre Dame

Language
  • English

Record Visibility Public
Content License
  • All rights reserved

Departments and Units

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