CoufalV072004.pdf (367.84 kB)
A Family Version of Lefschetz-Nielsen Fixed Point Theory
thesis
posted on 2004-07-08, 00:00 authored by Vesta Mai CoufalIn classical Lefschetz-Nielsen theory, one defines the Lefschetz invariantL(f) of an endomorphism f of a manifold M. The definition depends on thefundamental group of M, and hence on choosing a base point * in M and abase path from * to f(*). Our goal is to develop a family version ofLefschetz-Nielsen theory, i.e., for a smooth fiber bundle p:E--> B and afiber bundle endomorphism f:E--> E. A family version of the classicalapproach involves choosing a section s:B--> E of p and a path ofsections from s to fs. Not only is this artificial, but such apath does not always exist. To avoid this difficulty, we replace the fundamental group with the fundamentalgroupoid. This gives us a base point free version of the Lefschetzinvariant. In the family setting, we define the Lefschetz invariant using abordism theoretic construction, and prove a Hopf-Lefschetz theorem. We thendescribe our ideas for extending the algebraic base point free invariant toget an algebraic version of the Lefschetz invariant in the family setting.
History
Date Created
2004-07-08Date Modified
2018-10-25Defense Date
2004-06-22Research Director(s)
Bruce WilliamsCommittee Members
Bruce WilliamsDegree
- Doctor of Philosophy
Degree Level
- Doctoral Dissertation
Language
- English
Alternate Identifier
etd-07082004-165859Publisher
University of Notre DameProgram Name
- Mathematics
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