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Type of Document Dissertation Author Coufal, Vesta Mai Author's Email Address vcoufal@nd.edu URN etd-07082004-165859 Title A Family Version of Lefschetz-Nielsen Fixed Point Theory Degree Doctor of Philosophy Department Mathematics Advisory Committee
Advisor Name Title Bruce Williams Committee Member Keywords
- topology
Date of Defense 2004-06-22 Availability unrestricted Abstract In classical Lefschetz-Nielsen theory, one defines the Lefschetz invariantL(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.
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