A Family Version of Lefschetz-Nielsen Fixed Point Theory

In 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.