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.