Topology Seminar (Main Talk): Towers of regular self-covers and linear endomorphisms of tori

Seminar | February 15 | 4:10-5 p.m. | 3 Evans Hall

 Wouter van Limbeek, University of Michigan

 Department of Mathematics

Let M be a closed manifold that admits a nontrivial cover diffeomorphic to itself. What can we then say about M? Examples are provided by tori, in which case the covering is a linear endomorphism. Under the assumption that all iterates of the covering of M are regular, we show that any self-cover is closely related to a linear endomorphism on a torus, namely the covering is induced by a such a linear endomorphism on a quotient of the fundamental group. Under further hypotheses we show that a finite cover of M is a principal torus bundle. We use this to give an application to holomorphic self-covers of Kaehler manifolds.