Higher Reidemeister Torsion

John Klein, Wayne State University

Abstract. Reidemeister torsion is an algebraic invariant associated with a smooth manifold. It first gained attention in the early twentieth century in the works of Franz, Reidemeister and de Rham, and ultimately led to the classification of certain three dimensional manifolds called "lens spaces" (these are quotients of the three sphere by a free linear action of a finite cyclic group). In the 1970s, an analytical analog of the torsion was introduced by Ray and Singer. The analytical version was subsequently shown to coincide with the algebraic one by Cheeger and Mueller (independently). My talk will explain how, in the last fifteen years, these invariants have been generalized to families of manifolds. I will explain the known computations, various approaches and also some conjectures.