The Torsion of Automorphisms of Nilpotent Spaces

Abstract

We reprise a $K_1$-valued refinement of Whitehead torsion originally studied by Gersten. We use this Gersten torsion to show that for nilpotent spaces with infinite fundamental group, any self-equivalence which acts as the identity on the fundamental group has vanishing Whitehead torsion. We find two applications of our vanishing result. First, we provide many examples of spaces with infinitely many simple structures. Second, we conclude that the group of homotopy classes of simple self-equivalences of a connected nilpotent space that act as the identity on the fundamental group is commensurable to an arithmetic group, building on a theorem of Sullivan. We also give a corrected version of Sullivan's proof as an appendix.