The Torsion of Automorphisms of Nilpotent Spaces

TL;DR

This paper explores a refined Whitehead torsion for nilpotent spaces, showing that certain self-equivalences have vanishing torsion, leading to new examples of spaces with multiple simple structures and linking their automorphism groups to arithmetic groups.

Contribution

It introduces a $K_1$-valued refinement of Whitehead torsion and proves its vanishing for specific self-equivalences, extending understanding of automorphism groups of nilpotent spaces.

Findings

Self-equivalences acting trivially on fundamental group have vanishing Whitehead torsion.

Many spaces have infinitely many simple structures.

Automorphism groups are commensurable to arithmetic groups.

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.