The Torsion of Automorphisms of Nilpotent Spaces
Sacha Goldman
TL;DR
This work refines Whitehead torsion for nilpotent spaces via Gersten’s $K_1$-valued torsion in $A$-theory, establishing a vanishing result for self-equivalences acting trivially on $\pi_1$ and exploiting it to draw structural consequences. The Vanishing Theorem yields a robust obstruction-theoretic control that, combined with $A$-theory, reveals infinitely many simple structures in suitable settings (Moduli Theorem). The Arithmeticity Theorem further identifies the group of simple based self-equivalences as commensurable to an arithmetic group, grounded in Sullivan’s rational homotopy theory with a corrected, basepoint-aware appendix. Together, these results illuminate the relationship between simple homotopy types, automorphism groups, and arithmetic structures in nilpotent spaces, with concrete examples and consequences for moduli and $h$-cobordism phenomena.
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.