Generalised doubles and simple homotopy types of high dimensional manifolds

TL;DR

This paper characterizes fundamental groups of high-dimensional manifolds that are homotopy equivalent but not simple homotopy equivalent, expanding the understanding of manifold classifications in algebraic topology.

Contribution

It provides the first examples of even-dimensional manifolds with the same homotopy type but different simple homotopy types for a broad class of groups.

Findings

Existence of manifolds with specified fundamental groups that are h-cobordant but not simple homotopy equivalent.

A formula for Whitehead torsion of homotopy equivalences between doubles of thickenings.

Construction of examples for any finitely presented group with nontrivial involution on Whitehead group.

Abstract

We characterise the set of fundamental groups for which there exist $n$-manifolds that are $h$-cobordant (hence homotopy equivalent) but not simple homotopy equivalent, when $n$ is sufficiently large. In particular, for $n \ge 12$ even, we show that examples exist for any finitely presented group $G$ such that the involution on the Whitehead group $Wh(G)$ is nontrivial. This expands on previous work, where we constructed the first examples of even-dimensional manifolds that are homotopy equivalent but not simple homotopy equivalent. Our construction is based on doubles of thickenings, and a key ingredient of the proof is a formula for the Whitehead torsion of a homotopy equivalence between such manifolds.