Homology manifolds and euclidean bundles

TL;DR

This paper constructs a specific Poincaré complex demonstrating that vanishing surgery obstructions do not guarantee a reduction of the Spivak normal fibration to a Euclidean bundle, challenging existing assumptions.

Contribution

It provides a counterexample showing that vanishing total surgery obstruction does not imply the Spivak normal fibration reduces to a Euclidean bundle.

Findings

Constructed a Poincaré complex with vanishing surgery obstruction

Showed the Spivak normal fibration does not admit a Euclidean bundle reduction

Contradicts previous claims linking surgery obstructions and normal fibrations

Abstract

We construct a Poincar\'e complex whose periodic total surgery obstruction vanishes but whose Spivak normal fibration does not admit a reduction to a stable euclidean bundle. This contradicts the conjunction of two claims in the literature: Namely, on the one hand that a Poincar\'e complex with vanishing periodic total surgery obstruction is homotopy equivalent to a homology manifold, which appears in work of Bryant--Ferry--Mio--Weinberger, and on the other that the Spivak normal fibration of a homology manifold always admits a reduction to a stable euclidean bundle, which appears in work of Ferry--Pedersen.