Homology manifolds and euclidean bundles

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.