Reducibility of low dimensional Poincar\'e duality spaces

TL;DR

This paper investigates when low-dimensional Poincaré duality spaces have a reducible Spivak normal fibration, establishing existence results in dimensions less than 4 and orientable 4-manifolds, with counterexamples in non-orientable cases.

Contribution

It proves reducibility of the Spivak normal fibration for low-dimensional Poincaré duality spaces, clarifying the role of orientability in dimension 4.

Findings

Reduction always exists in dimensions less than 4.

In dimension 4, reduction exists if the space is orientable.

Counterexamples exist in non-orientable 4-dimensional spaces.

Abstract

We discuss Poincar\'e duality complexes X and the question whether or not their Spivak normal fibration admits a reduction to a vector bundle in the case where the dimension of X is at most 4. We show that in dimensions less than 4 such a reduction always exists, and in dimension 4 such a reduction exists provided X is orientable. In the non-orientable case there are counterexamples to reducibility by Hambleton--Milgram.