Reducibility of low dimensional Poincaré duality spaces

Abstract

We discuss Poincaré 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.

arXiv commentv2: 26 pages, largely rewritten. v1 has a mistake (Lemma 3.2 is wrong, this is explained in Proposition 3.9 and the warning preceding 3.9). v2 now contains a new proof of the main theorem. The approach of the previous version is still explained in section 3. In addition there is an appendix about PD complexes from an infinity-categorical point of view