${\rm PD}\sb 4$-complexes with free fundamental group

Abstract

We show that PD4-complexes with free fundamental group are determined by their intersection pairings and that every hermitian form on a finitely generated free module over the group ring of a free group is realized by some such complex. The purpose of this article is to show that some of the basic properties of PD4-complexes with free fundamental group can be derived homologically, without reference to the topology of 4-manifolds or stabilization by connected sums, as used in (4, 8, 13). We also avoid explicit calculations of obstructions, relying instead on the easily verified fact that the 3-skeletons of the complexes considered have su‰ciently many self homotopy equivalences. In particular we give a new proof of the fact that such complexes are determined by their intersection pairings, and that every hermitian form on a finitely generated free module over the group ring of a free group is realized by some such complex. In the final section we consider briefly the classification (up to s-cobordism) of closed 4-manifolds with free fundamental group.