On Poincaré four-complexes with free fundamental groups

Let X be a closed connected and oriented PL manifold whose fundamental group p1 is a free group of rank p. Let L be the integral group ring of p1. Then H2ðX ; LÞ is L-free (see (6)). We show that there is a bijective correspondence between homotopy equivalence classes of 4-dimensional Poincarecomplexes Y with Y ð3Þ G X ð3Þ and invertible hermitian matrices of type k over L, where k is the rank of H2ðX ; LÞ.