Poincaré duality complexes in dimension four

Abstract

We describe an algebraic structure on chain complexes yielding algebraic models which classify homotopy types of PD⁴-complexes. Generalizing Turaev's fundamental triples of PD³-complexes we introduce fundamental triples of PDⁿ-complexes and show that two PDⁿ complexes are orientedly homotopy equivalent if and only if their fundamental triples are isomorphic. As applications we establish a conjecture of Turaev and obtain a criterion for the existence of degree 1 maps between n-dimensional manifolds.