Poincaré duality complexes in dimension four

Abstract

Generalising Hendriks’ fundamental triples of ${PD}^3$–complexes, we introduce fundamental triples for ${PD}^n$–complexes and show that two ${PD}^n$–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. Another main result describes chain complexes with additional algebraic structure which classify homotopy types of ${PD}^4$–complexes. Up to $2$–torsion, homotopy types of ${PD}^4$–complexes are classified by homotopy types of chain complexes with a homotopy commutative diagonal.