Generalising Hendriks’ fundamental triples of –complexes, we introduce fundamental triples for –complexes and show that two –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 maps between –dimensional manifolds. Another main result describes chain complexes with additional algebraic structure which classify homotopy types of –complexes. Up to –torsion, homotopy types of –complexes are classified by homotopy types of chain complexes with a homotopy commutative diagonal.
"Poincaré duality complexes in dimension four." Algebr. Geom. Topol. 8 (4) 2355 - 2389, 2008. https://doi.org/10.2140/agt.2008.8.2355