Volume 8, issue 4 (2008)

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Poincaré duality complexes in dimension four

Hans Joachim Baues and Beatrice Bleile

Algebraic & Geometric Topology 8 (2008) 2355–2389
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.

Keywords
homotopy types of manifolds, PD complex, degree 1 map, chain complex, 4-dimensional manifold
Mathematical Subject Classification 2000
Primary: 57P10
Secondary: 55S35, 55S45