Poincare duality complexes in dimension four

Abstract

We describe an algebraic structure on chain complexes yielding algebraic models which classify homotopy types of Poincare duality complexes of dimension 4. Generalizing Turaev's fundamental triples of Poincare duality complexes of dimension 3, we introduce fundamental triples for Poincare duality complexes of dimension n > 2 and show that two Poincare duality 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.

27 pages, made changes concerning examples in the literature after recieving helpful comments

References in corpus (3)

• Strongly minimal PD4-complexes
• Poincare duality in dimension 3
• Topological 4-manifolds with geometrically 2-dimensional fundamental groups

Cited by in corpus (8)

• Strongly minimal PD4-complexes
• Some questions on subgroups of 3-dimensional Poincaré duality groups
• Homotopy classification of $P{D}_4$-complexes relative an order relation
• Poincaré duality complexes with highly connected universal cover
• Poincare Duality Complexes with Highly Connected Universal Covers
• s-Cobordism classification of $4$-manifolds through the group of homotopy self-equivalences
• Homotopy classification of 4-manifolds with finite abelian 2-generator fundamental groups
• $P{D}_4$-complexes and 2-dimensional duality groups