Poincare duality complexes in dimension four

TL;DR

This paper develops an algebraic framework to classify 4-dimensional Poincare duality complexes, extending previous models to higher dimensions and providing tools for understanding their homotopy types and mappings.

Contribution

It introduces fundamental triples for Poincare duality complexes in dimensions greater than two, generalizing Turaev's work, and establishes criteria for homotopy equivalence and existence of degree 1 maps.

Findings

Classifies 4-dimensional Poincare duality complexes using algebraic models.

Proves that fundamental triples determine homotopy types up to isomorphism.

Provides a criterion for the existence of degree 1 maps between manifolds.

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.