Poincaré duality complexes in dimension four

Baues, Hans Joachim¹ ; Bleile, Beatrice²

¹Max–Planck–Institut für Mathematik, Vivatsgasse 7, PO Box 7280, D–53111 Bonn, Germany
²School of Science and Technology, University of New England, NSW 2351, Australia

Algebraic and Geometric Topology, Tome 8 (2008) no. 4, pp. 2355-2389

Résumé

Generalising Hendriks’ fundamental triples of PD3–complexes, we introduce fundamental triples for PDn–complexes and show that two PDn–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 PD4–complexes. Up to 2–torsion, homotopy types of PD4–complexes are classified by homotopy types of chain complexes with a homotopy commutative diagonal.

Détail

DOI : 10.2140/agt.2008.8.2355

Keywords: homotopy types of manifolds, PD complex, degree 1 map, chain complex, 4-dimensional manifold

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