Volume 8, issue 4 (2008)

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ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
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 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.

Keywords
homotopy types of manifolds, PD complex, degree 1 map, chain complex, 4-dimensional manifold
Mathematical Subject Classification 2000
Primary: 57P10
Secondary: 55S35, 55S45
References
Publication
Received: 26 February 2008
Revised: 20 October 2008
Accepted: 26 October 2008
Published: 20 December 2008
Authors
Hans Joachim Baues
Max–Planck–Institut für Mathematik
Vivatsgasse 7
PO Box 7280
D–53111 Bonn
Germany
Beatrice Bleile
School of Science and Technology
University of New England
NSW 2351
Australia