Poincaré duality complexes in dimension four

Generalising Hendriks’ fundamental triples of ${P D}^{3}$ –complexes, we introduce fundamental triples for ${P D}^{n}$ –complexes and show that two ${P D}^{n}$ –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 ${P D}^{4}$ –complexes. Up to $2$ –torsion, homotopy types of ${P D}^{4}$ –complexes are classified by homotopy types of chain complexes with a homotopy commutative diagonal.

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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

Volume 8, issue 4 (2008)

Hans Joachim Baues and Beatrice Bleile

Abstract