## Algebraic & Geometric Topology

### Poincaré duality complexes in dimension four

#### 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.

#### Article information

Source
Algebr. Geom. Topol., Volume 8, Number 4 (2008), 2355-2389.

Dates
Revised: 20 October 2008
Accepted: 26 October 2008
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.agt/1513796937

Digital Object Identifier
doi:10.2140/agt.2008.8.2355

Mathematical Reviews number (MathSciNet)
MR2465744

Zentralblatt MATH identifier
1164.57008

Subjects
Primary: 57P10: Poincaré duality spaces
Secondary: 55S35: Obstruction theory 55S45: Postnikov systems, $k$-invariants

#### Citation

Baues, Hans Joachim; Bleile, Beatrice. Poincaré duality complexes in dimension four. Algebr. Geom. Topol. 8 (2008), no. 4, 2355--2389. doi:10.2140/agt.2008.8.2355. https://projecteuclid.org/euclid.agt/1513796937

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