## Algebraic & Geometric Topology

### Euler characteristics of generalized Haken manifolds

#### Abstract

Haken $n$–manifolds have been defined and studied by B Foozwell and H Rubinstein in analogy with the classical Haken manifolds of dimension $3$, based upon the theory of boundary patterns developed by K Johannson. The Euler characteristic of a Haken manifold is analyzed and shown to be equal to the sum of the Charney–Davis invariants of the duals of the boundary complexes of the $n$–cells at the end of a hierarchy. These dual complexes are shown to be flag complexes. It follows that the Charney–Davis conjecture is equivalent to the Euler characteristic sign conjecture for Haken manifolds. Since the Charney–Davis invariant of a flag simplicial $3$–sphere is known to be nonnegative it follows that a closed Haken $4$–manifold has nonnegative Euler characteristic. These results hold as well for generalized Haken manifolds whose hierarchies can end with compact contractible manifolds rather than cells.

#### Article information

Source
Algebr. Geom. Topol., Volume 14, Number 6 (2014), 3701-3716.

Dates
Received: 28 February 2014
Revised: 2 June 2014
Accepted: 9 June 2014
First available in Project Euclid: 19 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513716054

Digital Object Identifier
doi:10.2140/agt.2014.14.3701

Mathematical Reviews number (MathSciNet)
MR3302976

Zentralblatt MATH identifier
1314.57021

#### Citation

Davis, Michael W; Edmonds, Allan L. Euler characteristics of generalized Haken manifolds. Algebr. Geom. Topol. 14 (2014), no. 6, 3701--3716. doi:10.2140/agt.2014.14.3701. https://projecteuclid.org/euclid.agt/1513716054

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