Algebraic & Geometric Topology

Euler characteristics of generalized Haken manifolds

Michael W Davis and Allan L Edmonds

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57N65: Algebraic topology of manifolds
Secondary: 05E45: Combinatorial aspects of simplicial complexes 57N80: Stratifications

Charney–Davis conjecture Euler characteristic Haken manifold hierarchy orbifold flag triangulation generalized homology sphere boundary pattern aspherical manifold


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.

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