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Euler characteristics of generalized Haken manifolds

Michael W Davis and Allan L Edmonds

Algebraic & Geometric Topology 14 (2014) 3701–3716

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.

Charney–Davis conjecture, Euler characteristic, Haken manifold, hierarchy, orbifold, flag triangulation, generalized homology sphere, boundary pattern, aspherical manifold
Mathematical Subject Classification 2010
Primary: 57N65
Secondary: 05E45, 57N80
Received: 28 February 2014
Revised: 2 June 2014
Accepted: 9 June 2014
Published: 15 January 2015
Michael W Davis
Department of Mathematics
The Ohio State University
231 W 18th Ave
Columbus, OH 43210
Allan L Edmonds
Department of Mathematics
Indiana University
831 E 3rd St
Bloomington, IN 47401