Euler characteristics of generalized Haken manifolds

Davis, Michael W¹ ; Edmonds, Allan L²

¹Department of Mathematics, The Ohio State University, 231 W 18th Ave, Columbus, OH 43210, USA
²Department of Mathematics, Indiana University, 831 E 3rd St, Bloomington, IN 47401, USA

Algebraic and Geometric Topology, Tome 14 (2014) no. 6, pp. 3701-3716

Résumé

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.

Détail

DOI : 10.2140/agt.2014.14.3701

Classification : 57N65, 05E45, 57N80
Keywords: Charney–Davis conjecture, Euler characteristic, Haken manifold, hierarchy, orbifold, flag triangulation, generalized homology sphere, boundary pattern, aspherical manifold

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