The Euler Characteristic of a Haken 4-Manifold

2013/06/11 by Allan L. Edmonds, Edmonds, Allan L.
Mathematics · #05E45 #57M #Combinatorics (math.CO) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Geometry and complex manifolds #math.CO #math.GT #msc:05E45 #msc:57M

paper · pdf · doi:10.48550/arxiv.1306.2616

Substantially reorganized exposition and some minor corrections

openalex publication_date 2013/06/11 · arxiv created 2014/01/08 · arxiv updated 2014/01/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Haken n-manifolds are aspherical manifolds, defined and studied by B. Foozwell and H. Rubinstein, that can be successively cut open along essential codimension-one submanifolds until a disjoint union of n-cells is obtained. Such manifolds come equipped with a boundary pattern, a particular kind of decomposition of the boundary into codimension-zero submanifolds. We prove that there is a certain numerical function phi(X⁴) depending only on the boundary and boundary pattern of the compact Haken 4-manifold X⁴ (and vanishing if X⁴ has empty boundary), such that for any compact Haken 4-manifold X⁴ the Euler characteristic satisfies the inequality chi(X⁴) >= phi(X⁴). In particular, if X⁴ is a closed Haken 4-manifold, then chi(X⁴) >= 0.

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