Amalgamations of Heegaard splittings in 3–manifolds without some essential surfaces

Yang, Guoqiu¹ ; Lei, Fengchun²

¹Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China
²School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China

Algebraic and Geometric Topology, Tome 9 (2009) no. 4, pp. 2041-2054

Résumé

Let M be a compact, orientable, ∂–irreducible 3–manifold and F be a connected closed essential surface in M with g(F) ≥ 1 which cuts M into M1 and M2. In the present paper, we show the following theorem: Suppose that there is no essential surface with boundary (Qi,∂Qi) in (Mi,F) satisfying χ(Qi) ≥ 2 + g(F) − 2g(Mi) + 1, i = 1,2. Then g(M) = g(M1) + g(M2) − g(F). As a consequence, we further show that if Mi has a Heegaard splitting V i ∪SiWi with distance D(Si) ≥ 2g(Mi) − g(F), i = 1,2, then g(M) = g(M1) + g(M2) − g(F).

The main results follow from a new technique which is a stronger version of Schultens’ Lemma.

Détail

DOI : 10.2140/agt.2009.9.2041

Keywords: essential surface, Heegaard genus

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