Volume 9, issue 4 (2009)

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

Guoqiu Yang and Fengchun Lei

Algebraic & Geometric Topology 9 (2009) 2041–2054
Abstract

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.

Keywords
essential surface, Heegaard genus
Mathematical Subject Classification 2000
Primary: 57M99, 57N10
Secondary: 57M27
References
Publication
Received: 28 May 2009
Revised: 3 September 2009
Accepted: 5 September 2009
Published: 14 October 2009
Authors
Guoqiu Yang
Department of Mathematics
Harbin Institute of Technology
Harbin 150001
China
Fengchun Lei
School of Mathematical Sciences
Dalian University of Technology
Dalian 116024
China