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

Yang, Guoqiu; Lei, Fengchun

doi:10.2140/agt.2009.9.2041

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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

DOI: 10.2140/agt.2009.9.2041

Abstract

Let $M$ be a compact, orientable, $\partial$ –irreducible $3$ –manifold and $F$ be a connected closed essential surface in $M$ with $g (F) \geq 1$ which cuts $M$ into $M_{1}$ and $M_{2}$ . In the present paper, we show the following theorem: Suppose that there is no essential surface with boundary $(Q_{i}, \partial Q_{i})$ in $(M_{i}, F)$ satisfying $χ (Q_{i}) \geq 2 + g (F) - 2 g (M_{i}) + 1$ , $i = 1, 2$ . Then $g (M) = g (M_{1}) + g (M_{2}) - g (F)$ . As a consequence, we further show that if $M_{i}$ has a Heegaard splitting $V_{i} \cup_{S_{i}} W_{i}$ with distance $D (S_{i}) \geq 2 g (M_{i}) - g (F)$ , $i = 1, 2$ , then $g (M) = g (M_{1}) + g (M_{2}) - g (F)$ .

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

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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

Volume 9, issue 4 (2009)