#### Volume 11, issue 2 (2011)

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The intersecting kernels of Heegaard splittings

### Fengchun Lei and Jie Wu

Algebraic & Geometric Topology 11 (2011) 887–908
##### Abstract

Let $V{\cup }_{S}W$ be a Heegaard splitting for a closed orientable $3$–manifold $M$. The inclusion-induced homomorphisms ${\pi }_{1}\left(S\right)\to {\pi }_{1}\left(V\right)$ and ${\pi }_{1}\left(S\right)\to {\pi }_{1}\left(W\right)$ are both surjective. The paper is principally concerned with the kernels $K=Ker\left({\pi }_{1}\left(S\right)\to {\pi }_{1}\left(V\right)\right)$, $L=Ker\left({\pi }_{1}\left(S\right)\to {\pi }_{1}\left(W\right)\right)$, their intersection $K\cap L$ and the quotient $\left(K\cap L\right)∕\left[K,L\right]$. The module $\left(K\cap L\right)∕\left[K,L\right]$ is of special interest because it is isomorphic to the second homotopy module ${\pi }_{2}\left(M\right)$. There are two main results.

(1)  We present an exact sequence of $ℤ\left({\pi }_{1}\left(M\right)\right)$–modules of the form

$\left(K\cap L\right)∕\left[K,L\right]↪R\left\{{x}_{1},\dots ,{x}_{g}\right\}∕J\stackrel{{T}^{\varphi }}{\to }R\left\{{y}_{1},\dots ,{y}_{g}\right\}\stackrel{\theta }{\to }R\stackrel{ϵ}{↠}ℤ,$

where $R=ℤ\left({\pi }_{1}\left(M\right)\right)$, $J$ is a cyclic $R$–submodule of $R\left\{{x}_{1},\dots ,{x}_{g}\right\}$, ${T}^{\varphi }$ and $\theta$ are explicitly described morphisms of $R$–modules and ${T}^{\varphi }$ involves Fox derivatives related to the gluing data of the Heegaard splitting $M=V{\cup }_{S}W$.

(2)  Let $\mathsc{K}$ be the intersection kernel for a Heegaard splitting of a connected sum, and ${\mathsc{K}}_{1}$, ${\mathsc{K}}_{2}$ the intersection kernels of the two summands. We show that there is a surjection $\mathsc{K}\to {\mathsc{K}}_{1}\ast {\mathsc{K}}_{2}$ onto the free product with kernel being normally generated by a single geometrically described element.

##### Keywords
Heegaard splitting, intersecting kernel, $3$–manifold, mapping class, Riemann surface
##### Mathematical Subject Classification 2000
Primary: 57M27, 57M99, 20F38
Secondary: 57M05, 37E30
##### Publication
Received: 25 July 2010
Revised: 29 December 2010
Accepted: 12 January 2011
Published: 25 March 2011
##### Authors
 Fengchun Lei School of Mathematical Sciences Dalian University of Technology Dalian 116024 China Jie Wu Department of Mathematics National University of Singapore S17-06-02, 10 Lower Kent Ridge Road Singapore 119076 Singapore http://www.math.nus.edu.sg/~matwujie