The intersecting kernels of Heegaard splittings

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 $\mathbb{Z}\left({\pi }_1\left(M\right)\right)$–modules of the form

$$\left(K\cap L\right)∕\left[K,L\right]\hookrightarrow R\left\{x_1,\dots ,x_g\right\}∕J\stackrel{T^{\varphi }}{\to }R\left\{y_1,\dots ,y_g\right\}\stackrel{\theta }{\to }R\stackrel{ϵ}{\twoheadrightarrow }\mathbb{Z},$$

where $R=\mathbb{Z}\left({\pi }_1\left(M\right)\right)$, $J$ 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 $\mathcal{K}$ be the intersection kernel for a Heegaard splitting of a connected sum, and ${\mathcal{K}}_1$, ${\mathcal{K}}_2$ the intersection kernels of the two summands. We show that there is a surjection $\mathcal{K}\to {\mathcal{K}}_1\ast {\mathcal{K}}_2$ onto the free product with kernel being normally generated by a single geometrically described element.