Heegaard splitting - $\pi_1(T) \to M$ is surjective

Question

Let $M$ be a closed, orientable 3-manifold and let $(V_1, V_2)$ be a Heegaard splitting of $M$ with $T = \partial V_1 = \partial V_2$. I know that by Seifert-van Kampen we have a pushout diagram consisting of $\pi_1T, \pi_1V_1, \pi_1V_2, \pi_1M$ and I know that the maps induced by inclusions $\pi_1T \to \pi_1V_1$ and $\pi_1T \to \pi_1V_2$ are surjective. Why is the map induced by inclusion $\pi_1T \to \pi_1M$ surjective?

Answer 1

Here's a proof using CW-complexes.

First, there is a CW-decomposition of $M$ whose 1-skeleton $M^{(1)}$ is a subset of $T$. To see why, choose collections of discs $\{D_{1,1},...,D_{1,g}\}$ in $V_1$ whose complement in $V_1$ is a 3-ball, and $\{D_{2,1},...,D_{2,g}\}$ in $V_2$ whose complement in $V_2$ is a 3-ball; this can always be done where $g$ is the common genus of the handlebodies $V_1,V_2$. Make sure to choose these discs so that the collection of their boundaries is in general position in the surface $T$, hence the union of their boundaries is a graph in $T$. Now extend that graph, by adding some additional edges in $T$, to obtain a CW decomposition of the surface $T$. We then obtain the desired CW-decomposition of $M$ whose 2-skeleton is the union of $T$ with all the discs $D_{i,j}$.

Second, there is a general theorem in algebraic topology that for any CW-decomposition of any connected topological space $X$, the inclusion of the 1-skeleton $X^{(1)} \hookrightarrow X$ induces a surjection on fundamental groups.

Finally, the composition of inclusions $$M^{(1)} \hookrightarrow T \hookrightarrow M $$ induces homomorphisms $$\pi_1(M^{(1)}) \mapsto \pi_1(T) \mapsto \pi_1(M) $$ whose composition is surjective, hence the map $\pi_1(T) \mapsto \pi_1(M)$ is surjective.