# Reducible Heegaard splittings can be written as connected sums

Recall that a Heegaard splitting is a decomposition of a three manifold into a triple $(V,W,S)$ where V and W are solid handlebodies meeting along their common boundary S. A Heegaard splitting is called reducible if there is an essential closed curve $c \subset S$ so that there are disks $D_1 \subset V$ and $D_2 \subset W$ so that $\partial D_1 = \partial D_2 = c$. A theorem of Haken states that any Heegard Splitting of a reducible three manifold M is reducible.

I am wondering why this implies that the given Heegaard splitting can be written as a connected sum of two Heegaard splittings. Most authors state this consequence without proof or reasoning so it shouldn't be too hard and yet I can't quite see it. It seems clear enough to me that the sphere $D_1 \cup D_2$ should be the reducing sphere, but where do we cut the surface/handlebodies into summands.

• "but where do we cut the surface/handlebodies into summands?" Along $c$. Jan 10 '16 at 8:33

Then $P:=D_1 \cup D_2$ is a $2$-sphere intersecting $S$ in an essential simple closed curve (the existence of such $2$-sphere is in fact equivalent to reducibility of a splitting). Note that the only orientable connected $3$-manifolds with non-separating $2$-sphere have $S^2 \times S^1$ as a summand. So it is no harm to assume that the $2$-sphere $P$ is separating. In this case the simple closed curve $\partial D_1 = \partial D_2$ is separating. Note that $S$ needs to be of genus at least two.
By removing a small open regular neighborhood $N(P)$ of $P$ in $M$, each handlebody $V_i$ get cut into two handlebodies of lower genus. The two components of $M \setminus N(P)$ can be written as unions of these handlebodies with lower genus. Both components have a $2$-sphere as a boundary, which corresponds to non-identified disks in the boundary of the handlebodies. Each of these spherical boundaries can be capped by a $3$-ball, which gives two closed $3$-manifolds $M_1$ and $M_2$ with $M = M_1 \# M_2$. The hemispheres of these $3$-balls cap the non-identified disks in the boundary of the handlebodies. This has the same effect as gluing the non-identified disks together. Thus, we obtain Heegaard splittings of $M_1$ and $M_2$, which have lower genera than the splitting of $M$. In particular, if one of the summands is $S^3$, then we obtain a lower genus splitting of $M$ from the original one.
Note that we can also do the converse, i.e. we can construct a Heegaard splitting of $M_1 \# M_2$ from Heegaard splittings of $M_1$ and $M_2$. As a consequence, it suffices to study Heegaard splittings of irreduiclbe $3$-manifolds.