For a Heegaard splitting of $S^2 \times S^1$, we can take two copies of genus 1 handlebodies and glue boundaries with the identity map.
I want to generalize this a little bit.
In the case of $(S^2\times S^1)\# (S^2\times S^1)$, can we take two copies of genus 2 handlebodies and identity map of boundaries for a Heegaard splitting?
I think one way to prove this is to show the connected sum of handlebodies commute with gluing maps (the identity of boundaries in this case). How can I see this?
Or can you give me different approach to this problem? I want to know many approaches to this problem.