If $\phi$ is an automorphism of $\sum_g$, the closed compact surface of genus $g$, is there a "normal" way to heegaard split $\sum_g\times I/((x,0)\sim(\phi(x),1))$?
I was able to find a Heegaard splitting for $T^3$, and even more generally when $\phi=\text{id}$. I would appreciate a hint more than a full solution.
Thanks!
For $M(\phi) = S \times [0, 1]/ \sim $ fix two points $p, q \in S$ such that all $p, q, \phi(p), \phi(q)$ are different. Take $S \times 0 \sqcup S \times 1/2$ and glue them together by narrow horizontal tubes going around $p \times [o, 1/2]$ and $q \times [1/2, 1]$. So you obtain natural genus $2g + 1$ splitting. (I guess it's more of a hint because you need to prove that this surface is separarting, but it's pretty easy).
There's also pretty interesting paper: M. Scharlemann, A. Thompson. Heegaard splittings of (surface) $\times$ I are standard.