Let $H_g$ be a three dimensional handlebody bounded by a genus $g$ surface.
Let $M_g$ be a manifold obtained by gluing two copies of $H_g$ via an orientation reversing homeomorphism of the surface of $H_g$.
I would like to know what is a prime decomposition of the manifold $M_g$.
When $g=1$, we have $M_1$ is homeomorphic to $S^2 \times S^1$ and this is a prime decomposition.
What's the decomposition of $M_2$? Is it a connected sum of two $S^2 \times S^1$?
I appreciate any help. Thank you in advance.