Let $Y= S^1 \times T$ where $T$ is the torus. I need to define a manifold with boundary $X$ such that $\partial X = Y.$
I know that in general if $A$ is a manifold with boundary and $C$ is a manifold then we have $\partial (A \times C)=\partial A \times C.$ So it suffices to take $A=D^2$ the closed disk which has boundary $S^1$ and $C= T$ the torus. Does it work?
I prefer criticism on my solution first, and only secondly an alternative/ better solution.