An $n$-dimensional handlebody with $g$ handles is the result of attaching $g$ $n$ dimensional 1-handles to a single $n$-dimensional $0$-handle in such a way that the result is orientable. Is it in general true that if $H$ is an $n$-dimensional handlebody of genus $g$, then $H \times I$ is an $(n+1)$-dimensional handlebody of genus $g$?
I am not really sure how to identify manifold as being a handlebody in the first place. In dimension $3$, I know that when identifying a handlebody, you try and cut along disks and get a bunch balls and the number of disks and balls that you have allow you to determine $g$. Is a similar thing true in higher dimensions?
A "handlebody" in higher dimensions (be careful because in higher dimensions people usually use this word to mean any manifold with a handle decomposition) is a boundary connect sum of $S^1 \times B^{n-1}$. If you have an $n$-dimensional handlebody, then there is a set of codim 1 balls which you can cut along so every piece is just $S^1 \times B^{n-1}$. When you cross these balls with $I$ you get $n$-dimensional balls cutting your manifold into pieces which are just $S^1 \times B^{n-1}\times I = S^1 \times B^n$.