If $M$ is a manifold, then $\partial(\partial M)) = \emptyset.$ I've searched this question here and I did not find any solution. I know that this problem is equivalent to show that $\partial(\partial \mathbb{H}^n)) = \emptyset,$ but I still have no idea how solve.
Thanks
Suppose $M$ is $n$-dimensional and $x$ is an element of the boundary. Then there is an open set $U$ about $x$ and a homeomorphism $\phi \colon U \to [0,2) \times (0,2)^{n-1}$ such that $\phi(x) = (0,1,1, \ldots, 1)$. Observe that $\phi^{-1}(\{0\} \times (0,2)^{n-1} )$ is an open subset of the boundary homeomorphic to $(0,2)^{n-1}$. But since $(0,2)^{n-1}$ is open in $\mathbb R ^{n-1}$ this shows the boundary is an $(n-1)$-dimensional manifold.