I am reading some basic context books about topology (i.e. The Poincaré Conjecture, by Donal O'Shea between others) and following this open Topology and Geometry video lectures of the brilliant professor Tadashi Tokieda in the African Institute for Mathematical Sciences (for beginners in the matter, if you have time I would suggest you to have a look to them!).
I would like to ask the following question:
Is every $n$-manifold the boundary of an $(n+1)$-manifold? Is every compact $n$-manifold the boundary of a compact $(n+1)$-manifold?
Thank you!
p.s. This question was rewritten according to the suggestions in the comments and Meta (here), I hope now will be more accurate. Thanks to everybody for the suggestions!
I am not entirely sure what your question is, but here is my interpretation of it: is every $n$-manifold $X$ (without boundary) the boundary of some $(n+1)$-manifold with boundary $Y$? The answer is yes: just take $Y=X\times [0,\infty)$, identifying $X$ with $\partial Y=X\times\{0\}$.
(Mike Miller gave an answer to the contrary in the comments; however, his answer applies only if you demand that the manifolds be compact. That is, it is not true in general that every compact $n$-manifold is the boundary of some compact $(n+1)$-manifold with boundary. In particular, my answer does not work in that case, since $X\times [0,\infty)$ is never compact if $X$ is nonempty.)