TITLE: Examples of manifolds that are not boundaries
QUESTION [18 upvotes]: What are some examples of manifolds that do not have boundaries and are not boundaries of higher dimensional manifolds?
Is any $n$-dimensional closed manifold a boundary of some $(n+1)$-dimensional manifold?

REPLY [8 votes]: The circle $x^2+y^2=1$ in the punctured $(x,y)$-plane $\mathbb R^2\setminus \{(0,0)\}$ is a cycle but not a boundary .