According to the nLab entry for abstract cobordism categories, the natural way of axiomatizing the relation of two oriented manifolds being cobordant is the following: Definition 1 Two objects $M,N$ in a cobordism category $(D,\partial,i)$ are said to be cobordant if there are objects $U,V\in D$ such that $M+\partial U \simeq N+\partial V$. My question […]

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?

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