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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 is how to reconcile this definition with the classical definition of oriented cobordism:

Definition 2 Two $n$-dimensional manifolds $M,N$ are said to be cobordant if there is an $n+1$-dimensional manifold $W$ such that $\partial W\simeq \bar{M}+N$. Here $\bar{M}$ is defined as $M$ after an orientation reversal.

The problem only seems to arise when $M$ and $N$ are not cobordant to the empty manifold.

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These two definitions of oriented cobordism are equivalent (it's a kind of a variation on the theme $a-b=0\iff a+x=b+x$).

One direction is rather obvious: $\partial W\cong\bar M+N\implies M+\partial W=N+\partial(M\times[0;1])$.

The opposite direction is slightly more interesting. Observe that $A+\partial B\cong\partial C\implies \exists W:A\cong\partial W$ (just take $W=C\sqcup_{\partial\bar B}\bar B$). So $M+\partial U\cong N+\partial V\implies \partial(M\times[0;1]+ U)\cong(\bar M+N)+\partial V\implies\bar M+N\cong\partial W$.

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  • $\begingroup$ Such a nice and concise argument! Thank you very much! $\endgroup$ – Espen Nielsen Jun 4 '13 at 12:04

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