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.


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 $M+\partial U\cong N+\partial V\implies \partial(M\times[0;1]+ U)\cong(\bar M+N)+\partial V\implies\bar M+N\cong\partial W$.

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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.