# The problem with (co) bordism definition

According to one definition at Wiki:

An (n + 1)-dimensional cobordism is a quintuple (W; M, N, i, j) consisting of an (n + 1)-dimensional compact differentiable manifold with boundary, W; closed n-manifolds M, N; and embeddings i : M ⊂ ∂W, j : N ⊂ ∂W with disjoint images.

Note that n-manifolds are supposed to be closed, i.e. compact, without a boundary.

In 1 dimensional case you have a circle and it all makes sense.

However, I often see 0-dimensional manifold (a point) turning via bordism into a 1-dimensional manifold which is isomorphic to a closed interval in R. This 1-dimensional manifold has a border.

Then they construct 2-dimensional bordisms starting from it so what now happened to the definition that M and N are compact and have no border?

Edit based on comments: Is there an expanded cobordism definition where M and N are allowed to have a boundary or a definition that includes corners?

• It seems to me that what you are describing in the final paragraph are sometimes called cobordisms with corners. – Peter May 29 '17 at 19:27
• In this context, the author is most definitely talking about manifolds with corners, which require a bit of extra care than the ones without corners. You can not rely on the definition given on wikipedia for this case. – Peter May 29 '17 at 19:39
• The author is not claiming that Figure 17 is an example of a cobordism according to the original definition. Rather, it is an example of a more general notion that is what Section 5 is all about. – Eric Wofsey May 29 '17 at 20:20
• @Tony If memory serves, there is a categorical definition of cobordism in Robert Stong's book (assuming you know a little category theory). This would probably be the most general definition you could hope for. – Ben Sheller May 30 '17 at 5:21
• @Ben It's 1-categorical and not helpful for talking about the n-cobordism categories arising in the cobordism hypothesis. But I don't know a good reference for a careful definition of this cobordism category. – user98602 May 30 '17 at 7:53