I was going to say that there’s not much more to say about 1-cobordisms but then I realized I’ll probably return to these periodically to add some details. I’ll add a few now.
First there’s the notion of cobordism classes, which 0-manifolds are cobordant? The 0-manifolds in 1Cob are disjoint unions of oriented points. Certainly any + or – point is cobordant to itself. Notice that we don’t have any cobordism from a + to a -. When we oriented the boundary of the interval that way we ended up with either the death of a pair of points or birth (i.e maps to or from the empty set). What if we have two + and one -? There is a cobordism from those three points to one positively oriented point,
So (-) is cobordant to (+–). Similarly (+) is cobordant to (++-). This quickly generalizes. If we have n positively oriented points and n+1 negatively oriented points this is cobordant to (-).
So the cobordism equivalence classes of (-) and (+) are full of 0-manifolds. Now consider (++) what’s this cobordant to? We can’t annihilate these so it’s at least cobordant to itself and to any other 0-manifold with two more positively oriented points than negatively oriented points (because we can create an arbitrary number of pairs). This continues, the 0-manifold with three positively oriented points is cobordant to any 0-manifold with three more positively oriented points than negatively oriented points. In fact, let n be a positive integer, then the 0-manifold consisting of n positively oriented points is cobordant to any 0-manifold with n+m positively oriented points and m negatively oriented points, . Likewise we can reverse this and choose n to be a negative integer and then any 0-manifold with |n| negatively oriented points is cobordant to any 0-manifold with |n|+m negatively oriented points and m positively oriented points.
That’s a mouthful. What I’m trying to make clear is that for each integer there is a corresponding equivalence class of 0-manifolds that is labeled by the excess of either positively or negatively oriented points (the integer zero representing the empty cobordism between empty sets).
I know that there’s some resulting algebraic structure that comes out of this but I’m not there yet. Suppose I look at the equivalence classes labeled by 8 and -4. 8 would be the equivalence class of 0-manifolds with eight extra positively oriented points. -4 would be the equivalence class of 0-manifolds with 4 extra negatively oriented points. If you take a representative from each equivalence class and ‘add’ them via disjoint union you’ll end up with 8-4=4 extra positively oriented points. So the operation “+” on equivalence classes can be defined by taking disjoint union of representatives of the equivalence classes. So these equivalence classes add just like integers.
I don’t know much about this but I did see recently (I can’t remember where unfortunately) that the cobordism groups are for 1Cob, 0 for 2Cob, 0 for 3Cob and then for 4Cob with somehow generating it! I’ll have much more later on this aspect.
There are some more ideas we can add here, Morse functions and how they will give a notion of time on the cobordism, how to associate vector spaces to these manifolds (is that even possible for 1Cob as it would represent a 0+1 theory?). I’d like to talk about Morse functions in a separate post however so I’ll end this one.