## 1-Cobordism Part 3

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, $m>0$. 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 $n\in\mathbb{Z}$ 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 $\mathbb{Z}$ for 1Cob, 0 for  2Cob, 0 for 3Cob and then $\mathbb{Z}$ for 4Cob with $\mathbb{C}P^2$ 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.