As mentioned previously, Kock has a helical pedagogy in his book. This next post considers the decomposition of cobordisms. What follows isn’t rigorous, I think that’s coming up and will be much more detailed. Here’s the idea, suppose we have a pair of manifolds M and N and a cobordism between them W . Let f be a Morse function . (What’s a Morse function, read wikipedia and we’ll be even). We want f to be such that the inverse image of 0 is M and the inverse image of 1 is N. Now consider and it’s inverse image .
The idea is that now we have two cobordisms, one from M to and one from to N. The orientation of is chosen so that it’s an out-boundary on the ‘left’ and an in-boundary on the ‘right.’ Kock makes a statement like “…such that the positive normal points towards the out-boundaries, just as the positive normal of points towards 1.” This is certainly intuitively clear though hardly rigorous or prescriptive.
Question: The picture suggest that there’s a possible bad choice of t. The one where the inverse image would be a sort of figure-eight. That wouldn’t be a manifold and so we wouldn’t have a pair of cobordisms.
Partial Answer: As long as t is regular (see Morse functions) the inverse image seems like it should be a manifold. Regular values are not critical values and critical values are essentially max/min/saddle which is where you’d expect trouble.
But the above construction is tame compared to what follows. Let’s start with the cylinder this time (note: this can be a general cylinder of a (n-1)-manifold cross the unit interval, say ). The image below shows a simple decomposition.
Kock then challenges us with “But we could also reverse the orientation of the middle copy of M.”
Possible Answer: For a regular value of t we can think of the ‘chopped’ cylinder as two smaller cylinders and . What’s a cobordism? It’s an n-manifold (in this case with a pair of (n-1)-manifolds (in this case M) with embeddings (in this case the natural inclusion mappings) into the boundary. So we already have an embedding from M into the left end of and an embedding from M into the right end of . Because this is a cylinder the preimage of t is certainly a copy of M, but we’ll still need to embed it into the the cut ends of our cobordism. That gives us the choice of how we pick the embedding.
Kock tells us that the above decomposition of the cylinder isn’t valid. That make sense because if you look carefully you’ll see that it’s a cobordism (which is disconnected, it’s the disjoint union of n-manifolds) between a disjoint union of M’s to a disjoint union of M’s. See the figure,
So this decomposition turns the identity map on M into a pair of maps whose composition is the identity map on ! That shouldn’t happen. It’s analogous to factoring only to find out that ??
We don’t have to abandon this choice of embeddings though, we can just add some more decomposition. Choose another pair of t‘s (still regular) such that the following becomes the case,
The way this is draw is fine, but its easier to redraw some sections for human consumption,
Why this way? How do we know to assemble it in this fashion, remember that the decomposition, when reassembled, is glued by identifying boundaries. If you need to go back a couple images and label the boundaries and you’ll see that I haven’t scrambled them.
In this way the cobordism we started with is decomposed into corbodisms that compose (as morphisms) to give back the original corbordism. Kock ends the section with a bit of forshadowing, “This particular example will have important consequences for TQFTs.”
I wonder why? My first thought (and only thought) was that this decomposition reminds me of the sea of virtual particles I hear about from QFT. The above cobordism, decomposed or not, is just a cylinder. It’s an identity morphism from M to M and will be associated to the “do-nothing” map in the category of vector spaces. But after we decomposed the cylinder we ended up with a particular version of “do-nothing” in which if we peek in the middle of “nothing happening” we see a bunch of copies of M produced out of the emptiness (meaning cobordisms from empty manifolds to disjoint unions of M). These extra copies vanish before it matters and as I said, we’re left with just a cylinder. I don’t know if I’m onto something or incorrect. Time will tell.
The above is intuitive. There’ll be posts in the future on how to make ‘gluing’ more rigorous and why it results in the necessity of equivalence classes of cobordisms being arrows rather than cobordisms themselves. Though apparently instead of doing that one can move to higher categories. I suspect (but do not know yet) that ‘equivalent’ cobordisms, as 1-morphisms, would be related by the 2-morphisms that would be available in higher categories.