Reference request: gluing manifolds along pieces of boundary - MathOverflow

Reference request: gluing manifolds along pieces of boundary

2011-11-30T13:35:28Z

I've been asked for a reference for the following construction and since I didn't know one, I thought I'd ask here if anyone did.

Consider two smooth manifolds with boundary of the same dimension, $M$ and $N$. Suppose that we have a submanifold $S$ of the boundaries $\partial M$ and $\partial N$ of codimension $0$ ($S$ may have boundary as well). Then glue $M$ and $N$ together along $S$, smoothing out the corners (corresponding to the boundary of $S$) if necessary. To do this properly, one would have to add a collar to each of $M$ and $N$ which is "broken" at the boundary of $S$ (thus making manifolds-with-corners) and glue those.

Is there a standard reference for this that works through the details?

Answer by Igor Rivin for Reference request: gluing manifolds along pieces of boundary

2011-11-30T13:47:16Z

This seems to be discussed here. (particularly page 6)

Answer by Andrew Stacey for Reference request: gluing manifolds along pieces of boundary

2011-12-08T13:14:26Z

I did email Jim Davis and I have permission to post his reply (I'll summarise it). He taught a course which needed this result (which the notes that Igor Rivin links to are from). Being unable to find the precise statement (or proof) in the literature, he proved himself. He has ambitions to flesh out those notes to something fuller.

Since Kosinski's book was mentioned in the comments, it is perhaps worth pointing out that it contains the statement (p14):

Complications arise when more than one handle has been attached. When this happens some proofs have to rely strongly on the technique known as vigorous hand waving.

so for a precise statement/proof, then it would appear that Kosinski's book has to be discounted. (I should be honest and say that I haven't checked Kosinski's book myself; the original request was from a colleague and he has checked the book and is not happy with what is in there. The above quote was highlighted by Jim Davis.)

Answer by Chris Schommer-Pries for Reference request: gluing manifolds along pieces of boundary

2011-12-08T14:57:25Z

This was a bit too long for a comment, so I am posting it as an answer. You are sort of asking two things:

How to turn your manifolds M and S into an appropriate manifold with corners together with an appropriate notion of collar?
How to then glue these to obtain a new manifold?

To do (1) you'll need some assumptions on S, probably including compactness. In many cases though it might be clear that you can choose such collars. In that case you might be interested in Theorem 3.5 from my 2009 dissertation (arXiv:1112.1000, page 140). There I show that even if the collars are not specified, the glued manifold is still unique up to (non-canonical) diffeomorphism fixing S and restricting to the identity outside a neighborhood of S. In fact the construction shows that there is a canonical contractible family of these diffeomorphisms (and so there is a canoncial isotopy class of diffeomorphisms).

I used this to build one version of the 2-category of cobordisms, where you need to glue along parts of the boundary in the manner you describe, but where you also don't want to mod out by diffeomorphisms too early.

When S is a component of the boundary, you can find this result here:

James R. Munkres, Elementary differential topology, Lectures given at Massachusetts Institute of Technology, Fall, vol. 1961, Princeton University Press, Princeton, N.J., 1966.

I basically adapted this proof to cover the case of gluing manifolds along a portion of the boundary.