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It is easy to see (and presumably well-known) that the operation of attaching manifolds along a submanifold (as defined Characteristic Classes of a Fibered Sum) can be expressed as a push-out in the Smooth category, so long as the construction includes collars. The connected sum follows as a special case where the submanifold is a point. Furthermore, Thom's "shperical modifications" can also be treated as a connected sum. Basic facts about these operations (well-defined under isotopy of the embeddings, associative, etc.) can be proven as corollaries of theorems about push-outs.

I've done some work on my own developing this idea, but I'm wondering if there's some theory already in place that could help me save some time or point me in better directions. I haven't come across anything in my searches. What I'm looking for specifically are references which treat these operations as pushouts, and develop them as such. (What I am NOT looking for are modifications of the smooth category which make it closed under pushouts)

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Maybe you want to include a collar for gluing. –  S. Carnahan Mar 5 '12 at 7:19
    
In the smooth category, this isn't a pushout (as others have pointed out)... May I reinterpret your question as being about modifications to the definition of the smooth category so that this operation becomes a pushout? –  Daniel Moskovich Mar 5 '12 at 12:09
    
@Daniel No, that is not the question I am interested in. I am more interested in the cases where the topological pushout is actually a smooth pushout. Does this not happen even when the construction includes collars? –  William Mar 5 '12 at 17:58
    
It might help if, in addition to giving clarifying comments, you edited the question to reflect the more precise question you have in mind. –  S. Carnahan Mar 6 '12 at 7:13
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If you do not want to modify your category, then pushouts will be very rare. There are pushouts for pairs of open embeddings, but almost nothing else. You can say that the connected sum $M\#N$ is the pushout of $M\setminus\\{\text{point}\\}$ and $N\setminus\\{\text{point}\\}$ along $\mathbb{R}^n\setminus\\{0\\}$, but not that $M\#N$ is the pushout of $M\setminus(\text{open ball})$ and $N\setminus(\text{open ball})$ along $S^{n-1}$. This is one possible interpretation of "the construction includes collars". If you have a different one, you should explain it. –  Neil Strickland Mar 6 '12 at 22:45
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I don't think it is true that this kind of construction is a pushout in the smooth category. Consider the function $f:\mathbb{R}\to\mathbb{R}$ given by $f(x)=|x|$. This is smooth on $(-\infty,0]$ and on $[0,\infty)$. (There are some subtleties about how to define smoothness for functions on manifolds with boundary, but all possible variants are easily seen to be satisfied in this case.) If $\mathbb{R}$ were the pushout of $(-\infty,0]$ and $[0,\infty)$ along $\{0\}$ then $f$ would be smooth on $\mathbb{R}$, which is false.

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This is not an example of the construction I am considering. I am aware that a collar is needed in order to induce a smooth structure on an open overlap. I am not asserting that every topological pushout is a Smooth pushout, I am looking for a reference in the cases where it is. –  William Mar 5 '12 at 17:29
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In Chapter 3 of Chris Schommer-Pries's PhD thesis, surfaces with corners (more generally, manifolds with faces) are equipped with extra structure called a "halation", with precisely the goal of making the gluing operation a pushout. The topic of making the gluing operation into a pushout in the smooth category is discussed (and motivated) in some detail, quite lucidly.

Added: Alternatively, you could work in the microlinear category, where the problems we are discussing do not arise, and the naive gluing operation is a pushout. See e.g Basic Concepts of Synthetic Differential Geometry by Rene Lavendhomme.

Added 2: References for the "just add collars" approach are:

  • T. Kerler and V. Lyubashenko, Non-Semisimple Topological Quantum Field Theories for 3-Manifolds with Corners, Lecture Notes in Math. 1765, Springer-Verlag, Berlin (2001).
  • J. Morton, Extended TQFT's and Quantum Gravity, Ph.D Thesis (University of California, Riverside).
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For anyone else trying to search for this: that should be "halation" with one l, cognate with "halo". –  Neil Strickland Mar 5 '12 at 13:11
    
Typo corrected. –  Daniel Moskovich Mar 6 '12 at 0:00
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In the link the OP provided, the definition includes a collar, so the example given by Neil is not really a counter-example, since it does not satisfies the hypothesis. (This is more a comment than an answer, but I'm not allowed to comment yet, sorry.)

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The approach of equipping smooth manifolds with boundary with a choice of skeleton (or a distinguished smooth glued manifold) is discussed by Morton and Kerler-Lyubashenko (equivalent to including choices of collar). It's one approach; page 117-118 of Schommer-Pries's thesis explains drawbacks of this approach. In any event, it's not the category of smooth manifolds anymore, but some new category with this extra structure. –  Daniel Moskovich Mar 5 '12 at 12:05
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