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I feel like this problem has not been solved, but I'm interested in knowing any results on it. More specifically, I mean:

Let $B\stackrel{f}{\leftarrow} A \stackrel{g}{\rightarrow} C$ be a diagram of Smooth manifolds and Smooth functions, and let $B\stackrel{k}{\rightarrow} D \stackrel{l}{\leftarrow} C$ be the Topological pushout (that is, $D$ obtains a Set structure as the quotient of $B \coprod C$ by the images of $f$ and $g$, and receives the weak Topology induced by $k$ and $l$). GOAL: Determine necessary and sufficient conditions for the Topological manifold $D$ to receive a natural Smooth manifold structure making the diagram a Smooth pushout.

To make it seem less daunting, break it into steps:

1) When is $D$ a Topological manifold (that is, inherits a Topological atlas from the diagram)?

2) When does $D$ inherit Smooth transition functions? (or whatever your favorite definition of "smooth structure" is)

2') If $D$ does not inherit a Smooth structure from the diagram, can we give it one? (I believe this is just Hirsch-Mazur)

3) Can Smooth functions be defined piecewise?

I'm certain that #1 is most difficult. Using $A\subset\mathbb{R}=B=C$ and $f,g$ inclusion maps, $A=0$ gives a point with 4 arms (not locally Euclidean) and $A=\mathbb{R}\setminus0$ gives the line with two origins (not Hausdorff). I find it difficult to find concise conditions that rule out those two situations. My only idea is to assume that everything is also cellular and try to come up with an obstruction class somehow (hence this question: When is a finite cw-complex a compact topological manifold?)

I have a few ideas and results for #2. For example, attaching manifolds along a submanifold (as defined in Kosinski's "Differential Manifolds") can be expressed with a smooth pushout diagram, which I proved by making a more general sufficient condition for "passing" a Smooth structure from two (or $n$) Smooth manifolds onto a Topological manifold. My intuition says that the criteria for #3 will be almost the same as for #2

I'm interested in seeing what else is out there right now, and Google/arXiv searches aren't turning anything up. I'm half-expecting to see the usual "This is basically Surgery Theory" response....

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"Manifold" is not a structure on a topological space, it's a property. – Qiaochu Yuan Feb 23 '12 at 20:47
Edited accordingly – William Feb 23 '12 at 23:51

I think you should look at

MR2342857 (2008i:58016) Pradines, Jean In Ehresmann's footsteps: from group geometries to groupoid geometries. (English summary) Geometry and topology of manifolds, 87–157, Banach Center Publ., 76, Polish Acad. Sci., Warsaw, 2007.

(arXiv:0711.1608 ) which discusses these issues.

Compare also with

arXiv:0807.1704 [pdf, ps, other] Convenient Categories of Smooth Spaces John C. Baez, Alexander E. Hoffnung

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Thank you for the references. I was aware of "cartesean closed" expansions of the Smooth category, which your second reference goes into, but I am interested in pushouts that are specifically Smooth. This "diptych" idea in Pradines looks interesting, possibly providing a good way to phrase nec./suf. conditions for a smooth pushout. I'll keep looking at it, hopefully I find something – William Feb 24 '12 at 17:31

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