Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

We restrict ourselves to the category of smooth manifolds and smooth maps.

Suppose we have a pair of smooth maps $f:A \to X$ and $g:A \to Y$. A pushout is a pair of smooth maps $p:X \to Z$ and $q:Y \to Z$ satisfying $pf=qg$ and the following universal property:

($*$) For every pair of smooth maps $r:X \to W$ and $s:Y \to W$ satisfying $rf=sg$, there is a unique smooth map $u: Z \to W$ such that $up=r$ and $ur=s$.

Question: Are any (relatively convenient) conditions for the existence of pushout known?

In the case of pullback, it is known that we have a pullback if one of $f:X \to B$ and $g:Y \to B$ is a submersion (or $f$ and $g$ are transverse). I am asking about such a condition for pushout.

share|improve this question
1  
I don't have an answer for you, but this might be a starting point. Every pushout can be expressed as a coequalizer of a coproduct. Since there shouldn't be any problems with the existence of coproducts (as long as you allow disconnected manifolds), it might be fruitful to look at existence conditions for coequalizers (which are often a bit easier to get a handle on than pushouts). –  Unwisdom Feb 25 at 16:37
add comment

1 Answer 1

It's $X\coprod_A Y \simeq X \coprod Y /\sim $ where $\forall a,b \in A, f(a) \sim g(b)$

share|improve this answer
    
And why should this be a smooth manifold? –  Zhen Lin Feb 25 at 22:13
    
oops... one point for you! –  Léo Feb 25 at 22:17
1  
The relevant area seems to be Godement's theorem on quotients of differentiable spaces, and for a general look, see work of J. Pradines on Diptychs. (do a web search) –  Ronnie Brown Feb 26 at 16:18
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.