Consider all of your basic constructions/tools/theorems for manifolds: fundamental group, Euler characteristic, triangulations, orientation, smoothness, bundle structure, cobordisms, etc.. Viewing orbifolds as the natural generalization of manifolds by quotients of group actions (or just locally $\mathbb{R}^n/G\_i$), it *seems* that we can translate all of our notions for manifolds onto orbifolds with slight adjustments in the definitions. I am curious as to the extent of that statement's validity (which is important, because orbifolds are arising everywhere for me):  

**What basic constructions that exist on *manifolds*, cannot (or currently do not) have analogs for *orbifolds*?**  

Why? ($\leftarrow$ in case there is more to say than "orbifolds have singularities")


As pointed out nicely below, there is the question of whether such constructions will be *useful* to (all) orbifolds. So this thread also seeks known constructions which are either intractable or trivial.



What tools cannot work for orbifolds?
I'm not an expert and this might be wrong, but I think that [Cerf theory](http://en.wikipedia.org/wiki/Cerf_theory) should be impossible for orbifolds, and therefore all that comes from it, *e.g.* Kirby Calculus. Could somebody who knows please confirm this? I'm guessing that this comes **not** from orbifolds having singularities, **but rather** from orbifolds having built-in symmetries which mess up the necessary stratifications of the space of smooth functions to $\mathbb{R}$.


For example, [Kirby's Theorem](http://en.wikipedia.org/wiki/Kirby_calculus) comes from the fact that a generic path between Morse functions in the space of smooth functions to $\mathbb{R}$ involves only finitely many which are not Morse-Smale (these correspond to handleslides). But I think that this is just wrong for smooth orbifolds. In the orbifold case, Morse functions satisfying the Morse-Smale condition (transversality between stable and unstable manifolds at a critical point) are not dense among smooth functions to $\mathbb{R}$, so perhaps a generic path between two Morse functions in the space of smooth functions might contain all kinds of craziness, and I doubt that there exists a sensible finite set of local moves between handle decompositions to parallel the Kirby moves.




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Upon further thought, the answer above can be generalized. In topology, there are various constructions of manifolds as cell decompositions of various flavours (*e.g.* triangulations, surgery presentations...), along with which there are sets of moves telling us how to go between any two such constructions (*e.g.* Kirby moves, Pachner moves, subdivision...). These give us the possibility to use the constructions in order to construct manifold invariants, for example.


For orbifolds, you might have good analogues of the constructions, but I am not aware of good analogues for the "sets of moves" between them for any construction. I think that the issue is that orbifolds don't really form a category, but rather they form a 2-category of some sort. I suspect that the point now is something like that the relevant notion of a 2-functor makes the relevant diagrams commute not `on the nose', but up to some sort of morphism; and that such a setup can't give rise to a nice set of local moves which relate all of our constructions. I don't know the precise statement though. 


