What tools cannot work for orbifolds?

Question

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.

Answer 1

I'm not an expert and this might be wrong, but I think that 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 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.

Answer 2

The integral cohomology ring of orbifolds does not satisfy Poincaré duality.
Indeed, orbifolds do not even have finite cohomological dimension.

Answer 3

A note before: the communities of low-dimensional topologists and categorically minded people have different opinions on what an 'orbifold' is. Let $G$ be a discrete group that acts properly and smoothly on the manifold $M$. As far as I understand, low-dimensional topologists consider the quotient space $M/G$, together with the cleverly packaged information on the isotropy subgroups. For a categorist (and I am following them on this issue), an orbifold is a special type of 'stack', which in turn is an object of a suitable category of functors to groupoids (and can be represented by a certain Lie groupoid; in the case under discussion, it is the action groupoid of the $G$-action on $M$). I think of an orbifold as a space whose points can have automorphisms and (as a side remark) my personal opinion is that it is quite unfortunate that there does not seem to be a definition of a stack that does not appeal to the functor-of-points-philosophy.

Now to the real issue I wish to communicate. The Pontrjagin-Thom construction is fundamental in most branches of manifold topology. The basic case is when $f:E \to B$ is a proper smooth map of manifolds, $dim (E)=dim(B)+d$, $d \in \mathbb{Z}$. There is a stable normal bundle $\nu(f):= f^* TB - TE$ and the PT-construction is a map of spectra $\Sigma^{\infty} B_+ \to Th (\nu(f))$, the Thom spectrum of $\nu(f)$. From this and the Thom isomorphism, one deduces homological invariants, such as the Gysin map $f_!: H^i (E) \to H^{i-d} (B)$, which is moreover functorial , $(f \circ g)_! = f_! \circ g_!$.

My initial thought was that there should be an analogue for orbifolds. The tangent bundles make sense, and one can make sense out of stable vector bundles on a stack. Moreover, there is a homotopy type of stacks. If the stack is $M//G$, its homotopy type $Ho(M//G)$ is the Borel construction $EG \times_G M$, and it comes with a map to $M//G$ that is declared to be a weak homotopy equivalence. One can then define homotopy/(co)homology of stacks using this homotopy type (there are finer theories, due to Henriques-Gepner).

One can pull back stable vector bundles on stacks to the homotopy type and form the Thom spectra there. The expectation is that there is a PT map $\Sigma^{\infty} Ho(B)_+ \to Th (\nu(f))$ when $f$ is proper in a suitable sense. The unique map $\ast //G \to \ast$ should be a proper map.

Unfortunately, this does not work. Together with Jeffrey Giansiracusa, I wrote a paper that gives such a PT-construction, under the restriction that the map $f$ is ''representable'', which in the case $B=\ast$ amounts to saying that $E$ is a closed manifold. In his study of orbifold cobordism, Andres Angel used a different approach than the PT-construction.

Why is such a construction impossible? Well, let us suppose that there is such a construction that works at least for $\ast//G \to \ast$ (the quotient map $\ast \to \ast //G$ does not pose a problem). As in the case of the ordinary PT-construction, one defines the Gysin maps $f_!$ and proves that they are functorial. As the composition $\ast \to \ast//G \to \ast$ is the identity, the composition of the corresponding Gysin maps must be the identity. On the other hand, we know very well what the Gysin map of $\ast \to \ast//G$ does on $H^0$; it is multiplication by $|G|$. This leads us to the contradiction that the identity on $\mathbb{Z}$ factors through the multiplication by $|G|$ map.

Strangely enough, when we take real coefficients, the Gysin map is integration over the fibres, and this seems to work for orbifolds. I can also conceive an integration map in complex $K$-theory, but nevertheless the problem seems to be fundamental and severely restricts the application of manifold methods to the topology of orbifolds.