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Topologists study topological manifolds, differentiable manifolds and PL-manifolds (and some other flavors), with each class distinguished by the selection of a pseudogroup that restricts the transition functions that may occur in an atlas. Other pseudogroups exist and do receive study, but in many cases the extra structure they imbue has a geometrical rather than a topological character, e.g. on account of local invariants or continuously varying moduli.

The TOP, DIFF and PL pseudogroups each have their intrinsic importance for applications and their historical caché, so I understand why they attract so much attention. But I ask: do there exist theorems that characterize abstractly those pseudogroups that lead to reasonable theories of manifolds, theories with a topological character? In particular, does any such theorem explain the de facto privileged status of TOP, DIFF and PL (and a few others)? If no, then how big is the zoo of exotic topologically-flavored pseudogroups?

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I see an analogy (though that word might be too weak?) between arithmetic geometry and manifolds-with-pseudogroup-structure. In arithmetic geometry one asks whether a variety has a definition over a field, and also whether distinct varieties defined over one field become isomorphic over some larger field. This reminds one of how, say, passing from DIFF to TOP one meets new manifolds that didn't admit a DIFF-structure, but also sees new TOP-isomorphisms between distinct DIFF-manifolds.

The big difference, methodologically, seems that in arithmetic geometry the varieties (~manifolds) and the fields (~pseudogroups) will often enter on an equal footing. For example, one constructs important fields in class field theory as the fields of definition certain varieties. But in topology I have the impression that one always chooses at the start one or more pseudogroups, and then develops their theory; I don't know anywhere that the pseudogroups emerge naturally out the topological phenomena.

Perhaps I should start over and shape this into an autonomous question?

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    $\begingroup$ I feel like the privileged status of top, diff, PL, etc. has more to do with the associated sheaves of functions than the pseudogroup of transition maps. No notion of function on Euclidean space which is weaker than continuous or stronger than PL is geometrically sensible, there are no real obstructions to promoting $C^p$ to $C^{\infty}$, and analytic / algebraic manifolds are already well studied. $\endgroup$
    – Paul Siegel
    Commented Jan 16, 2011 at 19:57
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    $\begingroup$ Do you consider Haefliger's Gamma-structures (en.wikipedia.org/wiki/Haefliger_structure) as an example of these exotic pseudogroups? $\endgroup$
    – Maxime Bourrigan
    Commented Jan 17, 2011 at 0:44
  • $\begingroup$ Isn't the language of pseudogroups more general than that of sheaves? Given a sheaf of functions on ${\Bbb R}^n, one gets a pseudogroup by insisting that transition functions preserve the local sections. But I don't see how to attach a sheaf of 1-variable functions to an arbitrary pseudogroup sufficient to recover the pseudogroup from the sheaf. One can make intermediate pseudogroups for example by starting with smooth functions and then adding one non-smooth function (and everything it generates by composition). So in principle there seem to by many candidate examples. $\endgroup$
    – David Feldman
    Commented Jan 17, 2011 at 0:49

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Contact structures have a strong topoological character --- there are no local invariants for them, and no local moduli. They especially have a lot of known interaction with "standard" topology in dimension 3. They are quite flexible, with a rich group of automorphisms.

Symplectic structures do have local moduli, but the local parameter is just a cohomology class, which is itself straightforward topology. Symplectic structures have a rich group of automorphisms; it's a matter of word choice if you want to say they don't have a "topological" character. Obviously they're have undergone a vast amount of study, and they're very important.

Other "topological" examples: the pseudogroup of local bilipschitz homeomorpisms, and the pseudogroup of local quasiconformal homeomorphisms.

But really, for many purposes, I don't see the need to segregate the different kinds of pseudogroups: the relationships among them often tell us something about topology as well as the particular structure. When there is occasion to study actual diffeomorphisms and homeomorphisms between manifolds, rather than just isotopy classes of homeomorphisms, the subject is very closely related to foliations, which is very closely related to flat bundles with fiber a manifold. There is a good deal of machinery, classifying spaces and the like, that works in similar ways for many different cases.

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    $\begingroup$ Do symplectic structures had local moduli? Doesn't Darboux's theorem say they don't? $\endgroup$
    – Mariano Suárez-Álvarez
    Commented Jan 17, 2011 at 3:06
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    $\begingroup$ @Mariano: I didn't phrase it clearly: I meant "local moduli" in the sense of the space of structures, rather than local within the manifold. You're correct, Darboux' theorem says there are no local moduli in the latter sense. With this definition, many others qualify as well: even finite-dimensional cases, like hyperbolic structures, similarity structures, affine structures, projective structures, etc etc as well as foliations, multi-foliations, multi-foliations with additional transverse structure, complex structures, etc: some of them more toward the geometric end, some more toward floppy. $\endgroup$
    – Bill Thurston
    Commented Jan 17, 2011 at 9:49

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