# Pseudogroups and the flavors of the topology of manifolds

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?

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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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. –  Paul Siegel Jan 16 '11 at 19:57
Do you consider Haefliger's Gamma-structures (en.wikipedia.org/wiki/Haefliger_structure) as an example of these exotic pseudogroups? –  Maxime Bourrigan Jan 17 '11 at 0:44
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. –  David Feldman Jan 17 '11 at 0:49