
SIGMA 13 (2017), 098, 10 pages arXiv:1710.11440
https://doi.org/10.3842/SIGMA.2017.098
On the Generalization of Hilbert's Fifth Problem to Transitive Groupoids
Paweł Raźny
Institute of Mathematics, Faculty of Mathematics and Computer Science, Jagiellonian University in Cracow, Poland
Received December 02, 2017, in final form December 21, 2017; Published online December 31, 2017
Abstract
In the following paper we investigate the question: when is a transitive topological groupoid continuously isomorphic to a Lie groupoid? We present many results on the matter which may be considered generalizations of the Hilbert's fifth problem to this context. Most notably we present a ''solution'' to the problem for proper transitive groupoids and transitive groupoids with compact source fibers.
Key words:
Lie groupoids; topological groupoids.
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References

Engelking R., General topology, Sigma Series in Pure Mathematics, Vol. 6, 2nd ed., Heldermann Verlag, Berlin, 1989.

Gleason A.M., Groups without small subgroups, Ann. of Math. 56 (1952), 193212.

Lee J.M., Introduction to topological manifolds, Graduate Texts in Mathematics, Vol. 202, 2nd ed., Springer, New York, 2011.

Mackenzie K.C.H., Lie groupoids and Lie algebroids in differential geometry, London Mathematical Society Lecture Note Series, Vol. 124, Cambridge University Press, Cambridge, 1987.

Mackenzie K.C.H., General theory of Lie groupoids and Lie algebroids, London Mathematical Society Lecture Note Series, Vol. 213, Cambridge University Press, Cambridge, 2005.

Moerdijk I., Mrčun J., Introduction to foliations and Lie groupoids, Cambridge Studies in Advanced Mathematics, Vol. 91, Cambridge University Press, Cambridge, 2003.

Montgomery D., Zippin L., Small subgroups of finitedimensional groups, Ann. of Math. 56 (1952), 213241.

Müller C., Wockel C., Equivalences of smooth and continuous principal bundles with infinitedimensional structure group, Adv. Geom. 9 (2009), 605626, math.DG/0604142.

Palais R.S., On the existence of slices for actions of noncompact Lie groups, Ann. of Math. 73 (1961), 295323.

Siwiec F., Sequencecovering and countably biquotient mappings, General Topology and Appl. 1 (1971), 143154.

Tao T., Hilbert's fifth problem and related topics, Graduate Studies in Mathematics, Vol. 153, Amer. Math. Soc., Providence, RI, 2014.

Torres D.M., Proper Lie groupoids are real analytic, arXiv:1612.09012.

