Geodesic
The Solution of Hilbert's Fifth Problem for Transitive Groupoids
Symmetry, integrability and geometry: methods and applications, Tome 14 (2018) Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

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.
Keywords: Lie groupoids; topological groupoids. 
@article{SIGMA_2018_14_a69,
     author = {Pawe{\l} Ra\'zny},
     title = {The {Solution} of {Hilbert's} {Fifth} {Problem} for {Transitive} {Groupoids}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2018},
     volume = {14},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2018_14_a69/}
}
TY  - JOUR
AU  - Paweł Raźny
TI  - The Solution of Hilbert's Fifth Problem for Transitive Groupoids
JO  - Symmetry, integrability and geometry: methods and applications
PY  - 2018
VL  - 14
UR  - http://geodesic.mathdoc.fr/item/SIGMA_2018_14_a69/
LA  - en
ID  - SIGMA_2018_14_a69
ER  - 
%0 Journal Article
%A Paweł Raźny
%T The Solution of Hilbert's Fifth Problem for Transitive Groupoids
%J Symmetry, integrability and geometry: methods and applications
%D 2018
%V 14
%U http://geodesic.mathdoc.fr/item/SIGMA_2018_14_a69/
%G en
%F SIGMA_2018_14_a69
Paweł Raźny. The Solution of Hilbert's Fifth Problem for Transitive Groupoids. Symmetry, integrability and geometry: methods and applications, Tome 14 (2018). http://geodesic.mathdoc.fr/item/SIGMA_2018_14_a69/

[1] Bing R. H., “The cartesian product of a certain non-manifold and a line is $E_{4}$”, Bull. Amer. Math. Soc., 64 (1958), 82–84 | DOI | MR | Zbl

[2] Brown R., Hardy J. P. L., “Topological groupoids. I. Universal constructions”, Math. Nachr., 71 (1976), 273–286 | DOI | MR | Zbl

[3] Engelking R., Dimension theory, North-Holland Mathematical Library, 19, North-Holland Publishing Co., Amsterdam–Oxford–New York; PWN – Polish Scientific Publishers, 1978 | MR | Zbl

[4] Gleason A. M., “Groups without small subgroups”, Ann. of Math., 56 (1952), 193–212 | DOI | MR | Zbl

[5] Mackenzie K. C. H., Lie groupoids and Lie algebroids in differential geometry, London Mathematical Society Lecture Note Series, 124, Cambridge University Press, Cambridge, 1987 | DOI | Zbl

[6] Mackenzie K. C. H., General theory of Lie groupoids and Lie algebroids, London Mathematical Society Lecture Note Series, 213, Cambridge University Press, Cambridge, 2005 | DOI | Zbl

[7] Montgomery D., Zippin L., “Small subgroups of finite-dimensional groups”, Ann. of Math., 56 (1952), 213–241 | DOI | MR | Zbl

[8] Palais R. S., “On the existence of slices for actions of non-compact Lie groups”, Ann. of Math., 73 (1961), 295–323 | DOI | MR | Zbl

[9] Pasike E. E., Petunin Yu. I., Savkin V. I., “Continuous bijective mappings in topological and Banach manifolds”, J. Math. Sci., 58 (1992), 286–29 | DOI | MR

[10] Raźny P., “On the generalization of Hilbert's fifth problem to transitive groupoids”, SIGMA, 13 (2017), 098, 10 pp. | DOI | MR | Zbl

[11] Renault J., A groupoid approach to $C^{\ast} $-algebras, Lecture Notes in Math., 793, Springer, Berlin, 1980 | DOI | Zbl

[12] Tao T., Hilbert's fifth problem and related topics, Graduate Studies in Mathematics, 153, Amer. Math. Soc., Providence, RI, 2014 | DOI | Zbl