|
SIGMA 14 (2018), 070, 11 pages arXiv:1805.03066
https://doi.org/10.3842/SIGMA.2018.070
The Solution of Hilbert's Fifth Problem for Transitive Groupoids
Paweł Raźny
Institute of Mathematics, Faculty of Mathematics and Computer Science, Jagiellonian University in Cracow, Poland
Received May 11, 2018, in final form July 10, 2018; Published online July 17, 2018
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.
pdf (321 kb)
tex (16 kb)
References
-
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.
-
Brown R., Hardy J.P.L., Topological groupoids. I. Universal constructions, Math. Nachr. 71 (1976), 273-286.
-
Engelking R., Dimension theory, North-Holland Mathematical Library, Vol. 19, North-Holland Publishing Co., Amsterdam - Oxford - New York, PWN - Polish Scientific Publishers, Warsaw, 1978.
-
Gleason A.M., Groups without small subgroups, Ann. of Math. 56 (1952), 193-212.
-
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.
-
Montgomery D., Zippin L., Small subgroups of finite-dimensional groups, Ann. of Math. 56 (1952), 213-241.
-
Palais R.S., On the existence of slices for actions of non-compact Lie groups, Ann. of Math. 73 (1961), 295-323.
-
Pasike E.E., Petunin Yu.I., Savkin V.I., Continuous bijective mappings in topological and Banach manifolds, J. Math. Sci. 58 (1992), 286-29.
-
Raźny P., On the generalization of Hilbert's fifth problem to transitive groupoids, SIGMA 13 (2017), 098, 10 pages, arXiv:1710.11440.
-
Renault J., A groupoid approach to $C^{\ast} $-algebras, Lecture Notes in Math., Vol. 793, Springer, Berlin, 1980.
-
Tao T., Hilbert's fifth problem and related topics, Graduate Studies in Mathematics, Vol. 153, Amer. Math. Soc., Providence, RI, 2014.
|
|