### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 12 (2016), 026, 19 pages      arXiv:1408.1555      https://doi.org/10.3842/SIGMA.2016.026

### Basic Forms and Orbit Spaces: a Diffeological Approach

Yael Karshon a and Jordan Watts b
a) Department of Mathematics, University of Toronto, 40 St. George Street, Toronto Ontario M5S 2E4, Canada
b) Department of Mathematics, University of Colorado Boulder, Campus Box 395, Boulder, CO, 80309, USA

Received October 06, 2015, in final form February 16, 2016; Published online March 08, 2016

Abstract
If a Lie group acts on a manifold freely and properly, pulling back by the quotient map gives an isomorphism between the differential forms on the quotient manifold and the basic differential forms upstairs. We show that this result remains true for actions that are not necessarily free nor proper, as long as the identity component acts properly, where on the quotient space we take differential forms in the diffeological sense.

Key words: diffeology; Lie group actions; orbit space; basic differential forms.

pdf (449 kb)   tex (27 kb)

References

1. Baez J.C., Hoffnung A.E., Convenient categories of smooth spaces, Trans. Amer. Math. Soc. 363 (2011), 5789-5825, arXiv:0807.1704.
2. Blohmann C., Fernandes M.C.B., Weinstein A., Groupoid symmetry and constraints in general relativity, Commun. Contemp. Math. 15 (2013), 1250061, 25 pages, arXiv:1003.2857.
3. Bredon G.E., Introduction to compact transformation groups, Pure and Applied Mathematics, Vol. 46, Academic Press, New York - London, 1972.
4. Chen K.-T., Iterated integrals of differential forms and loop space homology, Ann. of Math. 97 (1973), 217-246.
5. Chen K.-T., On differentiable spaces, in Categories in Continuum Physics (Buffalo, N.Y., 1982), Lecture Notes in Math., Vol. 1174, Springer, Berlin, 1986, 38-42.
6. Crainic M., Struchiner I., On the linearization theorem for proper Lie groupoids, Ann. Sci. Éc. Norm. Supér. (4) 46 (2013), 723-746, arXiv:1103.5245.
7. Cushman R., Śniatycki J., Differential structure of orbit spaces, Canad. J. Math. 53 (2001), 715-755.
8. Donato P., Iglésias P., Exemples de groupes difféologiques: flots irrationnels sur le tore, C. R. Acad. Sci. Paris Sér. I Math. 301 (1985), 127-130.
9. Duistermaat J.J., Kolk J.A.C., Lie groups, Universitext, Springer-Verlag, Berlin, 2000.
10. Guillemin V., Ginzburg V., Karshon Y., Moment maps, cobordisms, and Hamiltonian group actions, Mathematical Surveys and Monographs, Vol. 98, Amer. Math. Soc., Providence, RI, 2002.
11. Haefliger A., Groupoï des d'holonomie et classifiants, Astérisque 116 (1984), 70-97.
12. Hochschild G., The structure of Lie groups, Holden-Day, Inc., San Francisco - London - Amsterdam, 1965.
13. Iglesias P., Karshon Y., Zadka M., Orbifolds as diffeologies, Trans. Amer. Math. Soc. 362 (2010), 2811-2831, math.DG/0501093.
14. Iglesias-Zemmour P., Diffeology, Mathematical Surveys and Monographs, Vol. 185, Amer. Math. Soc., Providence, RI, 2013.
15. Iglesias-Zemmour P., Karshon Y., Smooth Lie group actions are parametrized diffeological subgroups, Proc. Amer. Math. Soc. 140 (2012), 731-739, arXiv:1012.0107.
16. Karshon Y., Zoghi M., Orbifold groupoids and their underlying diffeology, Earlier version posted at http://www.math.toronto.edu/mzoghi/research/Groupoids.pdf and summarized in Zoghi's PhD thesis, University of Toronto, 2010.
17. Koszul J.L., Sur certains groupes de transformations de Lie, in Géométrie différentielle (Colloques Internationaux du Centre National de la Recherche Scientifique, Strasbourg, 1953), Centre National de la Recherche Scientifique, Paris, 1953, 137-141.
18. Lerman E., Meinrenken E., Tolman S., Woodward C., Nonabelian convexity by symplectic cuts, Topology 37 (1998), 245-259, dg-ga/9603015.
19. Palais R.S., On the existence of slices for actions of non-compact Lie groups, Ann. of Math. 73 (1961), 295-323.
20. Satake I., On a generalization of the notion of manifold, Proc. Nat. Acad. Sci. USA 42 (1956), 359-363.
21. Satake I., The Gauss-Bonnet theorem for $V$-manifolds, J. Math. Soc. Japan 9 (1957), 464-492.
22. Schwarz G.W., Smooth functions invariant under the action of a compact Lie group, Topology 14 (1975), 63-68.
23. Sjamaar R., A de Rham theorem for symplectic quotients, Pacific J. Math. 220 (2005), 153-166, math.SG/0208080.
24. Sjamaar R., Lerman E., Stratified symplectic spaces and reduction, Ann. of Math. 134 (1991), 375-422.
25. Śniatycki J., Differential geometry of singular spaces and reduction of symmetry, New Mathematical Monographs, Vol. 23, Cambridge University Press, Cambridge, 2013.
26. Souriau J.-M., Groupes différentiels, in Differential Geometrical Methods in Mathematical Physics (Proc. Conf., Aix-en-Provence/Salamanca, 1979), Lecture Notes in Math., Vol. 836, Springer, Berlin - New York, 1980, 91-128.
27. Watts J., The calculus on subcartesian spaces, M.Sc. Thesis, University of Calgary, Canada, 2006.
28. Watts J., Diffeologies, differential spaces, and symplectic geometry, Ph.D. Thesis, University of Toronto, Canada, 2012.
29. Watts J., The orbit space and basic forms of a proper Lie groupoid, arXiv:1309.3001.
30. Watts J., Wolbert S., Diffeology: a concrete foundation for stacks, arXiv:1406.1392.