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DIFFERENTIAL FORMS ON STRATIFIED SPACES

Part of: General theory of differentiable manifolds

Published online by Cambridge University Press:  13 July 2018

SERAP GÜRER Open the ORCID record for SERAP GÜRER [Opens in a new window]  and
PATRICK IGLESIAS-ZEMMOUR Open the ORCID record for PATRICK IGLESIAS-ZEMMOUR [Opens in a new window]
SERAP GÜRER*
Affiliation:
Galatasaray University, Ortaköy, Çiraǧan Cd. No. 36, 34349 Beşktaş/İstanbul, Turkey email sgurer@gsu.edu.tr
PATRICK IGLESIAS-ZEMMOUR
Affiliation:
I2M CNRS, Marseille, France The Hebrew University of Jerusalem, Israel email piz@math.huji.ac.il
*
✉serapgurer@gmail.com
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Abstract

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First, we extend the notion of stratified spaces to diffeology. Then we characterise the subspace of stratified differential forms, or zero-perverse forms in the sense of Goresky–MacPherson, which can be extended smoothly into differential forms on the whole space. For that we introduce an index which outlines the behaviour of the perverse forms on the neighbourhood of the singular strata.

Keywords

stratificationdiffeologydifferential forms

MSC classification

Primary: 58A35: Stratified sets
Secondary: 58A10: Differential forms
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 98 , Issue 2 , October 2018 , pp. 319 - 330
Copyright
© 2018 Australian Mathematical Publishing Association Inc. 

Footnotes

This research is partially supported by Tübi̇tak, Career Grant No. 115F410, Galatasaray University Research Fund Grant No. 15.504.001 and a 2017 grant of the French Embassy in Ankara, Turkey.

References

Brasselet, J.-P., Hector, G. and Saralegi, M., ‘Théorème de De Rham pour les variétés stratifiées’, Ann. Global Anal. Geom. 9(3) (1991), 211–243.Google Scholar
Brion, M., ‘Equivariant intersection cohomology of semi-stable points’, Amer. J. Math. 118(3) (1996), 595–610.Google Scholar
Brylinski, J.-L., ‘Equivariant intersection cohomology’, Prépublication de l’IHES, 1986; in Kazhdan–Lusztig Theory and Related Topics (Chicago, IL, 1989), Contemporary Mathematics, 139 (American Mathematical Society, Providence, RI, 1992), 5–32.Google Scholar
Donato, P. and Iglesias, P., ‘Exemple de groupes différentiels: flots irrationnels sur le tore’, C. R. Acad. Sci. A 301(4) (1985), 127–130.Google Scholar
Goresky, M., ‘Whitney stratified chains and cochains’, Trans. Amer. Math. Soc. 267(1) (1961), 175–196.Google Scholar
Goresky, M. and MacPherson, R., ‘La dualité de Poincaré pour les espaces singuliers’, C. R. Acad. Sci. A 284 (1977), 1549–1551.Google Scholar
Gürer, S. and Iglesias-Zemmour, P., ‘The diffeomorphisms of the square’, Blogpost, December 2016, http://math.huji.ac.il/∼piz/documents/DBlog-Rmk-DOTS.pdf.Google Scholar
Gürer, S. and Iglesias-Zemmour, P., ‘On manifolds with boundary and corners’, Preprint, 2017, http://math.huji.ac.il/∼piz/documents/OMWBAC.pdf.Google Scholar
Iglesias-Zemmour, P., Fibrations Difféologique et Homotopie, Thèse d’état, Université de Provence, Marseille, 1985. http://math.huji.ac.il/∼piz/documents/TheseEtatPI.pdf.Google Scholar
Iglesias-Zemmour, P., Diffeology, Mathematical Surveys and Monographs, 185 (American Mathematical Society, Providence, RI, 2013).Google Scholar
Iglesias-Zemmour, P., ‘Example of singular reduction in symplectic diffeology’, Proc. Amer. Math. Soc. 144(2) (2016), 1309–1324.Google Scholar
Iglesias-Zemmour, P., Karshon, Y. and Zadka, M., ‘Orbifolds as diffeology’, Trans. Amer. Math. Soc. 362(6) (2010), 2811–2831.Google Scholar
Iglesias-Zemmour, P. and Lachaud, G., ‘Espaces différentiables singuliers et corps de nombres algébriques’, Ann. Inst. Fourier (Grenoble) 40(1) (1990), 723–737.Google Scholar
Iglesias-Zemmour, P. and Laffineur, J.-P., ‘Noncommutative geometry and diffeology, the case of orbifolds’, J. Noncommut. Geom. (2017), to appear. http://math.huji.ac.il/∼piz/documents/CSAADTCOO.pdf.Google Scholar
Kloeckner, B., ‘Quelques notions d’espaces stratifiés’, Sémin. Théor. Spectr. Géom. 26 (2007–2008), 13–28.Google Scholar
Mather, J., Notes on Topological Stability, Mimeographed Lecture Notes (Harvard University, 1970).Google Scholar
Pflaum, M., Analytic and Geometric Study of Stratified Spaces, Lecture Notes in Mathematics, 1768 (Springer, Berlin–Heidelberg, 2001).Google Scholar
Pollini, G., ‘Intersection differential forms’, Rend. Sem. Math. Univ. Padova 113 (2005), 71–97.Google Scholar
Siebenmann, L., ‘Deformation of homeomorphisms on stratified sets’, Comment. Math. Helv. 47 (1972), 123–163.Google Scholar
Śniatycki, J., Differential Geometry of Singular Spaces and Reduction of Symmetry, New Mathematical Monographs, 23 (Cambridge University Press, Cambridge, 2013).Google Scholar
Souriau, J.-M., ‘Un algorithme générateur de structures quantiques’, Prétirage CPT-84/PE.1694, Centre de Physique Théorique, Luminy, 13288 Marseille cedex 9, 1984.Google Scholar
Thom, R., ‘La stabilité topologique des applications polynomiales’, Enseign. Math. 8 (1962), 24–33.Google Scholar
Whitney, H., ‘Complexes of manifolds’, Proc. Natl. Acad. Sci. USA 33 (1947), 10–11.Google Scholar