×

Morphisms between complete Riemannian pseudogroups. (English) Zbl 1147.57025

In this long, detailed and well-written paper the authors study mappings between Riemannian foliations and morphisms between Riemannian pseudogroups. To use the language of categories the authors generalize some classical notions of the theory. They introduce the notion of a morphism of pseudogroups which is more general than the étalé morphism defined by A. Haefliger in [in: A fête of topology, Pap. Dedic. Itiro Tamura, 3–32 (1988; Zbl 0667.57012)]. Elements of a morphism are required to be only continuous maps, not homeomorphisms of open subsets. This slight generalization permits to define a functor from the category of foliated spaces and their continuous foliated mappings into the category of pseudogroups and their morphisms, the holonomy functor. The authors’ interest concentrates on mappings between Riemannian foliations and on Riemannian pseudogroups, i.e. pseudogroups whose elements are isometries of some Riemannian metric, and their morphisms.
The main result of the paper is Theorem A which says that any morphism between complete Riemannian pseudogroups is complete, has a closure and is of class \(C^{0,\infty},\) i.e. of class \(C^{\infty}\) along the orbits. The theorem is a generalization of a well-known result on Lie groups saying that a continuous homomorphism of Lie groups is of class \(C^{0,\infty}.\) To supplement the theorem there are examples of non-complete morphisms between complete pseudogroups showing that the “Riemannian” assumption is essential.
To study in depth the problems of homotopy of foliated mappings of Riemannian foliations the authors introduce a so-called “strong adapted topology,” which is the same as the compact open topology if the leaf closures are compact. For this topology they prove a series of results on denseness, approximation and homotopy of \(C^{0,\infty}\)-foliated maps. As a corollary they obtain the following result:
Any foliated homotopy equivalence between transversely complete Riemannian foliations induces an isomorphism of \(E_i\), \(i \geq 2, \) of the associated spectral sequence, which, in particular, generalizes the result of A. El Kacimi Alaoui and M. Nicolau on the topological invariance of basic cohomology of Riemannian foliations on closed manifolds, [Math. Ann. 295, No. 4, 627–634 (1993; Zbl 0793.57016)].
Finally, they introduce a similar topology in the space of morphisms between Riemannian pseudogroups and prove similar properties of these morphisms.
It is a long paper divided into 31 sections. The first 15 sections are introductory. The main result is proved in Sections 16 and 17. Sections 28–23 are dedicated to foliated maps. The invariance of the spectral sequence is proved in Section 25, and Sections 26–29 are about morphisms of pseudogroups. Examples are presented in Section 30. The paper is supplemented by a section (31) devoted to open problems.

MSC:

57R30 Foliations in differential topology; geometric theory
58H05 Pseudogroups and differentiable groupoids
55R20 Spectral sequences and homology of fiber spaces in algebraic topology
53C12 Foliations (differential geometric aspects)
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Álvarez López, J. A., Duality in the spectral sequence of Riemannian foliations, Amer. J. Math., 111, 6, 905-926 (1989) · Zbl 0685.57017
[2] Álvarez López, J. A., A finiteness theorem for the spectral sequence of a Riemannian foliation, Illinois J. Math., 33, 1, 79-92 (1989) · Zbl 0644.57014
[3] J.A. Álvarez López, A. Candel, Equicontinuous foliated spaces, in preparation; J.A. Álvarez López, A. Candel, Equicontinuous foliated spaces, in preparation · Zbl 1177.53026
[4] J.A. Álvarez López, A. Candel, Generic geometry of leaves, in preparation; J.A. Álvarez López, A. Candel, Generic geometry of leaves, in preparation
[5] Álvarez López, J. A.; Masa, X. M., Morphisms of pseudogroups and foliated maps, (The Proceedings of the International Conference “Foliations 2005”. The Proceedings of the International Conference “Foliations 2005”, Łódź, Poland (2006), World Scientific: World Scientific Singapore), 1-19 · Zbl 1222.57027
[6] Arnol’d, V. I., Small denominators. I. Mapping the circle onto itself, Izv. Akad. Nauk SSSR Ser. Mat., 25, 21-86 (1961) · Zbl 0135.42603
[7] Bott, R.; Tu, L. W., Differential Forms in Algebraic Topology, Grad. Texts in Math., vol. 82 (1982), Springer: Springer New York · Zbl 0496.55001
[8] Bourbaki, N., Éléments de mathématique. Fasc. XXXVII. Groupes et algèbres de Lie. Chapitre II: Algèbres de Lie libres. Chapitre III: Groupes de Lie (1972), Hermann: Hermann Paris · Zbl 0244.22007
[9] Buffet, J.-P.; Lor, J.-C., Une construction d’un universal pour une classe assez large de \(Γ\)-structures, C. R. Acad. Sci. Paris Sér. A-B, 270, A640-A642 (1970)
[10] Candel, A.; Conlon, L., Foliations. I, Grad. Stud. in Math., vol. 23 (2000), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI · Zbl 0936.57001
[11] Colman, H., Transverse Lusternik-Schnirelmann category of Riemannian foliations, Topology Appl., 141, 1-3, 187-196 (2004) · Zbl 1052.55005
[12] Colman, H.; Hurder, S., LS-category of compact Hausdorff foliations, Trans. Amer. Math. Soc., 356, 4, 1463-1487 (2004) · Zbl 1038.55003
[13] Colman, H.; Macías-Virgós, E., Transverse Lusternik-Schnirelmann category of foliated manifolds, Topology, 40, 2, 419-430 (2001) · Zbl 0972.55002
[14] Godbillon, C., Feuilletages: Études Géométriques, Progr. Math., vol. 98 (1991), Birkhäuser: Birkhäuser Boston · Zbl 0724.58002
[15] Gromov, M., Carnot-Carathéodory spaces seen from within, (Sub-Riemannian Geometry. Sub-Riemannian Geometry, Progr. Math., vol. 144 (1996), Birkhäuser: Birkhäuser Basel), 79-323 · Zbl 0864.53025
[16] Haefliger, A., Homotopy and integrability, (Manifolds, Amsterdam, 1970. Manifolds, Amsterdam, 1970, Lecture Notes in Math., vol. 197 (1971)), 133-163 · Zbl 0215.52403
[17] Haefliger, A., Some remarks on foliations with minimal leaves, J. Differential Geom., 15, 2, 269-284 (1980) · Zbl 0444.57016
[18] Haefliger, A., Groupoïdes d’holonomie et classifiants, Astérisque, 116, 70-97 (1984) · Zbl 0562.57012
[19] Haefliger, A., Pseudogroups of local isometries, (Cordero, L. A., Differential Geometry, Santiago de Compostela, 1984. Differential Geometry, Santiago de Compostela, 1984, Res. Notes in Math., vol. 131 (1985), Pitman: Pitman Boston, MA), 174-197 · Zbl 0656.58042
[20] Haefliger, A., Leaf closures in Riemannian foliations, (Mizutani, T.; Matsumoto, Y.; Morita, S., A Fête of Topology (1988), Academic Press: Academic Press Boston, MA), 3-32 · Zbl 0667.57012
[21] Haefliger, A., Foliations and compactly generated pseudogroups, (Foliations: Geometry and Dynamics, Warsaw, 2000 (2002), World Scientific: World Scientific River Edge, NJ), 275-295 · Zbl 1002.57059
[22] Hatcher, A., Algebraic Topology (2002), Cambridge Univ. Press · Zbl 1044.55001
[23] Hector, G.; Hirsch, U., Introduction to the Geometry of Foliations, Part A, Aspects Math., vol. E1 (1981), Friedr. Vieweg & Sohn: Friedr. Vieweg & Sohn Braunschweig
[24] Hirsch, M. W., Differential Topology, Grad. Texts in Math., vol. 33 (1976), Springer: Springer New York · Zbl 0121.18004
[25] Hurder, S., Category and compact leaves, Topology Appl., 153, 2135-2154 (2006) · Zbl 1098.57016
[26] S. Hurder, P. Walczak, Compact foliations with finite transverse LS category, preprint, 2002; S. Hurder, P. Walczak, Compact foliations with finite transverse LS category, preprint, 2002 · Zbl 1404.57046
[27] El Kacimi-Alaoui, A.; Nicolau, M., On the topological invariance of the basic cohomology, Math. Ann., 295, 4, 627-634 (1993) · Zbl 0793.57016
[28] Kamber, F. W.; Tondeur, P., Foliations and metrics, (Differential Geometry, College Park, MD, 1981/1982. Differential Geometry, College Park, MD, 1981/1982, Progr. Math., vol. 32 (1983), Birkhäuser Boston: Birkhäuser Boston Boston, MA), 103-152 · Zbl 0207.22503
[29] Kechris, A. S., Classical Descriptive Set Theory, Grad. Texts in Math., vol. 156 (1995), Springer: Springer New York · Zbl 0819.04002
[30] Mac Lane, S., One universe as a foundation for category theory, (Reports of the Midwest Category Seminar, vol. III (1969), Springer: Springer Berlin), 192-200 · Zbl 0211.32202
[31] Masa, X. M., Cohomology of Lie foliations, (Differential Geometry, Santiago de Compostela, 1984. Differential Geometry, Santiago de Compostela, 1984, Res. Notes in Math., vol. 131 (1985), Pitman: Pitman Boston, MA), 211-214
[32] Masa, X. M., Duality and minimality in Riemannian foliations, Comment. Math. Helv., 67, 1, 17-27 (1992) · Zbl 0778.53029
[33] Masa, X. M., Alexander-Spanier cohomology of foliated manifolds, Illinois J. Math., 46, 4, 979-998 (2002) · Zbl 1027.57026
[34] McCleary, J., A User’s Guide to Spectral Sequences, Cambridge Stud. Adv. Math., vol. 58 (2001), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0959.55001
[35] Meigniez, G., Submersions, fibrations and bundles, Trans. Amer. Math. Soc., 354, 9, 3771-3787 (2002) · Zbl 1001.55016
[36] Molino, P.; Pierrot, M., Théorèmes de slice et holonomie des feuilletages riemanniens singuliers, Ann. Inst. Fourier (Grenoble), 37, 4, 207-223 (1987) · Zbl 0625.57016
[37] Molino, P., Riemannian Foliations, Progr. Math., vol. 73 (1988), Birkhäuser Boston: Birkhäuser Boston Boston, MA, Translated from the French by Grant Cairns. With appendices by Cairns, Y. Carrière, É. Ghys, E. Salem and V. Sergiescu
[38] Nomizu, K.; Ozeki, H., The existence of complete Riemannian metrics, Proc. Amer. Math. Soc., 12, 889-891 (1961) · Zbl 0102.16401
[39] Pears, A. R., Dimension Theory of General Spaces (1975), Cambridge Univ. Press: Cambridge Univ. Press Cambridge, England · Zbl 0312.54001
[40] Ramsay, A., Local product structure for group actions, Ergodic Theory Dynam. Systems, 11, 1, 209-217 (1991) · Zbl 0723.57029
[41] Reinhart, B. L., Foliated manifolds with bundle-like metrics, Ann. of Math. (2), 69, 119-132 (1959) · Zbl 0122.16604
[42] Salem, É., Une généralisation du théorème de Myers-Steenrod aux pseudogroupes d’isométries, Ann. Inst. Fourier (Grenoble), 38, 2, 185-200 (1988) · Zbl 0613.58041
[43] Sarkaria, K. S., A finiteness theorem for foliated manifolds, J. Math. Soc. Japan, 30, 4, 687-696 (1978) · Zbl 0398.57012
[44] Schwarz, G. W., Lifting smooth homotopies of orbit spaces, Inst. Hautes Études Sci. Publ. Math., 51, 37-135 (1980) · Zbl 0449.57009
[45] Spanier, E. H., Algebraic Topology (1981), Springer: Springer New York
[46] Stefan, P., Accessible sets, orbits, and foliations with singularities, Proc. London Math. Soc. (3), 29, 699-713 (1974) · Zbl 0342.57015
[47] Tarquini, C., Feuilletages conformes, Ann. Inst. Fourier (Grenoble), 54, 2, 453-480 (2004) · Zbl 1064.53014
[48] van Est, W. T., Rapport sur les \(S\)-atlas, Astérisque, 116, 235-292 (1984), Transversal structure of foliations (Toulouse, 1982) · Zbl 0543.58003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.