S. K. Proof and M. , We assume that the Killing foliation F is induced by an orientable action ? : G × M ? M with G connected. We fix a tamer K. We prove P(M) by induction on depth

J. A. Alvarez-lópez, On the first secondary invariant of Molino's central sheaf, Ann. Polon. Math, vol.64, pp.253-265, 1996.

W. M. Boothby, An introduction to differentiable manifolds and Riemannian geometry, of Pure and Applied Mathematics, 1986.

R. Bott and L. W. Tu, Differential forms in algebraic topology, Graduate Texts in Mathematics, vol.82, 1982.
DOI : 10.1007/978-1-4757-3951-0

G. E. Bredon, Introduction to compact transformation groups, Pure and Applied Mathematics, vol.46, 1972.

G. E. Bredon, Topology and geometry, volume 139 of Graduate Texts in Mathematics, 1997.

J. Brylinski, Equivariant intersection cohomology, Kazhdan-Lusztig theory and related topics, pp.5-32, 1989.
DOI : 10.1090/conm/139/1197827

Y. Carrì-ere, Flots riemanniens. Astérisque, Transversal structure of foliations, issue.116, pp.31-52, 1982.

A. Kacimi-alaoui and G. Hector, D??composition de Hodge basique pour un feuilletage riemannien, Annales de l???institut Fourier, vol.36, issue.3, pp.207-227, 1986.
DOI : 10.5802/aif.1066

M. Goresky and R. Macpherson, Intersection homology theory, Topology, vol.19, issue.2, pp.135-162, 1980.
DOI : 10.1016/0040-9383(80)90003-8

URL : http://doi.org/10.1016/0040-9383(80)90003-8

M. Goresky and R. Macpherson, Intersection homology II, Inventiones Mathematicae, vol.12, issue.1, pp.77-129, 1983.
DOI : 10.1007/BF01389130

URL : http://www.digizeitschriften.de/download/PPN356556735_0072/PPN356556735_0072___log10.pdf

F. W. Kamber and P. Tondeur, Duality theorems for foliations. Astérisque, Transversal structure of foliations, vol.116, pp.108-116, 1982.

S. Kobayashi, Transformation groups in differential geometry, Classics in Mathematics, 1995.
DOI : 10.1007/978-3-642-61981-6

X. Masa, Duality and minimality in Reimannian foliations, Commentarii Mathematici Helvetici, vol.67, issue.1, pp.17-27, 1992.
DOI : 10.1007/BF02566486

V. Miquel and R. Wolak, Minimal singular Riemannian foliations, Comptes Rendus Mathematique, vol.342, issue.1, pp.33-36, 2006.
DOI : 10.1016/j.crma.2005.10.031

P. Molino, Desingularisation Des Feuilletages Riemanniens, American Journal of Mathematics, vol.106, issue.5, pp.1091-1106, 1984.
DOI : 10.2307/2374274

P. Molino, Feuilletages riemanniens réguliers et singuliers, Géométrie différentielle, pp.173-201, 1986.

P. Molino, Riemannian foliations Translated from the French by Grant Cairns, Progress in Mathematics. Birkhäuser Boston, Inc, vol.73, 1988.

P. Molino, Orbit-like foliations, Geometric study of foliations, pp.97-119, 1993.

P. Molino and V. Sergiescu, Deux remarques sur les flots riemanniens, Manuscripta Mathematica, vol.36, issue.4, pp.145-161, 1985.
DOI : 10.1007/BF01168350

J. Ortega and T. S. Ratiu, Momentum maps and Hamiltonian reduction, Progress in Mathematics, vol.222, 2004.
DOI : 10.1007/978-1-4757-3811-7

D. Poguntke, Dense Lie Group Homomorphisms, Journal of Algebra, vol.169, issue.2, pp.625-647, 1994.
DOI : 10.1006/jabr.1994.1300

B. L. Reinhart, Foliated Manifolds with Bundle-Like Metrics, The Annals of Mathematics, vol.69, issue.1, pp.119-132, 1959.
DOI : 10.2307/1970097

J. and R. Prieto, Estudio cohomológico de Flujos Riemannianos, 2003.

J. R. Prieto, M. Saralegi-aranguren, and R. Wolak, Top-Dimensional Group of the Basic Intersection Cohomology for Singular Riemannian Foliations, Bulletin of the Polish Academy of Sciences Mathematics, vol.53, issue.4, pp.429-440, 2005.
DOI : 10.4064/ba53-4-8

URL : https://hal.archives-ouvertes.fr/hal-00371682

J. R. Prieto, M. Saralegi-aranguren, and R. Wolak, Tautness for riemannian foliations on non-compact manifolds, manuscripta mathematica, vol.75, issue.2, pp.177-200, 2008.
DOI : 10.1007/s00229-008-0172-0

URL : https://hal.archives-ouvertes.fr/hal-00371685

J. R. Prieto, M. Saralegi-aranguren, and R. Wolak, Cohomological tautness for Riemannian foliations. Russ, J. Math. Phys, vol.16, issue.3, pp.450-466, 2009.
URL : https://hal.archives-ouvertes.fr/hal-00371679

M. Saralegi, Homological properties of stratified spaces, Illinois J. Math, vol.38, issue.1, pp.47-70, 1994.
URL : https://hal.archives-ouvertes.fr/hal-00439195

M. Saralegi-aranguren and R. Wolak, Basic intersection cohomology of conical fibrations, Mathematical Notes, vol.9, issue.3, pp.235-257, 2005.
DOI : 10.1007/s11006-005-0022-2

M. Saralegi-aranguren and R. Wolak, The BIC of a singular foliation defined by an abelian group of isometries, Annales Polonici Mathematici, vol.89, issue.3, pp.203-246, 2006.
DOI : 10.4064/ap89-3-1

URL : https://hal.archives-ouvertes.fr/hal-00111281

M. Saralegi-aranguren and R. Wolak, Finiteness of the basic intersection cohomology of a Killing foliation, Mathematische Zeitschrift, vol.89, issue.1-2, pp.443-457, 2012.
DOI : 10.1007/s00209-011-0942-3

V. Sergiescu, Cohomologie basique et dualit?? des feuilletages riemanniens, Annales de l???institut Fourier, vol.35, issue.3, pp.137-158, 1985.
DOI : 10.5802/aif.1022

P. Tondeur, Geometry of foliations, Monographs in Mathematics. Birkhäuser Verlag, vol.90, 1997.
DOI : 10.1007/978-3-0348-8914-8