C. If, S. , S. , T. ?. Id, S. Id et al., S are two source strata of T . Thus using Property (C) for p gives

R. Two-strata-with-s-?-r, S. Singular, R. , S. , ?. Id et al., If S id is regular, we have Did * p(S) = t(S) ? p(S id ) = t(S) Since S is singular and S id is regular, we have codim S ? 2 by hypothesis. Thus, Did * p(S) ? 0. Suppose now that S id is singular. Since the map id is a stratified map, then R id is a regular stratum and codim S id ? codim S. We know that, To get (6.2), we have to verify Property (D) of Definition 6.8. Let S Thus, Property (D) for the perversity p implies

. Dp-(-s, The proof of Theorem C uses Theorem 5.1. For doing that, we need to make explicit the behavior of a conical neighborhood

J. Brylinski, Equivariant intersection cohomology, Kazhdan-Lusztig theory and related topics, Contemp. Math. Amer. Math. Soc, vol.13994, pp.5-32, 1989.

E. Sylvain, J. L. Cappell, and . Shaneson, Singular spaces, characteristic classes, and intersection homology, Ann. of Math, vol.13492, issue.2 2, pp.325-374, 1991.

D. Chataur, M. Saralegi-aranguren, and D. Tanré, Blown-up intersection cohomology, Contemp. Math. Amer. Math. Soc, vol.708, pp.45-102, 2018.
DOI : 10.1090/conm/708/14263
URL : https://hal.archives-ouvertes.fr/hal-01426143

. Soc, Poincaré duality with cap products in intersection homology MR 3758431 [6] Greg Friedman, Singular intersection homology Superperverse intersection cohomology: stratification (in)dependence) [8] , Singular chain intersection homology for traditional and super-perversities, Adv. Math. Math. Z. Trans. Amer. Math. Soc, vol.254, issue.252 1 5, pp.314-351, 2006.

G. Friedman and J. E. Mcclure, Cup and cap products in intersection (co)homology, Advances in Mathematics, vol.240, pp.383-426, 2013.
DOI : 10.1016/j.aim.2013.02.017
URL : http://arxiv.org/pdf/1106.4799

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

C. Henry and . King, Topological invariance of intersection homology without sheaves, Topology Appl, pp.149-160, 1985.

R. Macpherson, Intersection homology and perverse sheaves, Unpublished AMS Colloquium Lectures, 1991.

G. Padilla, Intersection cohomology of stratified circle actions, Illinois J. Math, vol.49, issue.2, pp.659-685, 2005.
DOI : 10.1016/j.topol.2007.06.001
URL : https://doi.org/10.1016/j.topol.2007.06.001

G. Padilla and M. Saralegi-aranguren, Intersection cohomology of the circle actions, Topology and its Applications, vol.154, issue.15, pp.2764-2770, 2007.
DOI : 10.1016/j.topol.2007.06.001
URL : https://doi.org/10.1016/j.topol.2007.06.001

J. Ignacio, R. Prieto, and M. E. Saralegi-aranguren, Equivariant intersection cohomology of the circle actions, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM, vol.108, issue.1, pp.49-62, 2014.

M. Saralegi-aranguren, de Rham intersection cohomology for general perversities, Illinois J. Math, vol.49, issue.3, pp.737-758, 2005.
URL : https://hal.archives-ouvertes.fr/hal-00001421

L. C. Siebenmann, Deformation of homeomorphisms on stratified sets, Commentarii Mathematici Helvetici, vol.47, issue.1, pp.123-136, 1972.
DOI : 10.1007/BF02566793

. Yu and . Smirnov, A necessary and sufficient condition for metrizability of a topological space, Doklady Akad, Nauk SSSR (N.S.), vol.7712, pp.197-200, 1951.

H. Edwin and . Spanier, Algebraic topology, pp.21011235-1007, 1966.

G. Valette and G. Lefschetz-duality-for-intersection-homology, A Lefschetz duality for intersection homology, Geometriae Dedicata, vol.231, issue.3???4, pp.283-299, 2014.
DOI : 10.1016/j.aim.2012.07.020
URL : https://link.springer.com/content/pdf/10.1007%2Fs10711-013-9856-z.pdf

U. Lafma, J. De-picardie, and . Verne, rue Saint-Leu, 80039 Amiens Cedex 1, France E-mail address: David.Chataur@u-picardie.fr Laboratoire de Mathématiques de Lens EA 2462, Fédération CNRS Nord-Pas-de-Calais FR2956, Université d'Artois, 62307 Lens Cedex, France E-mail address: martin.saraleguiaranguren@univ-artois, fr Département de Mathématiques