De Rham intersection cohomology for general perversities

Martintxo Saralegi-Aranguren (1)

1 LML - Laboratoire de Mathématiques de Lens

For a stratified pseudomanifold $X$, we have the de Rham Theorem $ \\lau{\\IH}{*}{\\per{p}}{X} = \\lau{\\IH}{\\per{t} - \\per{p}}{*}{X}, $ for a perversity $\\per{p}$ verifying $\\per{0} \\leq \\per{p} \\leq \\per{t}$, where $\\per{t}$ denotes the top perversity. We extend this result to any perversity $\\per{p}$. In the direction cohomology $\\mapsto$ homology, we obtain the isomorphism $$ \\lau{\\IH}{*}{\\per{p}}{X} = \\lau{\\IH}{\\per{t} -\\per{p}}{*}{X,\\ib{X}{\\per{p}}}, $$ where $ {\\displaystyle \\ib{X}{\\per{p}} = \\bigcup_{ S \\preceq S_{1} \\atop \\per{p} (S_{1})< 0}S = \\bigcup_{ \\per{p} (S)< 0}\\overline{S}.} $ In the direction homology $\\mapsto$ cohomology, we obtain the isomorphism $$ \\lau{\\IH}{\\per{p}}{*}{X}=\\lau{\\IH}{*}{\\max ( \\per{0},\\per{t} -\\per{p})}{X}. $$ In our paper stratified pseudomanifolds with one-codimensional strata are allowed.

Mots clés: Intersection cohomology; Stratifications

Domaines: Topologie algébrique [math.AT]

Martintxo Saralegi-Aranguren. De Rham intersection cohomology for general perversities. Illinois Journal of Mathematics, 2005, 49, pp.737-758. ⟨hal-00001421v2⟩