de Rham intersection cohomology for general perversities
Martintxo Saralegi-Aranguren · Illinois Journal of Mathematics · 2005
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 \[ \ib{X}{\per{p}} = \bigcup_{ S \preceq S_{1} \atop \per{p} (S_{1})< 0}S = \bigcup_{ \per{p} (S)< 0} \overline{S}. \] 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.