De Rham intersection cohomology for general perversities

Details De Rham intersection cohomology for general perversities De Rham intersection cohomology for general perversities

2004-04-06

arxiv.org/abs/math/0404130v2

Illinois Journal of Mathematics 49 (2005) 737-758

Mathematics

Algebraic Topology

Scientific paper

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}\bar{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.

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