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.
Categories: math.AT