TITLE: Is intersection homology the usual homology of something else?
QUESTION [15 upvotes]: Let $X$ be the sort of topological space for which it makes sense to talk about the intersection homology.  Fix a perversity $p$, or just take $p= 1/2$ if you like. 


Is there some naturally defined $X'$ such that ${}^p IH_* (X) = H_* (X')$ ?

REPLY [9 votes]: Such a space $X'$ cannot possibly depend functorially on $X$: suppose that there is a functor $F$ from the category of topological spaces (that have reasonable stratifications) and continuous maps to itself such that IH(X)=H(F(X)).  Then IH would be functorial, but considering the example 
$$(S^1 \hookrightarrow X \rightarrow S^1)=id_{S^1}$$ where $X$ is the pinched torus and $S^1$ is the non-collapsed circle (see pages 55 and 56 of Kirwan-Woolf "Introduction to Intersection Homology") would imply that the intersection homology of $S^1$ (=ordinary homology of $S^1$) imbeds in the intersection homology of $X$.  But the first IH group of $X$ (middle perversity) is $0$.