I'm learning about intersection homology, and I'm trying to write a proof of the following statement:
Take X,Y two filtered spaces with perversities $p$ and $q$ resp. , a continuous function $f:X\to Y$ such that for all stratum $S$ of Y, there is a strata $S^f$ of Y such that $f(S)\subset S^f$ and $codim_Y(S^f)\leq codim_X(S)$. Then if for all stratum $S$ of X,we have : $$p(S)\leq q(S^f)$$ Then $f$ induces $$C_*^p(X)\to C_*^q(Y), (\sigma:\Delta^n \to X)\mapsto (f \circ \sigma:\Delta^n \to Y)$$ And also we have in homology : $$f_*: I^pH_*(X)\to I^qH_*(Y) $$
First I tried to show that a p-admissible singular filtered simplex is sent to a q-admissible one, but I have a hard time doing this first step: I took a strata $T$ of Y such that $(f\circ \sigma)\cap T\neq \emptyset$ but then I don't know how to find a link between $T$ and the stratas $S^f$. Can someone help me?