What is the meaning of the symbol $\pitchfork$? I have seen this symbol in the formulation of this question. There, it is said: > Let $f: X \to Y$ be a smooth map with $f \pitchfork Z$. I was googling, but I haven't found any answers.

It means transversality. Spesifically, that the image of $f$ is transverse to $Z$. See <

Two submanifolds $U$ and $V$ of a manifold $M$ are called transversal at $x\in Z$ if $$T_xU+T_xV=T_xM$$ and simply transversal if they are transversal at all intersection points. Note that the sum is not direct. We identify $T_xU$ and $T_xV$ by the appropriate subspaces of $T_xM$.