  We describe a unified approach to estimating the dimension of $f^{-1}(A)$ for
any $G$-equivariant map $f \colon X \to Y$ and any closed $G$-invariant subset
$A\subseteq Y$ in terms of connectivity of $X$ and dimension of $Y$, where $G$
is either a cyclic group of order $p^k$, a $p$-torus ($p$ a prime), or a torus.
