  In this paper configuration spaces of smooth manifolds are considered. The
accent is made on actions of certain groups (mostly $p$-tori) on this spaces by
permuting their points. For such spaces the cohomological index, the genus in
the sense of Krasnosel'skii-Schwarz, and the equivariant
Lyusternik-Schnirelmann category are estimated from below, and some corollaries
for functions on configuration spaces are deduced.
