  The many body contributions to the nuclear anapole moment of $^{133}$Cs,
$^{205}$Tl, $^{207,209}$PB, and $^{209}$Bi from the core polarization are
calculated in the random-phase approximation with the effective residual
interaction. Strong reduction of a valence nucleon contribution was found
provided by the core polarization effects. The contribution of the core
particles to the anapole moment compensates this reduction to large extent
keeping the magnitude of nuclear anapole moment close to its initial single
particle value.
