  Suppose that $(\Phi, M^n)$ is a smooth $({\Bbb Z}_2)^k$-action on a closed
smooth $n$-dimensional manifold such that all Stiefel-Whitney classes of the
tangent bundle to each connected component of the fixed point set $F$ vanish in
positive dimension. This paper shows that if $\dim M^n>2^k\dim F$ and each
$p$-dimensional part $F^p$ possesses the linear independence property, then
$(\Phi, M^n)$ bounds equivariantly, and in particular, $2^k\dim F$ is the best
possible upper bound of $\dim M^n$ if $(\Phi, M^n)$ is nonbounding.
