$({\mathbb Z}_2)^k$-actions with $w(F)=1$

Suppose that $(\Phi, M^n)$ is a smooth $({\mathbb Z}_2)^k$-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.