(ℤ₂)^{𝕜}-actions with 𝕨(𝔽)=1

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