(Z₂)ᵏ-actions with w(F)=1

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Suppose that (Φ, Mⁿ) is a smooth (Z₂)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 Mⁿ>2ᵏ F and each p-dimensional part Fᵖ possesses the linear independence property, then (Φ, Mⁿ) bounds equivariantly, and in particular, 2ᵏ F is the best possible upper bound of Mⁿ if (Φ, Mⁿ) is nonbounding.