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.