Let
p
:
E
→
B
p:E \to B
be an
n
n
-sphere bundle,
q
:
V
→
B
q:V \to B
be an
R
n
{{\mathbf {R}}^n}
-bundle and
f
:
E
→
V
f:E \to V
be a fibre preserving map over a paracompact space
B
B
. Let
p
¯
:
E
¯
→
B
\overline p :\overline E \to B
be the projectivized bundle obtained from
p
p
by the antipodal identification and let
A
¯
f
{\overline A _f}
be the subset of
E
¯
\overline E
consisting of pairs
{
e
,
−
e
}
\{ e, - e\}
such that
f
e
=
f
(
−
e
)
fe = f( - e)
. If the cohomology dimension
d
d
of
B
B
is finite then the map
(
p
¯
|
A
¯
f
)
∗
(\bar {p} | \overline {A}_f)^*
is injective for a continuous cohomology theory
H
∗
{H^*}
. Moreover, if the
j
j
th Stiefel-Whitney class of
q
q
is zero for
1
⩽
j
⩽
r
1 \leqslant j \leqslant r
then
(
p
¯
|
A
¯
f
)
∗
(\bar {p} | \overline {A}_f)^*
is injective in degrees
i
⩾
d
−
r
i \geqslant d - r
. If all the Stiefel-Whitney classes of
q
q
are zero then
(
p
¯
|
A
¯
f
)
∗
(\bar {p} | \overline {A}_f)^*
is injective in every degree.