Let
I
n
i
I_n^i
be the set of
mod
-
2
\bmod {\text { - }}2
characteristic classes which are of dimension i, and they are zero for all n-dimensional smooth manifolds. Let
I
n
,
k
i
I_{n,k}^i
be the set of i-dimensional
mod
-
2
\bmod {\text { - }}2
characteristic classes which are zero for all n-dimensional smooth manifolds which immerse in codimension k, (we are talking about normal characteristic classes). Let K be the (graded) ideal in
H
∗
(
B
O
,
Z
2
)
{H^ \ast }(BO,{Z_2})
generated by
w
k
+
1
,
w
k
+
2
,
…
{w_{k + 1}},{w_{k + 2}}, \ldots
. Then if
i
⩽
(
n
+
k
)
/
2
i \leqslant (n + k)/2
, we have
I
n
,
k
i
=
I
n
i
+
K
i
I_{n,k}^i = I_n^i + {K^i}
. We have some related results for imbedded manifolds, and also for manifolds which immerse or imbed with an SO, U, SU, Spin, etc. structure on the normal bundle.