Imbeddings, Immersions, and Characteristic Classes of Differentiable Manifolds

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Let $I_n^i$ be the set of $\bmod {\text { - }}2$ characteristic classes which are of dimension i, and they are zero for all n-dimensional smooth manifolds. Let $I_{n,k}^i$ be the set of i-dimensional $\bmod {\text { - }}2$ 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 ${H^ \ast }(BO,{Z_2})$ generated by ${w_{k + 1}},{w_{k + 2}}, \ldots$. Then if $i \leqslant (n + k)/2$, we have $I_{n,k}^i = I_n^i + {K^i}$. 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.