Relations among characteristic classes of $n$-manifolds imbedded in $\textbf {R}^{n+k}$

Abstract:

Let ${I_n} \subseteq {H^ \ast }(BO;{Z_2})$ be the (graded) set of those normal characteristic classes which are zero on all compact, closed ${C^\infty }$ manifolds. Let ${I_{n,k}} \subseteq {H^ \ast }(BO;{Z_2})$ be the set of those characteristic classes which are zero on all n-manifolds which imbed in ${R^{n + k}}$. Let K be the (graded) ideal in ${H^ \ast }(BO;{Z_2})$ generated by the Stiefel-Whitney classes ${w_k},{w_{k + 1}},{w_{k + 2}},{w_{k + 3}}, \ldots$. We will prove the following result: If $1 \leqslant i \leqslant \min \{ (2k - 2),(n + k - 1)/2\}$, then $I_{n,k}^i = I_n^i + {K^i}$ . Also, we will prove an analogous result for manifolds with an extra structure.