d will be denoted by $L^{n-4k}_z$. The structure group of the framing of this component is $\I_{2,z}$. Lemma 1 is proved.

### The last part of the proof of the Theorem 1 {#the-last-part-of-the-proof-of-the-theorem-1 .unnumbered}

Let us construct a pair of polyhedra $(P',Q') \subset \R^n$, $dim(P') =2s-n=n-2k-q-2$, $dim(Q') = dim(P')-1$. Obviously, $dim(P')< 2k -1$. Take a generic mapping $d': \RP^s \to \R^n$. Let us consider the submanifold with boundary $(\Delta'^{reg},
\partial \Delta'^{reg}) \subset \R^n$ (see the denotation in Lemma 1). Let $\eta_{\Delta'^{reg}}: (\Delta'^{reg},\partial
\Delta'^{reg}) \to (K(\D_4,1),K(\I_b,1))$ be the classifying mapping for the double point self-intersection manifold of $d$.

By a standard argument we may modify the mapping $d$ into $d'$ such that the mapping $\eta_{\Delta^{reg}}$ is a homotopy equivalence of pairs up to the dimension $q+1$. After this modification $d' \to d$ we define $(P,Q)=(\Delta^{reg},\partial \Delta^{reg}) \subset \R^n$ and the mapping $\eta_{\Delta^{reg}}$ is a $(q+1)$-homotopy equivalence.

The subpolyhedron $Q$ is equipped with two cohomology classes $\kappa_{Q,1}, \kappa_{Q,2} \in H^1(Q;\Z/2)$. Because $\Sigma$ is a submanifold in $\RP^s$, the restriction of the characteristic class $\kappa \in H^1(\RP^s;\Z/2)$ to $H^1(\Sigma;\Z/2)$ is well-defined. The inclusion $i_Q: Q \subset U_{\Sigma}$ determines the cohomology class $(i_Q)^{\ast}(\kappa) \in H^1(Q;\Z/2)$. The cohomology class $\kappa_{Q,1}$ is defined as the characteristic class of the canonical double points covering over $\Sigma$. The class $\kappa_{Q,2}$ is defined by the formula $\kappa_{Q,2} = (i_Q)^{\ast}(\kappa) + \kappa_{Q,1}$.

The immersed manifold (with boundary) $(N^{n-2k} \cap U_{\Sigma})
\looparrowright U_{\Sigma}$ is equipped with an $\I_b$-framing. Obviously the classes $\kappa_{Q,1}, \kappa_{Q,2} \in
H^1(U_{\Sigma};\Z/2)=H^1(Q;\Z/2)$ restricted to $H^1(g_2(N^{n-2k}_{ext});\Z/2)$ ( recall that $g_2(N^{n-2k}_{ext})=g_2(N^{n-2k}) \cap (\R^n \setminus
U_{\Delta})$) agree with t