s of the restriction of the given immersion $f$ ($g$) to the submanifold in $M^{n-1}$ ($N^{n-2}$) dual to $w_1(\kappa)^{k-1} \in H^{k-1}(M^{n-1};\Z/2)$ ($w_2(\eta)^{k-1} \in H^{2k-2}(N^{n-2};\Z/2)$).

Let $(g,\Xi,\eta)$ be a $\D_4$-framed (generic) immersion in the codimension $2k$. Let $h: L^{n-4k} \looparrowright \R^n$ be the immersion of the self-intersection (double points) manifold of $g$. The normal bundle $\nu_h$ of the immersion $h$ is decomposed into a direct sum of $k$ isomorphic copies of a 4-dimensional bundle $\zeta$ with the structure group $\Z/2 \int \D_4$. This decomposition is given by the isomorphism $\Psi: \nu_h \cong k \zeta$. The bundle $\nu_h$ itself is classified by the mapping $\zeta:
L^{n-4k} \to K(\Z/2 \int \D_4,1)$.

All the triples $(h,\zeta,\Psi)$ described above (we do not assume that a triple is realized as the double point manifold for a $\D_4$-framed immersion) up to the standard cobordism relation form the cobordism group $Imm^{\Z/2 \int \D_4}(n-4k,4k)$. The self-intersection of an arbitrary $\D_4$-framed immersion is a $\Z/2 \int \D_4$-framed immersed manifold and the cobordism class of this manifold well-defines the natural homomorphism $$\delta_{\D_4}^k : Imm^{\D_4}(n-2k,2k) \to Imm^{\Z/2 \int \D_4}(n-4k,4k). \eqno(6)$$

The subgroup $\D_4 \oplus \D_4 \subset \Z/2 \int \D_4$ of index 2 induces the double cover $\bar L^{n-4k} \to L^{n-4k}$. This double cover corresponds with the canonical double cover over the double point manifold.

Let $\bar \zeta: \bar L^{n-4k} \to K(\D_4,1)$ be the classifying mapping induced by the projection homomorphism $\D_4 \oplus \D_4
\to \D_4$ to the first factor. Let $\bar \zeta \to L^{n-4k}$ be the 2-dimensional $\D_4$–bundle defined as the pull-back of the universal 2-dimensional bundle with respect to the classifying mapping $\bar \zeta$.

### Definition 2 {#definition-2 .unnumbered}

The Kervaire invariant $\Theta_{\Z/2 \int \D_4}^k: Imm^{\Z/2 \int \D_4}(n-4k,4k) \to \Z/2$ for a $\Z/2
\int \D_4$-framed immersion $(h,\Psi,\zeta)$ is defined by 