Geometric approach to stable homotopy groups of spheres. Kervaire invariants. II

We present an approach to the Kervaire-invariant-one problem. The notion of the geometric $(\mathbb Z/2\oplus\mathbb Z/2)$-control of self-intersection of a skew-framed immersion and the notion of the $(\mathbb Z/2\oplus\mathbb Z/4)$-structure on the self-intersection manifold of a $\mathbf D_4$-framed immersion are introduced. It is shown that a skew-framed immersion $f\colon M^{\frac{3n+q}4}\looparrowright\mathbb R^n$, $0<q\ll n$ (in the $(\frac{3n}4+\varepsilon)$-range) admits a geometric $(\mathbb Z/2\oplus\mathbb Z/2)$-control if the characteristic class of the skew-framing of this immersion admits a retraction of the order $q$, i.e., there exists a mapping $\kappa_0\colon M^{\frac{3n+q}4}\to\mathbb R\mathrm P^{\frac{3(n-q)}4}$ such that this composition $I\circ\kappa_0\colon M^{\frac{3n+q}4}\to\mathbb R\mathrm P^{\frac{3(n-q)}4}\to\mathbb R\mathrm P^\infty$ is the characteristic class of the skew-framing of $f$. Using the notion of $(\mathbb Z/2\oplus\mathbb Z/2)$-control, we prove that for a sufficiently large $n$, $n=2^l-2$, an arbitrary immersed $\mathbf D_4$-framed manifold admits in the regular cobordism class (modulo odd torsion) an immersion with a $(\mathbb Z/2\oplus\mathbb Z/4)$-structure.