Geodesic
Geometric approach to stable homotopy groups of spheres. Kervaire invariants.~II
Fundamentalʹnaâ i prikladnaâ matematika, Tome 13 (2007) no. 8, pp. 17-41.

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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 $f\colon M^{\frac{3n+q}4}\looparrowright\mathbb R^n$, $0$ (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. 
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     author = {P. M. Akhmet'ev},
     title = {Geometric approach to stable homotopy groups of spheres. {Kervaire} {invariants.~II}},
     journal = {Fundamentalʹna\^a i prikladna\^a matematika},
     pages = {17--41},
     publisher = {mathdoc},
     volume = {13},
     number = {8},
     year = {2007},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FPM_2007_13_8_a1/}
}
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P. M. Akhmet'ev. Geometric approach to stable homotopy groups of spheres. Kervaire invariants.~II. Fundamentalʹnaâ i prikladnaâ matematika, Tome 13 (2007) no. 8, pp. 17-41. http://geodesic.mathdoc.fr/item/FPM_2007_13_8_a1/

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