Geometric approach towards stable homotopy groups of spheres. The Kervaire invariant II
arXiv:0801.1417
Abstract
The notion of the geometrical --control of self-intersection of a skew-framed immersion and the notion of the -structure (the cyclic structure) on the self-intersection manifold of a $\D_4$-framed immersion are introduced. It is shown that a skew-framed immersion , (in the -range) admits a geometrical --control if the characteristic class of the skew-framing of this immersion admits a retraction of the order , i.e. there exists a mapping $κ_0: M^{\frac{3n+q}{4}} \to \RP^{\frac{3(n-q)}{4}}$, such that this composition $I \circ κ_0: M^{\frac{3n+q}{4}} \to \RP^{\frac{3(n-q)}{4}} \to \RP^{\infty}$ is the characteristic class of the skew-framing of . Using the notion of -control we prove that for a sufficiently great , , an arbitrary immersed $\D_4$-framed manifold admits in the regular cobordism class (modulo odd torsion) an immersion with a -structure. In the last section we present an approach toward the Kervaire Invariant One Problem.