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Petr M. Akhmet'ev

Publications and source records attributed to Petr M. Akhmet'ev.

14 recordsLinked to original sources
arXiv2025

Geometric approach to stable homotopy groups of spheres II; Arf-Kervaire Invariants

The Kervaire Invariant 1 Problem until recently was an open problem in algebraic topology. Hill-Hopkins-Ravenel theorem clams a negative solution of the problem for all dimensions $n=2^l-2$, $l \ge 8$. We prove the statement of Hill-Hopkins-Ravenel theorem for all dimensions $2^l-2$, $l \ge l_0$, where $l_0$ is a sufficiently great positive integer. The proof is based on the Hirsh control principle and the Compression theorem by the author. A notion internal symmetry: of Abelian (for skew-framed immersions), bi-cyclic (for $\mathbb{Z}/2^{[3]}$-framed immersions) and quaternion-cyclic structure (for $\mathbb{Z}/2^{[4]}$-framed immersions) are introduced.

math.AT↗
arXiv2024

Minimum Quadratic Helicity States

Building on previous results on the quadratic helicity in magnetohydrodynamics (MHD) we investigate particular minimum helicity states. Those are eigenfunctions of the curl operator and are shown to constitute solutions of the quasi-stationary incompressible ideal MHD equations. We then show that these states have indeed minimum quadratic helicity.

physics.flu-dyn↗
arXiv2023

A generalized Kervaire Problem in stable homotopy groups of spheres

We will give an elementary self-contained description of the Mahowald element in the stable homotopy group of spheres $Π_{2^l}$, $l \ge 3$. Using this construction, we prove that a generalized Kervaire Problem, formulated by the author in \cite{A} is solved positively.

math.AT↗
arXiv2017

Calculations for the Practical Applications of Quadratic Helicity in MHD

For the quadratic helicity $χ^{(2)}$ we present a generalization of the Arnol'd inequality which relates the magnetic energy to the quadratic helicity, which poses a lower bound. We then introduce the quadratic helicity density using the classical magnetic helicity density and its derivatives along magnetic field lines. For practical purposes we also compute the flow of the quadratic helicity and show that for an $α^2$-dynamo setting it coincides with the flow of the square of the classical helicity. We then show how the quadratic helicity can be extended to obtain an invariant even under compressible deformations. Finally, we conclude with the numerical computation of $χ^{(2)}$ which show cases the practical usage of this higher order topological invariant.

physics.plasm-ph↗
arXiv2014

Geometric approach to stable homotopy groups of spheres. I. The Hopf invariant

A geometric approach to the stable homotopy groups of spheres is developed in this paper, based on the Pontryagin-Thom construction. The task of this approach is to obtain an alternative proof of the Hill-Hopkins-Ravenel theorem [H-H-R] on Kervaire invariants in all dimensions, except, possibly, a finite number of dimensions. In the framework of this approach, the Adams theorem on the Hopf invariant is studied, for all dimensions with the exception of 15, 31, 63, 127. The new approach is based on the methods of geometric topology.

math.AT↗
arXiv2013

On asymptotic higher analogs of the magnetic helicity invariant in MHD

A solution of a problem by V.I.Arnol'd about higher analog of the asymptotic Hopf invariant of divergence-free vector fields is presented. A higher invariant of magnetic fields, which is not expressed from the asymptotic linking numbers of magnetic lines is constructed and examples of an asymptotic invariants is constructed.

math.GT↗
arXiv2012

Geometric approach to stable homotopy groups of spheres II. The Kervaire invariant

A solution to the Kervaire invariant problem is presented. We introduce the concepts of abelian structure on skew-framed immersions, bicyclic structure on $\Z/2^{[3]}$--framed immersions, and quaternionic-cyclic structure on $\Z/2^{[4]}$--framed immersions. Using these concepts, we prove that for sufficiently large $n$, $n=2^{\ell}-2$, an arbitrary skew-framed immersion in Euclidean $n$-space $\R^n$ has zero Kervaire invariant. Additionally, for $\ell \ge 12$ (i.e., for $n \ge 4094$) an arbitrary skew-framed immersion in Euclidean $n$-space $\R^n$ has zero Kervaire invariant if this skew-framed immersion admits a compression of order 16.

math.AT↗
arXiv2011

Quadratic helicities and the energy of magnetic fields

Two non-local asymptotic invariants of magnetic fields for the ideal magnetohydrodynamics are introduced. The velocity of variation of the invariants for a non-ideal magnetohydrodynamics with a small magnetic dissipation is estimated. By means of the invariants the spectra of electromagnetic fields are investigated. A possible role of higher magnetic helicities during a relaxation of magnetic fields is discussed.

math-ph↗
arXiv2009

Geometric approach towards stable homotopy groups of spheres. The Hopf invariant

We develop a geometric approach to stable homotopy groups of spheres in the spirit of the work of Pontrjagin and Rokhlin. A new proof of the Hopf Invariant One Theorem by J.F.Adams is obtained in all dimensions except 15 and 31. To prove that the stable Hopf invariant H: Π_n \to Z/2 vanishes for n>31, we apply methods of geometric topology. The Pontrjagin-Thom construction along with Hirsch's compression lemma identify every α\in Π_n with the framed bordism class of a framed immersion of a closed n-manifold into R^{n+k}, for any given k>0. Its self-intersection M projects to an immersion f: M \to R^n which is framed by k copies of a line bundle κ. It is well-known that H(α) = . The self-intersection N of f is framed by k copies of a plane bundle with structure group D_4. We observe that H(α) = , where i immerses the double cover \bar N of N into M. The hardest part of the proof is to show that, after modifying f in its skew-framed bordism class, the classifying map g: N \to K(D_4,1) factors through K(Z/4,1), provided that n=2^l-1, l>5 and n-2k=15. This is achieved by analyzing immersions in the regular homotopy class of f that approximate the composition of the classifying map M \to RP^{n-k}, the projection of RP^{n-k} onto the join of copies of S^1/(Z/4) (the standard sphere), and an embedding of this join in R^n. The last step is proved with the quaternions.

math.AT↗
arXiv2008

Geometric approach towards stable homotopy groups of spheres. The Steenrod-Hopf invariant I

In this paper a geometric approach toward stable homotopy groups of spheres, based on the Pontrjagin-Thom construction is proposed. From this approach a new proof of Hopf Invariant One Theorem by J.F.Adams for all dimensions except $15,31,63,127$ is obtained. It is proved that for $n>127$ in the stable homotopy group of spheres $Π_n$ there is no elements with Hopf invariant one. The new proof is based on geometric topology methods. The Pontrjagin-Thom Theorem (in the form proposed by R.Wells) about the representation of stable homotopy groups of the real projective infinite-dimensional space (this groups is mapped onto 2-components of stable homotopy groups of spheres by the Khan-Priddy Theorem) by cobordism classes of immersions of codimension 1 of closed manifolds (generally speaking, non-orientable) is considered. The Hopf Invariant is expressed as a characteristic number of the dihedral group for the self-intersection manifold of an immersed codimension 1 manifold that represents the given element in the stable homotopy group. In the new proof the Geometric Control Principle (by M.Gromov) for immersions in a given regular homotopy classes based on Smale-Hirsch Immersion Theorem is required.

math.GT↗
arXiv2008

Geometric approach towards stable homotopy groups of spheres. The Kervaire invariant II

The notion of the geometrical $\Z/2 \oplus \Z/2$--control of self-intersection of a skew-framed immersion and the notion of the $\Z/2 \oplus \Z/4$-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 $f:M^{\frac{3n+q}{4}} \looparrowright \R^n$, $0 < q <<n$ (in the $\frac{3n}{4}+ε$-range) admits a geometrical $\Z/2 \oplus \Z/2$--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 $κ_0: M^{\frac{3n+q}{4}} \to \RP^{\frac{3(n-q)}{4}}$, 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 $f$. Using the notion of $\Z/2 \oplus \Z/2$-control we prove that for a sufficiently great $n$, $n=2^l-2$, an arbitrary immersed $\D_4$-framed manifold admits in the regular cobordism class (modulo odd torsion) an immersion with a $\Z/2 \oplus \Z/4$-structure. In the last section we present an approach toward the Kervaire Invariant One Problem.

math.GT↗