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Petr M. Akhmet'ev's work on the Arf-Kervaire invariant problem, June, 2009.

Akhmet'ev has some preprints leading to a theorem that θj does not exist for large j. The key papers are in Russian with shorter summaries in English. His approach is quite different from ours, and uses a geometric method suggested by some results of Eccles.

This page provides some links and discussion. Suggestions for additional material are welcome.



A geometric solution to the Kervaire Invariant One problem,
slides for a talk given by him at Princeton on May 19, 2009.
My understanding of Akhmet'ev's program,
a two page synopsis.
Akhmet'ev's Compression Theorem (6 pages, July 8)
This result is stated without proof on page 19 of Akhmet'ev's slides and at the end of my overview document. It is referred to there as the "desuspension theorem", and in some versions of the paper as the "retraction theorem". It is critical to determining which θjs can be excluded by his method. The proof given in his 2009 paper (in Russian) is over 20 pages.
This document gives a complete proof and some explicit computations of some of the numbers referred to in the theorem.
Akhmet'ev's arXiv preprints, some of which are listed below.
Geometric approach towards stable homotopy groups of spheres II. Kervaire Invariant (Russian, 81 pages, May 2009) June, 2009 version (Russian, 85 pages), P. M. Akhmetiev. Partial translation (July 21 edition, 5 sections with comments on the remaing three, 30 pages) by Peter Landweber. Here is his translation of the abstract with some comments. This is his main paper.
"It is proved that there exists an integer L such that a framed manifold of dimension 2l-2, l >= L has the trivial Kervaire Invariant."
My commentary on Landweber's partial translation, July 8, 2009.
Geometric approach towards stable homotopy groups of spheres. The Steenrod-Hopf Invariant (English, 16 pages, January 2008), P. M. Akhmetiev. Russian version, 99 pages, May 2009.
The purpose of this paper is to reprove a weaker version Adams' Hopf invariant one theorem using geometric methods similar to those used to deal with the Arf-Kervaire invariant.
  • My commentary on the English version of this paper.
  • On June 22, Akhmet'ev told me that Lemma 3 in the English version was incorrect and that the problem had been corrected in the subsequent Russian version.
Geometric approach towards stable homotopy groups of spheres. Kervaire Invariant (English, 27 pages, January, 2008), P. M. Akhmetiev.
Earlier published papers
Codimension one immersions and the Kervaire invariant one problem, Peter J. Eccles, 1981. The paper provides the basis of Akhmet'ev's program.
A geometrical proof of Browder's result on the vanishing of the Kervaire invariant, (English, 6 pages, 1998)  Pyotr M. Akhmetiev and Peter J. Eccles.
The relationship between framed bordism and skew-framed bordism, (English, 9 pages, 2005), Pyotr M. Akhmetiev and Peter J. Eccles
Geometric approach towards stable homotopy groups of spheres. Kervaire Invariants. II (English translation of a paper published in 2007, 16 pages), P. M. Akhmetiev. My commentary on this paper.