Akhmet'ev has some preprints leading to a theorem that θ_{j} 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.  
This document gives a complete proof and some explicit computations of some of the numbers referred to in the theorem. 

(January 18, 2010, 6 pages) by Peter S.Landweber.  
Akhmet'ev's arXiv preprints, some of which are listed below.  
Geometric approach towards stable homotopy groups of
spheres. I. The Hopf Invariant (English, 48 pages, August, 2011), P. M. Akhmetiev.

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 ArfKervaire
invariant.

Geometric approach towards stable homotopy groups of spheres II. Kervaire Invariant, Janaury, 2012 version (English, 104 pages), P. M. Akhmetiev. It was the subject of the lecture he gave at the Edinburgh conference of April, 2011.  This is his main
paper. "We prove that for sufficiently large n, n = 2^{j}  2, an arbitrary skewframed immersion in Euclidean nspace R^{n} has zero Kervaire invariant. Additionally, for j >= 12 (i.e., for n >= 4094) an arbitrary skewframed immersion in Euclidean n space R^{n} has zero Kervaire invariant if this skewframed immersion admits a compression of order 16." My commentary on Landweber's partial translation, July 8, 2009. 
Earlier published papers  
Codimension one immersions and the Kervaire invariant one problem, Peter J. Eccles, 1981.  This 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.  Abstract: Browder proved that the Kervaire invariant of a framed manifold of dimension n=4k+2 vanishes if n+2 is not a power of 2. We give a geometric proof using a characterization of it in terms of multiple points of immersions. 
The relationship between framed bordism and skewframed 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. 