Mike Hill, Mike Hopkins and I have recently solved the ArfKervaire invariant problem. Our main theorem states that θ_{j} does not exist for j > 6. θ_{j} is a hypothetical element of order 2 in the stable homotopy groups of spheres in the (2^{j+1}2)stem. It was previously known to exist for j < 6. The status of θ_{6} (in the 126stem) remains open. Talks since April, 2009 ON THE NONEXISTENCE OF ELEMENTS OF KERVAIRE INVARIANT ONE (221 pages) ON THE NONEXISTENCE OF ELEMENTS OF KERVAIRE INVARIANT ONE (220 pages) ON THE NONEXISTENCE OF ELEMENTS OF KERVAIRE INVARIANT ONE (158 pages) arXiv link ON THE NONEXISTENCE OF ELEMENTS OF KERVAIRE INVARIANT ONE (99 pages) arXiv link Three expository accounts THE ARFKERVAIRE INVARIANT PROBLEM IN ALGEBRAIC TOPOLOGY: INTRODUCTION, CDM Conference Harvard, 2009.s THE ARFKERVAIRE INVARIANT PROBLEM IN ALGEBRAIC TOPOLOGY: SKETCH OF THE PROOF, CDM Conference Harvard, 2010. THE ARFKERVAIRE INVARIANT PROBLEM IN ALGEBRAIC TOPOLOGY, Gokova Conference, 2010.


Mike Hill, myself and Mike Hopkins 
Photo taken by Bill Browder, February 11, 2010 
 Our result was first announced on April 21, 2009,
in a lecture by Hopkins at
the Geometry and
Physics: Atiyah80 conference. (Here
is another
link to the conference.) His title was Applications of algebra
to a problem in topology. Here are
the slides for that talk; the file is 44MB. There is
also a video.
Here is a streaming video of two talks he gave in Kyoto a few months later.  My graduate course on this topic, Spring 2010,
including
handwritten
lecture notes.
 Victor
Snaith's new book
Stable Homotopy Around the ArfKervaire Invariant (2009)
was written shortly before we solved the problem and thus says nothing
about our work. It does give a great deal of historical background.
He has followed up with a survey
article The ArfKervaire
invariant of framed manifolds which comments briefly on our work
and Akhmeti'ev's.
From his preface:As ideas for progress on a particular mathematics problem atrophy it can disappear. Accordingly I wrote this book to stem the tide of oblivion.
For a brief period overnight we were convinced that we had the method to make all the sought after framed manifolds  a feeling which must have been shared by many topologists working on this problem. All in all, the temporary high of believing that one had the construction was sufficient to maintain in me at least an enthusiastic spectator's interest in the problem.
In the light of the above conjecture and the failure over fifty years to construct framed manifolds of ArfKervaire invariant one this might turn out to be a book about things which do not exist. This [is] why the quotations which preface each chapter contain a preponderance of utterances from the pen of Lewis Carroll.Victor Snaith and William Browder in 1981  Petr
M. Akhmet'ev's work on the ArfKervaire invariant
problem.
Akhmet'ev has some preprints leading to a theorem that θ_{j} does not exist for large j. The two key papers are in Russian with shorter summaries in English (early 2008) and English translations provided by the author (December, 2009). They have been carefully studied by Peter Landweber. His approach is quite different from ours, and uses a geometric method suggested by some results of Peter Eccles. This page provides some links and discussion. Suggestions for additional material are welcome.
 HarvardMIT Summer (2009) Seminar on the Kervaire
Invariant. This page contains lecture notes and links to some
other papers on the problem by Mark
Mahowald and Fred
Cohen.
 Links to "Kervaire invariant" on MathSciNet, Google and Google Scholar.
 Scientific American article Hypersphere
Exotica: Kervaire Invariant Problem Has a Solution! of August,
2009.
 Simons Foundation article
Mathematicians solve 45yearold Kervaire invariant puzzle of July
20, 2009.
 Nature News article Hidden
riddle of shapes solved of May 1, 2009.
Return to Ravenel's home page .
Created April 30, 2009.
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