Arf-Kervaire invariant problem

Return to Ravenel's home page .

Drawing by Carolyn Snaith published in 1982,
see footnote. Larger picture

The Arf invariant Arf(q) of a quadratic form q is defined
on the 2009 banknote shown above. Larger picture

A solution to the Arf-Kervaire invariant problem, April, 2009.
Mike Hill, Mike Hopkins and I have recently solved the Arf-Kervaire 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 θ₆ (in the 126-stem) remains open.

Preprint of August 26, 2009
ON THE NON-EXISTENCE OF ELEMENTS OF KERVAIRE INVARIANT ONE
arXiv link


Mike Hill	Mike Hopkins

Here are some relevant links.

Our result was first announced on April 21, 2009, in a lecture by Hopkins at the Geometry and Physics: Atiyah80 conference. (This link has some more information.) 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.
More talks we have given since April.
A collection of earlier papers on the problem.
Victor Snaith's new book Stable Homotopy Around the Arf-Kervaire 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 Arf-Kervaire 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 Arf-Kervaire 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 Arf-Kervaire invariant problem.
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.
Harvard-MIT Summer 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 45-year-old 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; last modified November 22, 2009.

Footnote: Jig/reel image, taken from page xiii of Current trends in algebraic topology. Part 1 (Proceedings of the Conference held at the University of Western Ontario, London, Ont., June 29--July 10, 1981, edited by Richard M. Kane, Stanley O. Kochman, Paul S. Selick and Victor P. Snaith), rediscovered by Andy Baker and scanned by Bob Bruner.