Mike Hill, Mike Hopkins and I have recently solved the Arf-Kervaire invariant problem.
Our main theorem states that θj does not exist for j > 6. θj is a hypothetical element of order 2 in the stable homotopy groups of spheres in the (2j+1-2)-stem. It was previously known to exist for j < 6. The status of θ6 (in the 126-stem) remains open.
Talks since April, 2009
It was accepted for publication on September 30, 2015.
Preprint of May 25, 2015
ON THE NON-EXISTENCE OF ELEMENTS OF KERVAIRE INVARIANT ONE (221 pages)
ON THE NON-EXISTENCE OF ELEMENTS OF KERVAIRE INVARIANT ONE (220 pages)
ON THE NON-EXISTENCE OF ELEMENTS OF KERVAIRE INVARIANT ONE (158 pages)
ON THE NON-EXISTENCE OF ELEMENTS OF KERVAIRE INVARIANT ONE (99 pages)
Three expository accounts
THE ARF-KERVAIRE INVARIANT PROBLEM IN ALGEBRAIC TOPOLOGY: INTRODUCTION,
CDM Conference Harvard, 2009.s
THE ARF-KERVAIRE INVARIANT PROBLEM IN ALGEBRAIC TOPOLOGY: SKETCH OF THE PROOF,
CDM Conference Harvard, 2010.
THE ARF-KERVAIRE 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
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,
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
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
M. Akhmet'ev's work on the Arf-Kervaire invariant
Akhmet'ev has some preprints leading to a theorem that θj 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.
- Harvard-MIT 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
- Links to "Kervaire invariant" on MathSciNet, Google and Google Scholar.
- Scientific American article Hypersphere
Exotica: Kervaire Invariant Problem Has a Solution! of August,
- Simons Foundation article
Mathematicians solve 45-year-old Kervaire invariant puzzle of July
- Nature News article Hidden
riddle of shapes solved of May 1, 2009.
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Created April 30, 2009.
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