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.
Tentative title: Equivariant stable homotopy theory and the Kervaire invariant This is a draft of a work in progress. Please do not quote in public. Latest revision: December 1, 2019. READ AT YOUR OWN RISK. The color of this book has yet to be determined. Video (slides) of Hopkins' lecture announcing the theorem on April 21, 2009, in Edinburgh, at the conference in honor of Sir Michael Atiyah's 80th birthday Other talks since April, 2009 It was accepted for publication on September 30, 2015, and appeared in print the July, 2016 issue. MathSciNet review of the above by Paul Goerss. 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 
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 
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Created April 30, 2009.
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