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
including 2013 NWU talk
ON THE NON-EXISTENCE OF ELEMENTS OF KERVAIRE INVARIANT ONE (158 pages)
ON THE NON-EXISTENCE OF ELEMENTS OF KERVAIRE INVARIANT ONE (99 pages)
|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 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|
Return to Ravenel's home page .
Created April 30, 2009; last modified March 27, 2013.