A solution to the Arf-Kervaire invariant problem, April 2009

Mike Hill, Mike Hopkins and I have 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) remained open for many years after our work; see the Quanta story of May 5, 2025 for how it was settled.

The book

The purple book
Equivariant Stable Homotopy Theory and the Kervaire Invariant Problem, with Mike Hill and Mike Hopkins, Cambridge University Press, 2021.
Prepublication draft

Papers and talks

• A collection of earlier and more recent papers on the problem
• 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

Our paper

Annals of Mathematics link to our paper. It was accepted for publication on September 30, 2015, and appeared in print in the July 2016 issue.

MathSciNet review of the above by Paul Goerss:

“This important and remarkable paper largely ends one story, but gets us started on another. First, as the title announces, the authors almost completely settle the Kervaire invariant problem, a long-standing and important question in algebraic topology and homotopy theory that has directed and focused the research of several generations of mathematicians. Second, this paper marks the renaissance and reinvigoration of equivariant stable homotopy theory. While this has been an important subfield since at least the 1970s, the unexpected application of equivariant techniques to such an important problem has brought the study of group actions in stable homotopy theory to the front of the stage.”

Preprints: On the non-existence of elements of Kervaire invariant one

• Preprint of May 25, 2015 (221 pages)
• Preprint of June 11, 2014 (220 pages)
• Preprint of November 22, 2010 (158 pages) · arXiv link
• Preprint of August 26, 2009 (99 pages) · arXiv link

Three expository accounts

• The Arf-Kervaire invariant problem in algebraic topology: introduction, CDM Conference, Harvard, 2009.
• The Arf-Kervaire invariant problem in algebraic topology: sketch of the proof, CDM Conference, Harvard, 2010.
• The Arf-Kervaire invariant problem in algebraic topology, Gökova Conference, 2010.

Workshops

Workshop on the Kervaire invariant and stable homotopy theory, ICMS Edinburgh, 25–29 April, 2011.
Pictured: Michel Kervaire (1927–2007)

MSRI Hot Topics Workshop: the Kervaire Invariant, Berkeley, California, October 25–29, 2010.
Streaming videos are available at the link above. Slides and notes from talks

Some relevant links

• 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 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 Akhmet'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.

  

• 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 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 Cohen.
• Links to “Kervaire invariant” on MathSciNet, Google and Google Scholar.

In the press

• 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.