Current Developments in Mathematics

Volume 2009

The Arf-Kervaire problem in algebraic topology: an introduction

Pages: 23 – 58



Michael A. Hill (Department of Mathematics, University of Virginia)

Michael J. Hopkins (Department of Mathematics, Harvard University)

Douglas C. Ravenel (Department of Mathematics, University of Rochester, New York)


This paper gives the history and background of one of the oldest problems in algebraic topology, along with a short summary of our solution to it and a description of some of the tools we use. More details of the proof will be provided in our second paper, “The Arf-Kervaire invariant problem in algebraic topology: Sketch of the proof.” A rigorous account can be found in our preprint “The non-existence of elements of Kervaire invariant one,” on the arXiv and on the third author’s home page. The latter also has numerous links to related papers and talks we have given on the subject since announcing our result in April 2009.

Published 1 January 2009