Current Developments in Mathematics

Volume 2010

The Arf-Kervaire problem in algebraic topology: Sketch of the proof

Pages: 1 – 44



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)


We provide a sketch of the proof of the non-existence of Kervaire Invariant one manifolds using equivariant homotopy theory. A treatment of the statement and history of the problem can be found in “The Arf-Kervaire problem in algebraic topology: Introduction”. Our goal here is to introduce the reader to the techniques used to prove the result and to familiarize them with the kinds of computations needed.

Published 1 January 2010