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.

These notes are supplementary to the author’s lectures at the 2010 conference on Current Developments in Mathematics, held on November 19-20 in Cambridge Massachusetts.