Equivariant Stable Homotopy Theory and the Kervaire Invariant Problem

This is the definitive account of the resolution of the Kervaire invariant problem, a major milestone in algebraic topology. It develops all the machinery that is needed for the proof, and details many explicit constructions and computations performed along the way, making it suitable for graduate students as well as experts in homotopy theory.

The long-standing Kervaire invariant problem in homotopy theory arose from geometric and differential topology in the 1960s and was quickly recognised as one of the most important problems in the field. In 2009 the authors of this book announced a solution to the problem, which was published to wide acclaim in a landmark Annals of Mathematics paper. The proof is long and involved, using many sophisticated tools of modern (equivariant) stable homotopy theory that are unfamiliar to non-experts. This book presents the proof together with a full development of all the background material to make it accessible to a graduate student with an elementary algebraic topology knowledge. There are explicit examples of constructions used in solving the problem. Also featuring a motivating history of the problem and numerous conceptual and expository improvements on the proof, this is the definitive account of the resolution of the Kervaire invariant problem.