Stable Homotopy Around the Arf-Kervaire Invariant (Progress in Mathematics Book 273)

This monograph describes important techniques of stable homotopy theory, both classical and brand new, applying them to the long-standing unsolved problem of the existence of framed manifolds with odd Arf-Kervaire invariant. Opening with an account of the necessary algebraic topology background, it proceeds in a quasi-historical manner to draw from the author’s contributions over several decades. A new technique entitled “upper triangular technology” is introduced which enables the author to relate Adams operations to Steenrod operations and thereby to recover most of the important classical Arf-Kervaire invariant results quite simply. The final chapter briefly relates the book to the contemporary motivic stable homotopy theory of Morel-Voevodsky. Excerpt from a review: This takes the reader on an unusual mathematical journey. The problem referred to in the title, its history and the author's relationship with it are lucidly described in the book. The book does not offer a solution, but a new and interesting way of looking at it. The stated purpose of the book is twofold: (1) To rescue the Kervaire invariant problem from oblivion. (2) To introduce the "upper triangular technology" to approach the problem. This is very useful, since this method is not widely known. It is not an introduction to stable homotopy theory but rather a guide for experts along a path to a prescribed destination. In taking us there it assembles material from widely varying sources and offers a perspective that is not available anywhere else. This is a case where the whole is much greater than the sum of its parts. The manuscript is extremely well written. The author's style is engaging and even humorous at times. (Douglas Ravenel) Were I to take an iron gun, And ?re it o? towards the sun; I grant 'twould reach its mark at last, But not till many years had passed. But should that bullet change its force, And to the planets take its course, 'Twould never reach the nearest star, Because it is so very far. from FACTS by Lewis Carroll [55] Let me begin by describing the two purposes which prompted me to write this monograph. This is a book about algebraic topology and more especially about homotopy theory. Since the inception of algebraic topology [217] the study of homotopy classes of continuous maps between spheres has enjoyed a very exc- n n tional, central role. As is well known, for homotopy classes of maps f : S ?? S with n? 1 the sole homotopy invariant is the degree, which characterises the homotopy class completely. The search for a continuous map between spheres of di?erent dimensions and not homotopic to the constant map had to wait for its resolution until the remarkable paper of Heinz Hopf [111]. In retrospect, ?nding 3 an example was rather easy because there is a canonical quotient map from S to 3 1 1 2 theorbitspaceofthe freecircleactionS /S =CP = S . This book provides a clean, self-contained treatment of a long-standing piece of algebraic topology: the Kervaire invariant one problem. Beginning with the historical background, framing the problem in a stable homotopy theoretical point of view (using work of Browder to recast the geometric problem of manifolds into one of the survival of elements in the Adams spectral sequence), the book quickly moves into more modern approaches, looking at the stable homotopy groups of [italic]BO and then the relationship with the image of [italic]J. The book also makes careful use of the author's "upper triangular technology'', which provides a nice connection between various operations in connective [italic]K-theory and upper triangular 2-adic matrices Preface -- The Arf-kervaire Invariant Via Qx -- The Upper Triangular Technology -- A Brief Glimpse Of Algebraic K-theory -- The Matrix Corresponding To -- Real Projective Space -- Hurewicz Images, Bp-theory And The Arf-kervaire Invariant -- Upper Triangular Technology And The Arf-kervaire Invariant -- Futuristic And Contemporary Stable Homotopy. Victor P. Snaith. Includes Bibliographical References (p. [217]-233) And Index.