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

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 .

From the reviews:

“This book is concerned with homotopy theoretical approaches to the study of the Arf-Kervaire invariant one problem ... . The last chapter is an extra one in which some current themes related to the subject are described. ... The bibliography contains 297 titles. ... this book an excellent guide to the classical problem above.” (Haruo Minami, Zentralblatt MATH, Vol. 1169, 2009)

“This book provides a clean, self-contained treatment of a long-standing piece of algebraic topology: the Kervaire invariant one problem, and the reviewer found it a very interesting and helpful reference. ... The book itself is a very pleasant read. ... The reviewer found the opening quotations for each chapter especially droll. ... Finally, the chapter (and book) ends with some suggestions for further reading.”­­­ (Michael A. Hill, Mathematical Reviews, Issue 2011 d)