Victor P. Snaith Boeken

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 volume originated from a graduate course at the Fields Institute in Autumn 1993, part of a program titled "Artin L-functions." The final chapter introduced a method to construct class-group valued invariants from Galois actions on algebraic K-groups in dimensions two and three of number rings, inspired by Chinburg invariants that pertain to dimensions zero and one. The classical Chinburg invariants assess the Galois structure of classical objects like units in rings of algebraic integers. During the "Galois Module Structure" workshop in February 1994, discussions following my lecture on my invariant (0,1 (L/K, 3) from Chapter 5) revealed that similar higher-dimensional cohomological and motivic invariants were emerging in the work of various authors. Motivated by this development and believing that K-theory represents a fundamental motivic cohomology theory, I embraced the chance to collaborate on computing and generalizing these K-theoretic invariants. These generalizations took various forms—both local and global—as I engaged with aspects of number theory and the ongoing trends in arithmetic geometry related to "Galois Module Structure."