Foundations of Stable Homotopy Theory: (Series Number 185 Cambridge Studies in Advanced Mathematics)

The beginning graduate student in homotopy theory is confronted with a vast literature on spectra that is scattered across books, articles and decades. There is much folklore but very few easy entry points. This comprehensive introduction to stable homotopy theory changes that. It presents the foundations of the subject together in one place for the first time, from the motivating phenomena to the modern theory, at a level suitable for those with only a first course in algebraic topology. Starting from stable homotopy groups and (co)homology theories, the authors study the most important categories of spectra and the stable homotopy category, before moving on to computational aspects and more advanced topics such as monoidal structures, localisations and chromatic homotopy theory. The appendix containing essential facts on model categories, the numerous examples and the suggestions for further reading make this a friendly introduction to an often daunting subject.

Table of Contents:
Introduction; 1. Basics of stable homotopy theory; 2. Sequential spectra and the stable homotopy category; 3. The suspension and loop functors; 4. Triangulated categories; 5. Modern categories of spectra; 6. Monoidal structures; 7. Left Bousfield localisation; Appendix. Model categories; References; Index.

About the Author :
David Barnes is Senior Lecturer in Mathematics at Queen's University Belfast. His work focuses on stable homotopy theory, usually with either a monoidal or equivariant flavour, often using algebra to describe the structures in question. Constanze Roitzheim is Senior Lecturer in Mathematics at the University of Kent, Canterbury. Her work focuses on localisations of the stable homotopy category and related questions in algebra.

Review :
'The authors have made great efforts to ensure that the book is accessible to those who are not already experts in the area. The topics have been carefully chosen, and the exposition includes not just the technical details but also provides historical and motivational context for many of the important ideas.' Dan Isaksen, MAA Reviews 'The stated goal of the authors is to provide an accessible entry point to stable homotopy theory for first-year graduate students. The necessary prerequisites are good undergraduate knowledge of point-set topology and algebraic topology. Barnes and Roitzheim achieve their goal within the first three chapters by discussing a large collection of examples. Included among them are the Spanier-Whitehead category, sequential spectra, the stable homotopy category, and two important functors, namely the suspension and the loop functors.' M. Bona, Choice 'This is a useful contribution to the literature. As well as nurturing budding stable homotopy theorists, it could also serve as a resource for researchers whose primary interest is not stable homotopy theory, but who seek an understanding of such techniques.' Geoffrey M. L. Powell, Mathematical Reviews 'Especially as it seems to be very carefully written, I expect that it will become a standard textbook in the field.' Julie Bergner, zbMATH