Topics in Topology and Homotopy Theory

TL;DR

This book provides a comprehensive overview of advanced topics in algebraic and general topology, including homotopy theory, localization, operads, and stable homotopy, highlighting recent developments and key theorems.

Contribution

It compiles and explains significant recent progress and foundational results in topology, offering a unified perspective on complex topics like localization, operads, and model categories.

Findings

Explores the role of nilpotency in homotopy theory

Details Bousfield's localization theory for spaces and spectra

Discusses the global perspective of stable homotopy by Hopkins

Abstract

This book is an account of certain topics in general and algebraic topology. Instead of laying out a synopsis of each chapter, here is a sample of some of what is taken up: 1) Nilpotency and its role in homotopy theory. 2) Bousfield's theory of the localization of spaces and spectra. 3) Homotopy limits and colimits and their applications. 4) The James construction, symmetric products, and the Dold$-$Thom theorem. 5) Brown and Adams representability in the setting of triangulated categories 6) Operads and the May$-$Thomason theorem on the uniqueness of infinite loop space machines. 7) The plus construction and theorems A and B of Quillen. 8) Hopkins' global picture of stable homotopy theory. 9) Model categories, cofibration categories, and Waldhausen categories. 10) The Dugundji extension theorem and its consequences.