Global homotopy theory

TL;DR

This book develops a comprehensive framework for global homotopy theory, unifying equivariant theories across all compact Lie groups with detailed models, operations, and examples.

Contribution

It introduces a self-contained approach to global homotopy theory, unifying equivariant phenomena for all compact Lie groups within a single theoretical framework.

Findings

Develops models for unstable and stable global homotopy theory

Analyzes operations on equivariant homotopy groups

Provides explicit examples and calculations

Abstract

This book introduces a new context for global homotopy theory, i.e., equivariant homotopy theory with universal symmetries. Many important equivariant theories naturally exist not just for a particular group, but in a uniform way for all groups in a specific class. Prominent examples are equivariant stable homotopy, equivariant $K$-theory or equivariant bordism. Global equivariant homotopy theory studies such uniform phenomena, i.e., the adjective `global' refers to simultaneous and compatible actions of all compact Lie groups. We give a self-contained treatment of unstable and stable global homotopy theory, modeled by orthogonal spaces respectively orthogonal spectra under global equivalences. Specific topics include the global stable homotopy category, operations on equivariant homotopy groups, global model structures, and ultra-commutative multiplications. The book includes many explicit examples and detailed calculations.