Global homotopy theory

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.

References in corpus (23)

• Algebras and modules in monoidal model categories
• Rings, modules, and algebras in infinite loop space theory
• Quillen model structures for relative homological algebra
• Diagram spaces and symmetric spectra
• Model Structures on Commutative Monoids in General Model Categories
• Equivariant Kasparov theory and generalized homomorphisms
• On an extension of the notion of Reedy category
• Homotopy Theory of Orbispaces
• A uniqueness theorem for stable homotopy theory
• Diagram spaces, diagram spectra, and spectra of units
• Equivariant homotopy theory for pro-spectra
• Continuous functors as a model for the equivariant stable homotopy category
• Global orthogonal spectra
• Equivariant properties of symmetric products
• Complex cobordism of involutions
• Equivariant Structure on Smash Powers
• Realising formal groups
• The geometric realization of a simplicial Hausdorff space is Hausdorff
• Symmetric spectra model global homotopy theory of finite groups
• Tambara Functors and Commutative Ring Spectra
• Real Equivariant Bordism for elementary abelian 2-groups
• Classifying spaces for 1-truncated compact Lie groups
• Global Mackey functors with operations and n-special lambda rings

Cited by in corpus (54)

• The six operations in equivariant motivic homotopy theory
• Enriched $\infty$-categories via non-symmetric $\infty$-operads
• Nilpotence and descent in equivariant stable homotopy theory
• Equivariant Cohomotopy implies orientifold tadpole cancellation
• Global orthogonal spectra
• Equivariant properties of symmetric products
• G-symmetric monoidal categories of modules over equivariant commutative ring spectra
• On the Balmer spectrum for compact Lie groups
• Symmetric spectra model global homotopy theory of finite groups
• Enough vector bundles on orbispaces
• The Left Localization Principle, completions, and cofree $G$-spectra
• Proper equivariant stable homotopy theory
• Eilenberg-MacLane spectra as equivariant Thom spectra
• Orbispaces, orthogonal spaces, and the universal compact Lie group
• Global algebraic K-theory
• Homotopical and operator algebraic twisted K-theory
• Smith Ideals of Operadic Algebras in Monoidal Model Categories
• Global homotopy theory via partially lax limits
• Comparing cyclotomic structures on different models for topological Hochschild homology
• Conjugation Spaces are Cohomologically Pure
• Categories and orbispaces
• Power operations in the Stolz--Teichner program
• Equivariant cohomology for cyclic groups of square-free order
• Splittings of global Mackey functors and regularity of equivariant Euler classes
• Orbifold bordism and duality for finite orbispectra
• Proper Orbifold Cohomology
• Operads in Unstable Global Homotopy Theory
• The completion and local cohomology theorems for complex cobordism for all compact Lie groups
• G-Global Homotopy Theory and Algebraic K-Theory
• On the global homotopy theory of symmetric monoidal categories
• A multiplicative Tate spectral sequence for compact Lie group actions
• Orbifolds, Orbispaces and Global Homotopy Theory
• Algebraic models of change of groups functors in (co)free rational equivariant spectra
• Homotopy invariance of convolution products
• Commuting matrices and Atiyah's Real K-theory
• The Equivariant Cobordism Category
• Symmetric products and subgroup lattices
• Semistable models for modular curves and power operations for Morava E-theories of height 2
• Global $N_{\infty }$-operads
• Equivariant higher twisted K-theory of SU(n) for exponential functor twists
• External Spanier-Whitehead duality and homology representation theorems for diagram spaces
• Global transfer systems of abelian compact Lie groups
• Augmented Homotopical Algebraic Geometry
• Representation stability and outer automorphism groups
• Naive-commutative structure on rational equivariant $K$-theory for abelian groups
• Oriented and unitary equivariant bordism of surfaces
• Rational extensions of the representation ring global functor and a splitting of global equivariant $K$-theory
• Parametrized stability and the universal property of global spectra
• Local Gorenstein duality in chromatic group cohomology
• $G_{\infty }$-ring spectra and Moore spectra for $\beta$-rings
• Homotopy theory of monoid actions via group actions and an Elmendorf style theorem
• Bundles of strongly self-absorbing $C^{\ast }$-algebras with a Clifford grading
• Stabilization and costabilization with respect to an action of a monoidal category
• Realising ${\pi }_{\ast }^{e}R$-algebras by global ring spectra