13.1 Equivariant K-theory and its applications

Equivariant K-theory extends ordinary K-theory by incorporating group actions on spaces. It provides a powerful framework for studying symmetries in topology, geometry, and representation theory. This generalization allows for deeper insights into the structure of spaces with group actions.

The applications of equivariant K-theory are wide-ranging. From classifying vector bundles and computing indices to analyzing orbifolds and quotient spaces, it offers valuable tools for understanding complex mathematical objects through the lens of group symmetries.

Equivariant K-theory: Definition and Relation to Ordinary K-theory

Fundamental Concepts and Definitions

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• Equivariant K-theory generalizes ordinary K-theory by incorporating group actions on topological spaces
• K_G(X) denotes the equivariant K-group defined as the Grothendieck group of G-equivariant vector bundles over a G-space X
• Natural forgetful map exists from equivariant K-theory to ordinary K-theory omitting the group action
• Atiyah-Segal completion theorem captures the relationship between equivariant and ordinary K-theory
• Equivariant K-theory satisfies analogues of fundamental properties of ordinary K-theory
  • Homotopy invariance preserves K-groups under continuous deformations
  • Bott periodicity establishes a cyclic behavior in K-groups
• K_G(*) isomorphism to representation ring R(G) of group G connects equivariant K-theory to representation theory
• Equivariant K-theory extends to G-C*-algebras bridging topology and operator algebras

Properties and Extensions

• Equivariant K-theory incorporates group symmetries into topological invariants
• Forgetful map allows comparison between equivariant and non-equivariant settings
• Atiyah-Segal completion theorem relates equivariant K-theory to completed representation rings
• Homotopy invariance ensures stability of K-groups under continuous deformations (elastic transformations)
• Bott periodicity establishes cyclic behavior in K-groups ($K^{n+2}(X) \cong K^n(X)$ for complex K-theory)
• Representation ring connection provides algebraic tools for analyzing group actions
• G-C*-algebra extension allows application to noncommutative geometry and operator algebras

Computing Equivariant K-groups

Calculation Techniques for Specific Spaces

• Calculate equivariant K-theory of spheres with various group actions
  • Rotations ($SO(n)$ action on $S^{n-1}$)
  • Reflections ($\mathbb{Z}/2\mathbb{Z}$ action on $S^n$)
• Determine equivariant K-groups for torus actions on complex projective spaces
  • Standard $T^n$ action on $\mathbb{CP}^{n-1}$
• Compute equivariant K-theory for finite group actions on surfaces
  • Quotient singularities (cyclic group actions on $\mathbb{C}^2$)
• Analyze equivariant K-theory of Lie group actions on homogeneous spaces
  • $SU(n)$ action on flag manifolds

Advanced Computational Methods

• Apply Atiyah-Segal completion theorem to compute equivariant K-groups
  • Express K_G(X) in terms of completed representation rings
• Utilize spectral sequences for complex space computations
  • Atiyah-Hirzebruch spectral sequence relates equivariant cohomology to K-theory
• Employ localization techniques to simplify equivariant K-group calculations
  • Localization theorem reduces computations to fixed point sets
• Use character formulas and representation theory to analyze equivariant K-groups
  • Weyl character formula for compact Lie group representations

Applications of Equivariant K-theory

Topological and Geometric Applications

• Study equivariant vector bundles and their classification over G-spaces
  • Classify G-equivariant line bundles over a G-manifold
• Apply Atiyah-Singer G-index theorem to compute equivariant indices
  • Calculate equivariant index of Dirac operator on a spin manifold with group action
• Analyze group actions on noncommutative spaces using equivariant K-theory
  • Study crossed product C*-algebras arising from group actions
• Employ equivariant K-theory to investigate equivariant cohomology theories
  • Compare equivariant K-theory to Borel equivariant cohomology

Representation Theory and Invariants

• Apply equivariant K-theory to problems in geometric representation theory
  • Analyze character sheaves on reductive groups
  • Study orbital integrals in harmonic analysis on Lie groups
• Use equivariant K-theory to investigate equivariant topological invariants
  • Compute equivariant Euler characteristic of a G-manifold
  • Analyze equivariant signature for oriented G-manifolds
• Analyze role of equivariant K-theory in equivariant stable homotopy theory
  • Study G-spectra and their relationship to equivariant cohomology theories

Equivariant K-theory for Orbifolds and Quotient Spaces

Orbifolds and Equivariant K-theory

• Orbifolds resemble quotients of Euclidean space by finite group actions locally
• Equivariant K-theory provides framework for studying vector bundles on orbifolds
  • Analyze orbifold vector bundles using equivariant techniques
• Inertia orbifold and twisted sectors refine orbifold invariants in equivariant K-theory
  • Study Chen-Ruan cohomology using equivariant K-theory
• Equivariant K-theory investigates stringy topology of orbifolds
  • Analyze quantum cohomology of symplectic orbifolds

Quotient Spaces and Group Actions

• Equivariant K-theory of G-space X relates to K-theory of quotient space X/G for free actions
  • Study Atiyah-Segal completion theorem for free actions
• Analyze equivariant K-theory behavior under various quotient constructions
  • Proper actions (locally compact transformation groups)
  • Stacky quotients (Deligne-Mumford stacks)
• Apply equivariant K-theory to orbifold Gromov-Witten theory
  • Study quantum cohomology rings of symplectic orbifolds
• Examine role of equivariant K-theory in mirror symmetry for orbifolds
  • Analyze Landau-Ginzburg models for orbifolds using equivariant techniques