🪡K-Theory Unit 8 – Equivariant K–Theory and its Properties

Introduction to Equivariant K-Theory

• Equivariant K-theory extends classical K-theory by incorporating group actions on spaces and vector bundles
• Studies the K-theory of spaces equipped with an action of a group $G$, known as $G$-spaces
• Provides a powerful tool for understanding the interaction between topology and group actions
• Allows for the computation of K-theory invariants that take into account the symmetries of the space
• Has applications in various areas of mathematics, including representation theory, algebraic geometry, and mathematical physics (string theory, quantum field theory)
• Connects to other important theories, such as equivariant cohomology and equivariant stable homotopy theory
• Offers a framework for studying the relationship between the topology of a space and the symmetries it possesses

Group Actions and G-Spaces

• A group action of a group $G$ on a topological space $X$ is a continuous map $G \times X \to X$ satisfying certain axioms
  • Identity: $e \cdot x = x$ for all $x \in X$, where $e$ is the identity element of $G$
  • Compatibility: $(gh) \cdot x = g \cdot (h \cdot x)$ for all $g, h \in G$ and $x \in X$
• A $G$-space is a topological space $X$ equipped with a group action of $G$
• Examples of $G$-spaces include:
  • Trivial action: $G$ acts on $X$ by $g \cdot x = x$ for all $g \in G$ and $x \in X$
  • Rotation action: $G = S^1$ acts on $X = S^2$ by rotation about the vertical axis
• Orbits and fixed points play a crucial role in understanding the structure of $G$-spaces
  • The orbit of a point $x \in X$ is the set $G \cdot x = \{g \cdot x \mid g \in G\}$
  • The fixed point set of a subgroup $H \leq G$ is $X^H = \{x \in X \mid h \cdot x = x \text{ for all } h \in H\}$
• Quotient spaces $X/G$ and orbit spaces $X_{G}$ capture important information about the $G$-space $X$

Equivariant Vector Bundles

• An equivariant vector bundle over a $G$-space $X$ is a vector bundle $\pi: E \to X$ with a $G$-action on $E$ compatible with the action on $X$
  • The action on $E$ is linear on each fiber $E_x = \pi^{-1}(x)$
  • The projection map $\pi$ is $G$-equivariant: $\pi(g \cdot e) = g \cdot \pi(e)$ for all $g \in G$ and $e \in E$
• Equivariant vector bundles can be constructed from non-equivariant vector bundles using the associated bundle construction
• The direct sum, tensor product, and dual of equivariant vector bundles are again equivariant vector bundles
• Equivariant K-theory studies the isomorphism classes of equivariant vector bundles over a $G$-space
• The pullback of an equivariant vector bundle along a $G$-equivariant map is an equivariant vector bundle
• Equivariant vector bundles can be restricted to fixed point sets $X^H$ for subgroups $H \leq G$

The Equivariant K-Theory Ring

• The equivariant K-theory ring $K_G(X)$ of a $G$-space $X$ is the Grothendieck group of isomorphism classes of equivariant vector bundles over $X$
  • Addition in $K_G(X)$ corresponds to the direct sum of equivariant vector bundles
  • Multiplication in $K_G(X)$ is induced by the tensor product of equivariant vector bundles
• $K_G(X)$ is a contravariant functor from the category of $G$-spaces to the category of commutative rings
  • A $G$-equivariant map $f: X \to Y$ induces a ring homomorphism $f^*: K_G(Y) \to K_G(X)$
• The equivariant K-theory ring captures both topological and representation-theoretic information about the $G$-space $X$
• For a point space $\{\text{pt}\}$, $K_G(\{\text{pt}\}) \cong R(G)$, the representation ring of $G$
• The equivariant K-theory ring satisfies various functorial properties, such as pullback and pushforward maps
• Equivariant K-theory can be extended to generalized equivariant cohomology theories, such as equivariant KO-theory and equivariant KR-theory

Computational Techniques

• The Atiyah-Segal completion theorem relates the equivariant K-theory of a compact $G$-space $X$ to the non-equivariant K-theory of the Borel construction $EG \times_G X$
• The Mayer-Vietoris sequence allows for the computation of equivariant K-theory using a cover of the $G$-space by $G$-invariant open subsets
• The equivariant Chern character provides a connection between equivariant K-theory and equivariant cohomology
  • It is a ring homomorphism $\text{ch}_G: K_G(X) \to H^*_G(X; \mathbb{Q})$, where $H^*_G(X; \mathbb{Q})$ is the equivariant cohomology with rational coefficients
• Spectral sequences, such as the Atiyah-Hirzebruch spectral sequence and the Segal spectral sequence, can be used to compute equivariant K-theory
• Equivariant index theory, including the equivariant index theorem and the equivariant Riemann-Roch theorem, provides powerful tools for computing equivariant K-theory invariants
• Representation-theoretic methods, such as the character theory of compact Lie groups, can be employed to study equivariant K-theory
• Computational software, such as Sage and Macaulay2, can be used to perform explicit calculations in equivariant K-theory

Key Properties and Theorems

• The equivariant K-theory ring $K_G(X)$ is a module over the representation ring $R(G)$
• The equivariant Bott periodicity theorem states that $K_G(X) \cong K_G(X \times \mathbb{R}^2)$, providing a periodic structure in equivariant K-theory
• The equivariant Thom isomorphism theorem relates the equivariant K-theory of a $G$-equivariant vector bundle to the equivariant K-theory of its base space
• The equivariant Künneth theorem describes the equivariant K-theory of a product of $G$-spaces in terms of the equivariant K-theory of the factors
• The equivariant Atiyah-Segal completion theorem expresses the equivariant K-theory of a compact $G$-space in terms of the non-equivariant K-theory of the Borel construction
• The Segal conjecture, proved by Carlsson, relates the equivariant stable homotopy theory of a classifying space to the completion of its Burnside ring
• The equivariant index theorem, developed by Atiyah and Singer, computes the equivariant index of an elliptic operator on a compact $G$-manifold in terms of topological data

Applications in Mathematics and Physics

• Equivariant K-theory has applications in representation theory, providing a geometric approach to studying representations of groups
  • The Borel-Weil-Bott theorem uses equivariant K-theory to construct irreducible representations of compact Lie groups
• In algebraic geometry, equivariant K-theory is used to study the geometry of schemes with group actions
  • The equivariant Riemann-Roch theorem relates the equivariant K-theory of a scheme to its equivariant Chow groups
• Equivariant K-theory plays a role in the study of orbifolds and quotient singularities
• In string theory, equivariant K-theory is used to classify D-brane charges and to study the geometry of orbifolds
  • The equivariant K-theory of loop spaces is related to the K-theory of string orbifolds
• Equivariant K-theory has applications in quantum field theory, particularly in the study of gauge theories and anomalies
  • The equivariant index theorem is used to compute anomalies in gauge theories with symmetries
• In condensed matter physics, equivariant K-theory is used to study topological phases of matter with symmetries
  • The classification of symmetry-protected topological phases involves equivariant K-theory invariants

Advanced Topics and Current Research

• Equivariant twisted K-theory incorporates twists by cohomology classes or gerbes, leading to a more general theory
• Equivariant differential K-theory combines equivariant K-theory with differential forms, providing a refinement that captures geometric information
• Equivariant KK-theory, developed by Kasparov, is a bivariant version of equivariant K-theory that has applications in noncommutative geometry and index theory
• The Baum-Connes conjecture relates the equivariant K-theory of a group to the K-theory of its reduced group C*-algebra
  • The conjecture has been proven for various classes of groups, including amenable groups and hyperbolic groups
• Equivariant K-theory has connections to other areas of current research, such as:
  • Equivariant stable homotopy theory and equivariant spectra
  • Equivariant derived algebraic geometry and equivariant perfect complexes
  • Equivariant topological cyclic homology and trace methods
• Ongoing research explores the relationship between equivariant K-theory and other equivariant cohomology theories, such as equivariant Morava K-theory and equivariant elliptic cohomology
• Computational methods and algorithms for equivariant K-theory are an active area of research, aiming to develop efficient tools for explicit calculations in various settings