8.1 Definition and basic properties of equivariant K-Theory

Equivariant K-theory extends topological K-theory by incorporating group actions on spaces. It's defined as the Grothendieck group of equivariant vector bundles, capturing the interplay between group actions and vector bundle structures.

This powerful tool allows us to study spaces with symmetries, connecting representation theory and topology. It satisfies key properties like homotopy invariance and has a rich ring structure, making it a versatile framework for understanding group actions on spaces.

Equivariant K-theory for Group Actions

Definition and Grothendieck Group Construction

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• Equivariant K-theory generalizes topological K-theory by incorporating the action of a compact Lie group G on a compact Hausdorff space X
• The equivariant K-theory of X, denoted K_G(X), is defined as the Grothendieck group of G-equivariant vector bundles over X
  • A G-equivariant vector bundle consists of a vector bundle π: E → X and a G-action on E compatible with the G-action on X and the vector bundle structure
  • The Grothendieck group construction applies to the semigroup of isomorphism classes of G-equivariant vector bundles under the direct sum operation
• Examples:
  • For a trivial G-space (a single point with the trivial G-action), K_G(X) is isomorphic to the representation ring R(G)
  • For the unit sphere S^n with the antipodal Z/2-action, K_{Z/2}(S^n) ≅ R(Z/2) for even n and K_{Z/2}(S^n) ≅ R(Z/2)/(1-t) for odd n, where t is the generator of R(Z/2) corresponding to the sign representation

Homotopy Invariance and Ring Structure

• Equivariant K-theory satisfies the homotopy invariance property
  • If f_0, f_1: X → Y are G-equivariant homotopic maps between compact G-spaces, then the induced maps on equivariant K-theory are equal: f_0^* = f_1^*: K_G(Y) → K_G(X)
• The equivariant K-theory ring K_G(X) has a natural ring structure induced by the tensor product of G-equivariant vector bundles
  • The tensor product of G-equivariant vector bundles is compatible with the G-action and yields a new G-equivariant vector bundle
  • The ring structure on K_G(X) is well-defined due to the properties of the tensor product and the Grothendieck group construction

Functorial Properties of Equivariant K-theory

Contravariant Functor and Induced Homomorphisms

• Equivariant K-theory is a contravariant functor from the category of compact G-spaces to the category of rings
  • A G-equivariant map f: X → Y between compact G-spaces induces a ring homomorphism f^*: K_G(Y) → K_G(X) by pulling back G-equivariant vector bundles
  • The composition of G-equivariant maps corresponds to the composition of the induced ring homomorphisms in the opposite order
• Examples:
  • The inclusion map i: X → Y of a G-invariant subspace X into a compact G-space Y induces a ring homomorphism i^*: K_G(Y) → K_G(X)
  • The projection map p: X × Y → X from the product of compact G-spaces X and Y to X induces a ring homomorphism p^*: K_G(X) → K_G(X × Y)

Additivity Axiom and External Product

• Equivariant K-theory satisfies the additivity axiom
  • If X is a disjoint union of compact G-spaces X_1 and X_2, then K_G(X) ≅ K_G(X_1) ⊕ K_G(X_2)
  • The isomorphism is given by the direct sum of G-equivariant vector bundles over X_1 and X_2
• The external product is a natural transformation K_G(X) ⊗ K_H(Y) → K_{G×H}(X×Y) for compact G-space X and compact H-space Y
  • The external product satisfies associativity and is compatible with the ring structures on equivariant K-theory
  • It allows for the construction of G×H-equivariant vector bundles over the product space X×Y from G-equivariant vector bundles over X and H-equivariant vector bundles over Y

Equivariant K-theory for Simple Spaces

Trivial G-spaces and Representation Rings

• For a trivial G-space (a single point with the trivial G-action), the equivariant K-theory is isomorphic to the representation ring R(G)
  • The representation ring R(G) is the Grothendieck group of finite-dimensional complex representations of G, with the ring structure given by the tensor product of representations
  • The isomorphism sends a G-equivariant vector bundle over the trivial G-space to its fiber, which is a G-representation
• Example: For the trivial group G = {e}, the equivariant K-theory of a point is isomorphic to the integers: K_{G}(pt) ≅ R({e}) ≅ Z

Spheres with Group Actions

• For the unit sphere S^n with the antipodal Z/2-action, the equivariant K-theory is given by:
  • K_{Z/2}(S^n) ≅ R(Z/2) for even n
  • K_{Z/2}(S^n) ≅ R(Z/2)/(1-t) for odd n, where t is the generator of R(Z/2) corresponding to the sign representation
• For the unit sphere S^(2n-1) with the standard S^1-action by complex multiplication, the equivariant K-theory is:
  • K_{S^1}(S^(2n-1)) ≅ Z[t, t^(-1)]/(1-t^n), where t is the generator of R(S^1) corresponding to the standard representation
  • The ring Z[t, t^(-1)] is the Laurent polynomial ring over the integers, and the quotient by the ideal (1-t^n) captures the periodicity of the S^1-action on S^(2n-1)

Equivariant vs Non-equivariant K-theory

Forgetful Map and Short Exact Sequence

• There is a forgetful map f: K_G(X) → K(X) from equivariant K-theory to non-equivariant K-theory, obtained by forgetting the group action on the vector bundles
  • The forgetful map is a ring homomorphism that sends the class of a G-equivariant vector bundle to the class of its underlying vector bundle
• The forgetful map fits into the following short exact sequence: 0 → K_G(X) → K(X) → K(X/G) → 0
  • X/G is the orbit space of X under the G-action
  • The sequence relates equivariant K-theory, non-equivariant K-theory, and the K-theory of the orbit space
• Example: For a free G-action on X, the forgetful map K_G(X) → K(X) is an isomorphism, as every vector bundle over X can be uniquely equipped with a G-action compatible with the free G-action on X

Atiyah-Segal Completion Theorem

• The Atiyah-Segal completion theorem relates equivariant K-theory to non-equivariant K-theory of the Borel construction
  • K_G(X) ≅ K(X ×_G EG)^∧_I, where:
    • EG is a contractible space with a free G-action
    • X ×_G EG is the quotient of X × EG by the diagonal G-action
    • (-)^∧_I denotes the I-adic completion with respect to the augmentation ideal I of R(G)
• The theorem provides a way to compute equivariant K-theory in terms of non-equivariant K-theory of a related space (the Borel construction) and the representation ring of the group G
• Example: For a compact connected Lie group G acting on itself by conjugation, the Atiyah-Segal completion theorem yields an isomorphism K_G(G) ≅ R(G)^∧_I, where the completion is taken with respect to the augmentation ideal I of the representation ring R(G)