8.3 Equivariant Bott periodicity and localization theorems

Equivariant Bott periodicity extends topological K-theory to spaces with group actions. It establishes a natural isomorphism between G-equivariant K-theory groups of a space X and its double suspension. This powerful result connects equivariant and non-equivariant K-theory through the Borel construction.

Localization theorems link equivariant K-theory to fixed point subspaces. The Atiyah-Segal theorem recovers equivariant K-theory from fixed points by inverting elements in the representation ring. These results bridge equivariant and non-equivariant K-theory, showcasing connections to representation theory and geometry.

Equivariant Bott Periodicity

Statement and Proof

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• Equivariant Bott periodicity extends the fundamental result in topological K-theory relating the K-theory of a space X to the K-theory of its suspension ΣX, to spaces with a group action
• For a compact Lie group G and a compact G-space X, there exists a natural isomorphism between the G-equivariant K-theory groups $KG(X)$ and $KG(Σ^2X)$, where $Σ^2X$ represents the double suspension of X with the induced G-action
• The proof involves constructing an explicit homotopy equivalence between the G-equivariant K-theory spectra of X and $Σ^2X$, utilizing equivariant vector bundles and the equivariant Thom isomorphism theorem
  • The equivariant Thom isomorphism establishes a connection between the G-equivariant K-theory of a G-equivariant vector bundle and the K-theory of its Thom space, playing a crucial role in establishing the periodicity isomorphism
• Equivariant Bott periodicity can also be derived from the non-equivariant case by considering the Borel construction and employing the spectral sequence associated with the fibration $EG ×G X → BG$

Alternative Derivation and Spectral Sequences

• The Borel construction, which associates a classifying space $EG ×G X$ to a G-space X, provides an alternative approach to deriving equivariant Bott periodicity from the non-equivariant case
• The spectral sequence associated with the fibration $EG ×G X → BG$ relates the equivariant K-theory of X to the non-equivariant K-theory of the Borel construction $EG ×G X$
• By applying the non-equivariant Bott periodicity theorem to the space $EG ×G X$ and analyzing the spectral sequence, one can deduce the equivariant Bott periodicity isomorphism between $KG(X)$ and $KG(Σ^2X)$
• This approach highlights the interplay between equivariant and non-equivariant K-theory, and demonstrates the power of spectral sequence techniques in studying equivariant cohomology theories

Computing Equivariant K-theory

Applications of Bott Periodicity

• Equivariant Bott periodicity serves as a powerful tool for computing equivariant K-theory groups by reducing the problem to understanding the K-theory of simpler spaces related by suspension
• For a compact Lie group G acting freely on a sphere $S^{2n-1}$, the equivariant K-theory of the quotient space $S^{2n-1}/G$ can be determined using Bott periodicity and the equivariant K-theory of a point, $KG(pt)$, which is isomorphic to the representation ring $R(G)$
• When G acts trivially on a space X, equivariant Bott periodicity implies that the equivariant K-theory $KG(X)$ is isomorphic to the tensor product of the non-equivariant K-theory $K(X)$ with the representation ring $R(G)$
  • This result allows for the computation of equivariant K-theory in terms of non-equivariant K-theory and representation theory in the case of trivial group actions

Künneth Theorem and Product Spaces

• For a compact Lie group G acting on a product space $X × Y$, the equivariant Künneth theorem, in conjunction with Bott periodicity, can be employed to express $KG(X × Y)$ in terms of the equivariant K-theory groups of X and Y
• The equivariant Künneth theorem provides a decomposition of the equivariant K-theory of a product space in terms of tensor products of the equivariant K-theory groups of the individual factors
• By iteratively applying the equivariant Künneth theorem and Bott periodicity, one can compute the equivariant K-theory of higher-dimensional product spaces and relate it to the equivariant K-theory of the constituent spaces
• This computational technique is particularly useful in studying the equivariant K-theory of toric varieties and other spaces with a torus action that can be decomposed into simpler pieces

Localization Theorems for Equivariant K-theory

Atiyah-Segal Localization Theorem

• The Atiyah-Segal localization theorem establishes a connection between the equivariant K-theory of a G-space X and the equivariant K-theory of its fixed point subspaces $X^H$, where H is a subgroup of G
• For a compact Lie group G acting on a compact space X, the theorem asserts that the equivariant K-theory of X can be recovered from the equivariant K-theory of the fixed point subspaces $X^H$, as H ranges over the closed subgroups of G, by inverting certain elements in the representation ring $R(G)$
  • Specifically, the localization theorem states that the map $KG(X) → ∏H KH(X^H)$, induced by the inclusion of fixed point subspaces, becomes an isomorphism after inverting the elements $1 - [V]$ in $R(G)$, where V ranges over the non-trivial irreducible representations of G
• The proof of the localization theorem typically involves analyzing the equivariant K-theory spectral sequence associated with the filtration of X by the fixed point subspaces $X^H$ and applying the localization theorem for the representation ring $R(G)$

Variations and Generalizations

• Variations and generalizations of the localization theorem, such as the Atiyah-Bott-Berline-Vergne localization formula, provide more refined information about the relationship between the equivariant K-theory of a space and its fixed point subspaces
• The Atiyah-Bott-Berline-Vergne localization formula is particularly useful in the case of a compact Lie group G acting on a compact manifold X with isolated fixed points
  • The formula expresses the equivariant K-theory of X as a sum of local contributions from the fixed points, involving the non-equivariant K-theory of the fixed points and the representations of G on the tangent spaces at the fixed points
• Other variations of the localization theorem, such as the equivariant Riemann-Roch theorem and the equivariant index theorem, relate equivariant K-theory to other geometric and topological invariants, such as characteristic classes and indices of elliptic operators
• These generalizations demonstrate the rich interplay between equivariant K-theory, representation theory, and geometry, and provide powerful tools for computing and understanding equivariant invariants in various settings

Equivariant vs Non-Equivariant K-theory

Relationship via Localization Theorems

• Localization theorems provide a bridge between equivariant K-theory and non-equivariant K-theory by expressing the equivariant K-theory of a G-space X in terms of the ordinary K-theory of its fixed point subspaces
• In the case of a trivial G-action on X, the localization theorem implies that the equivariant K-theory $KG(X)$ is isomorphic to the tensor product of the non-equivariant K-theory $K(X)$ with the localized representation ring $R(G)_I$, where I is the ideal generated by the elements $1 - [V]$ for non-trivial irreducible representations V of G
  • This result allows for the computation of equivariant K-theory in terms of non-equivariant K-theory and representation theory in the case of trivial group actions
• For a compact Lie group G acting on a compact manifold X with isolated fixed points, the Atiyah-Bott-Berline-Vergne localization formula expresses the equivariant K-theory of X as a sum of local contributions from the fixed points, involving the non-equivariant K-theory of the fixed points and the representations of G on the tangent spaces at the fixed points

Connections to Representation Theory

• Localization theorems can be used to compute the equivariant K-theory of homogeneous spaces $G/H$ in terms of the representation theory of the subgroup H, by considering the fixed points of the action of a maximal torus of G
  • This technique is particularly effective for flag varieties and other homogeneous spaces of Lie groups, where the fixed points of the torus action are isolated and can be described explicitly
• The relationship between equivariant K-theory and representation theory can be further explored using the Borel-Weil-Bott theorem, which relates the equivariant K-theory of flag varieties to the irreducible representations of the corresponding Lie group
  • The Borel-Weil-Bott theorem provides a geometric realization of irreducible representations in terms of equivariant line bundles on flag varieties, and allows for the computation of the equivariant K-theory of these spaces in terms of representation-theoretic data
• The interplay between equivariant K-theory, representation theory, and geometric representation theory is a rich and active area of research, with applications to a wide range of topics in mathematics and mathematical physics, including gauge theory, string theory, and the Langlands program