Equivariant K-Theory

Introduction

$K$-theory is the cohomology theory generated by vector bundles, and its equivariant form is generated by the vector bundles that carry an action of the group. For a compact group $G$ acting on a compact space $X$, the complex $G$-vector bundles over $X$ form a monoid under direct sum, its Grothendieck group is $K_G(X)$, and the pair of groups $K_G^0(X)$ and $K_G^1(X)$ extends to a $\mathbb{Z}/2$-periodic equivariant cohomology theory. The theory is richer than the ordinary $K$-theory of the underlying space in exactly the way the action is: the value at a point is the representation ring $R(G) = K_G(\mathrm{pt})$, the free abelian group on the irreducible representations with the tensor product as multiplication, so the equivariant theory carries the representation theory of the group, and the computation of $K_G(X)$ is the computation of a module over that ring. The two structural theorems are the Bott periodicity of the theory, $K_G^0(X)\cong K_G^2(X)$, which is inherited from the ordinary theory by the action, and the Atiyah–Segal completion theorem, which identifies the equivariant $K$-theory completed at the augmentation ideal of $R(G)$ with the ordinary $K$-theory of the Borel construction $EG\times_G X$; the theorem is the $K$-theoretic analogue of the Borel model of equivariant cohomology.

The article develops the theory. It defines the equivariant bundles and the Grothendieck group, computes the representation ring and the values on the orbits and on the free and trivial actions, establishes Bott periodicity and the six-term exact sequence, states the completion theorem and its relation to the Borel cohomology of Equivariant Cohomology, develops the equivariant Chern character and the Atiyah–Segal localization theorem, and closes with the computations for the point, the orbits and the spheres with a circle action. The ordinary, non-equivariant theory — the Grothendieck group, Bott periodicity, the Chern character, the clutching construction and the exact sequences — is that of Topological K-Theory, and this article is its equivariant refinement; the representation ring is built from Character Theory and Representations of Groups; the classifying spaces are those of Classifying Spaces and Cohomology Operations; the Borel construction and its cohomology are those of Equivariant Cohomology and Two-Fold Coverings and the Borel Construction; and the operator-algebraic $K$-theory, the crossed product $\mathrm{C}^*(X)\rtimes G$ and the identification $K_G^*(X)\cong K_*(C_0(X)\rtimes G)$, is that of K-Theory of Operator Algebras and KK-Theory. The index-theoretic use of the equivariant theory, the equivariant index and the equivariant index theorem, belongs to the analysis of Part IV and to The Atiyah–Singer Index Theorem and K-Theory.

Nothing analytic and nothing geometric is used. The vector bundles are topological, the group is compact Lie or finite, the Grothendieck group and the representation ring are algebra, and no connection, no curvature, no distance and no measure is chosen; the analysis appears only in the cited operator-algebraic identification and in the index theory that is deferred. Throughout, $G$ is a compact Lie group (with the finite group as the discrete case of reference) acting on the left on a compact Hausdorff $G$-space $X$, all bundles are complex and finite-dimensional, $K_G^0(X)$ is the Grothendieck group of the equivariant bundles and $K_G^{-n}(X) = K_G^0(\Sigma^n X_+)$ with $\Sigma$ the reduced suspension of The Suspension Operator; the representation ring is $R(G) = K_G^0(\mathrm{pt})$, and the Borel construction $X_G = EG\times_G X$ and its cohomology are those of Equivariant Cohomology. The irreducibles and the character theory used for $R(G)$ are those of Character Theory.

Equivariant Vector Bundles

The Equivariant Bundles

Definition. A $G$-vector bundle over a $G$-space $X$ is a complex vector bundle $p : E \to X$ together with a left action of $G$ on $E$ by bundle maps covering the action on $X$: for every $g$ the bundle map commutes with the projections, $p\circ g = g\circ p$, and the induced map on the fibres $g : E_x \to E_{gx}$ is a linear isomorphism. A morphism of $G$-bundles is an equivariant bundle map; the $G$-bundles over $X$ form a category, and the direct sum and the tensor product of bundles extend to it with the diagonal action.

Proposition. The isomorphism classes of $G$-bundles over $X$ form a commutative monoid $\mathrm{Vect}_G(X)$ under direct sum, with the trivial bundle $X\times\mathbb{C}$ as unit; a $G$-bundle is trivial as an equivariant bundle exactly when it has an equivariant trivialisation, which is a strictly stronger condition than the existence of a trivialisation of the underlying bundle.

Proof. Direct sum is associative and commutative up to canonical isomorphism compatible with the action, and the pullback along a $G$-map is a monoid homomorphism; the last statement is the definition of an equivariant trivialisation, and the standard example is the product bundle $X\times V$ with a non-trivial representation of $G$ on the fibre $V$, trivial as a bundle and non-trivial as an equivariant bundle. $\square$

The Grothendieck Group and the Representation Ring

Definition. The equivariant $K$-group is the Grothendieck group of the monoid of $G$-bundles,

$$ K_G^0(X) = \mathrm{Groth}\bigl(\mathrm{Vect}_G(X)\bigr), $$

the abelian group of formal differences $[E]-[F]$ of $G$-bundles with the relations $[E\oplus F]=[E]+[F]$; the representation ring is

$$ R(G) = K_G^0(\mathrm{pt}), $$

the Grothendieck group of the finite-dimensional complex representations of $G$, with the multiplication induced by the tensor product.

Theorem. $K_G^0$ is a contravariant functor on the $G$-spaces: an equivariant map $f : X \to Y$ gives the pullback $f^* : K_G^0(Y)\to K_G^0(X)$, functorially, and $K_G^0(\mathrm{pt}) = R(G)$. The tensor product of bundles makes $K_G^0(X)$ a commutative ring and a module over $R(G)$; the module structure is the pullback along $X\to\mathrm{pt}$, and the ring is the equivariant analogue of the ordinary $K^0$ of Topological K-Theory.

Proof. The pullback of a bundle along a $G$-map is a $G$-bundle, and the pullback is functorial and preserves direct sums, hence descends to the Grothendieck groups; the tensor product is bilinear and respects the relations, giving the ring and the module structure. $\square$

The value at a point is the representation ring, and the whole theory is thus a family of rings and modules over $R(G)$: this is the sense in which the equivariant $K$-theory contains the representation theory of the group, and it is why the theory is not determined by the ordinary $K$-theory of the underlying space.

The Cohomology Theory and Periodicity

The Graded Groups and Bott Periodicity

Definition. For $n \geq 0$ the higher equivariant $K$-groups are

$$ K_G^{-n}(X) = K_G^0(\Sigma^n X_+), \qquad K_G^{-n}(X,Y) = \tilde K_G^0(\Sigma^n (X/Y)), $$

with the reduced groups defined by the base point, and $K_G^{-n} = K_G^n$ by the periodicity below; the pair of functions $K_G^0$ and $K_G^1$ together with the connecting maps is an equivariant generalised cohomology theory on the category of compact $G$-spaces.

Theorem (Bott periodicity). There is a natural isomorphism

$$ K_G^0(X) \xrightarrow{\ \cong\ } K_G^2(X), \qquad K_G^0(X) \cong K_G^{-2}(X), $$

for every compact $G$-space $X$, periodic of period two; the isomorphism is the multiplication by the Bott class of the equivariant line bundle over $\mathbb{P}^1_{\mathbb{C}} = S^2$ of the canonical line bundle, and it is equivariant for the natural action of $G$ on $S^2$, so the periodicity of the ordinary theory of Topological K-Theory holds with all the operators acting equivariantly.

Proof. The clutching construction of Topological K-Theory classifies the bundles over $\Sigma X_+$ by the bundles over $X$ up to an equivalence given by a loop, and the Bott class of the canonical line bundle over $S^2$ provides the periodicity element; the construction is natural and the action of $G$ on $S^2$ commutes with it, so the resulting isomorphism is equivariant. $\square$

The Exact Sequence

Theorem. For a closed $G$-invariant pair $(X,Y)$ there is a natural cyclic six-term exact sequence

$$ K_G^0(X,Y) \to K_G^0(X) \to K_G^0(Y) \xrightarrow{\ \delta\ } K_G^1(Y) \to K_G^1(X) \to K_G^1(X,Y) \xrightarrow{\ \delta\ } K_G^0(X,Y), $$

the equivariant form of the exact sequence of Topological K-Theory, the last map returning to the first term; and for a closed subgroup $H \leq G$ the induction and restriction functors relate $K_G^*$ and $K_H^*$ in a six-term sequence. The sequence is the one that makes the equivariant $K$-theory computable on the equivariant cell attachments of Equivariant Homotopy Theory.

Proof. The six-term sequence is the standard exact sequence of a generalised cohomology theory and a pair, with the periodicity identifying the degrees; the induction–restriction sequence is the equivariant form of the classical one. $\square$

The exact sequence is the computational engine of the theory: the equivariant $K$-groups of a $G$-CW complex are computed cell by cell, and the induction functor $K_H^*\to K_G^*$ together with the orbit decomposition of Equivariant Homotopy Theory reduces the computation to the representation rings of the isotropy groups.

The Computations from the Representations

The Representation Ring

Theorem. The representation ring of a finite group is the free abelian group on the irreducible complex characters,

$$ R(G) = \bigoplus_{\rho \in \hat G}\mathbb{Z}\,[\rho], $$

with the multiplication given by the tensor product and the decomposition into irreducibles; it has an involution induced by the dual representation, and its complexification is the ring of class functions. For a compact Lie group the same description holds with the irreducible representations of the compact group, and for the circle $R(S^1) = \mathbb{Z}[z,z^{-1}]$.

Proof. The Grothendieck group of the representations is free on the irreducibles by the semisimplicity of the group algebra of a finite group, and the tensor product of the representations decomposes into irreducibles; the identification with the class functions is the character theory of Character Theory, and the circle case is the Fourier decomposition. $\square$

So the coefficient ring of the equivariant $K$-theory is the representation ring, and its augmentation ideal $I = \ker(\varepsilon : R(G)\to\mathbb{Z})$ with $\varepsilon$ the dimension is the ideal that the completion theorem completes.

Orbits, Free and Trivial Actions

Theorem. The following computations are the anchors of the theory.

The orbit. $K_G^0(G/H) = R(H)$ for the closed subgroup $H$, and $K_G^0(G/H) \cong R(H)$ as a module over $R(G)$ through the restriction; for the free orbit $H=1$ this is $\mathbb{Z}$, and for the homogeneous space the value is the representation ring of the isotropy group.
The free action. If $G$ acts freely on $X$, then $K_G^0(X)\cong K^0(X/G)$, the ordinary $K$-theory of the orbit space, with the periodicity and the ring structure transported.
The trivial action. If $G$ acts trivially on $X$, then $K_G^0(X)\cong K^0(X)\otimes_{\mathbb{Z}} R(G)$ as a ring, the fibre contributing the ordinary $K$-theory of the space and the group contributing the representation ring.
The point with a group action. The general point is $K_G^0(\mathrm{pt}) = R(G)$; for the orbit $G/H$ the point is the case of the first item, and the whole theory is assembled from the representation rings of the isotropy groups by the exact sequences.

Proof. The first statement is the equivalence between $G$-bundles on $G/H$ and representations of $H$, by evaluation at the coset $H$; the second is the descent of the equivariant bundles to the quotient for a free action, which is the bundle version of the homotopy equivalence $X_G\simeq X/G$ of Equivariant Cohomology; the third is the product decomposition of the equivariant bundles when the action is trivial. $\square$

These four computations, with the six-term sequence and the orbit decomposition, compute the equivariant $K$-theory of a $G$-CW complex from the representation rings of its isotropy groups, which is the strategy of the theory in practice.

The Atiyah–Segal Completion Theorem

The Statement

Theorem (Atiyah–Segal completion). Let $G$ be a compact Lie group acting on a compact $G$-CW complex $X$, let $R(G) = K_G^0(\mathrm{pt})$ be the representation ring and let $I \subseteq R(G)$ be the augmentation ideal. Then the completion of the equivariant $K$-theory at $I$ is isomorphic to the ordinary $K$-theory of the Borel construction,

$$ K_G^*(X)^{\wedge}_I \;\cong\; K^*(X_G) = K^*(EG\times_G X), $$

the maps being the natural comparison map followed by completion, and the isomorphism is natural in the equivariant map. Equivalently, the Atiyah–Segal completion theorem identifies the Borel model of $K$-theory with the $I$-adic completion of the equivariant theory.

Proof sketch. The theorem is proved by a comparison of the Atiyah–Hirzebruch spectral sequences of the two theories, or by an induction over the skeleta of the $G$-CW complex with the induction functors and the completion of the representation rings of the isotropy groups; the statement is quoted from the literature, and the proof uses the exact sequences and the periodicity developed above with the spectral sequence of The Leray–Serre Spectral Sequence for the Borel fibration of Equivariant Cohomology. $\square$

The Relation to Borel Cohomology

Corollary. The completion theorem is the $K$-theoretic analogue of the identification of the Borel cohomology of Equivariant Cohomology with the cohomology of the homotopy quotient: the equivariant theory computed internally, from the representation rings and the exact sequences, agrees after completion with the theory computed externally from the single space $X_G$. For a free action the completion is unnecessary and the theorem reduces to $K_G^*(X)\cong K^*(X/G)$; for the trivial action it reduces to $K^*(X)\otimes R(G)$ completed at $I$, which is $K^*(X)\otimes \mathbb{Z} = K^*(X)$ in the connected case, consistent with the product formula.

Proof. Apply the theorem to the two extreme cases, using the free-action and trivial-action computations above and the fact that the completion of $R(G)$ at the augmentation ideal is $\mathbb{Z}$ for a connected compact Lie group. $\square$

The completion theorem is the reason the Borel model is the operative definition of equivariant $K$-theory in the applications: it says that the Borel construction computes the equivariant theory up to a completion that is invisible on the free part, and it makes the equivariant $K$-theory computable by the ordinary $K$-theory of a single space.

Characteristic Classes and Localization

The Equivariant Chern Character

Definition. The equivariant Chern character is the natural transformation

$$ \mathrm{ch}_G : K_G^0(X)\otimes\mathbb{Q} \longrightarrow \bigoplus_n H_G^{2n}(X;\mathbb{Q}), $$

built from the equivariant Chern classes of the $G$-bundles and the Chern character of Topological K-Theory, with values in the Borel cohomology of Equivariant Cohomology, and it is a ring homomorphism after the tensor product with the rationals.

Theorem. For a compact connected Lie group $G$ acting on $X$, the equivariant Chern character is an isomorphism after tensoring with $\mathbb{Q}$ onto the Weyl-invariant part of the Borel cohomology,

$$ \mathrm{ch}_G : K_G^0(X)\otimes\mathbb{Q} \;\xrightarrow{\ \cong\ }\; H_G^{\mathrm{even}}(X;\mathbb{Q})^{W}, $$

with $W$ the Weyl group of $G$; for the torus $W$ acts trivially and the Chern character is the isomorphism $K_T^0(X)\otimes\mathbb{Q}\cong H_T^{\mathrm{even}}(X;\mathbb{Q})$. The theorem is the equivariant refinement of the Chern character isomorphism of Topological K-Theory and the bridge between the equivariant $K$-theory of this article and the equivariant cohomology of Equivariant Cohomology.

Proof. The ordinary Chern character is a rational isomorphism, and the equivariance restricts it to the invariants under the action of the Weyl group on the Borel cohomology; the statement is quoted, and it is the equivariant form of the Chern character theorem. $\square$

The Localization Theorem

Theorem (Atiyah–Segal localization). Let $T$ be a torus acting on a compact $T$-space $X$ and let $F = X^T$ be the fixed set. Then the inclusion $F\hookrightarrow X$ induces an isomorphism after localizing the representation ring $R(T)$ at the multiplicative set generated by the nontrivial characters that occur in the normal directions to $F$:

$$ K_T^*(X)\otimes_{R(T)} R(T)[\mathcal{S}^{-1}] \;\cong\; K_T^*(F)\otimes_{R(T)} R(T)[\mathcal{S}^{-1}] \;\cong\; K^*(F)\otimes R(T)[\mathcal{S}^{-1}] . $$

In particular the equivariant $K$-theory of a torus action is determined by the fixed set, and the Euler classes of the normal bundles are the localization denominators. The theorem is the $K$-theoretic counterpart of the localization theorem of Equivariant Cohomology and the tool by which the equivariant index is computed from the fixed set in The Atiyah–Singer Index Theorem and K-Theory.

Proof. The localization is proved by the equivariant Bott periodicity and the exact sequence of the pair $(X, X\smallsetminus F)$, in which the composite of the restriction to the fixed set and the Euler class of the normal bundle is an isomorphism after inverting the classes; the statement is quoted. $\square$

Examples

Example (the point and the orbit). $K_G^0(\mathrm{pt}) = R(G)$ and $K_G^0(G/H) = R(H)$; for $G$ finite, $R(G)$ is free on the irreducibles, and the tensor product with the module structure over $R(G)$ is the representation theory of the group. The Atiyah–Segal completion of $R(G)$ at the augmentation ideal is $\mathbb{Z}$, so the Borel construction of a point has the $K$-theory of a point, as it must since $EG$ is contractible.

Example (spheres with a circle action). Let $S^1$ act on $S^{2n+1}\subset\mathbb{C}^{n+1}$ by scalar multiplication, the free action of Equivariant Cohomology; then $K_{S^1}^0(S^{2n+1})\cong K^0(\mathbb{CP}^n)$, a free abelian group of rank $n+1$ generated by the powers of the tautological line bundle, and it is the quotient $R(S^1)/(z^{n+1}-1)$-module corresponding to the truncation of the polynomial ring; the completion theorem identifies this with the $K$-theory of the Borel construction, and the Chern character maps it to the truncated cohomology ring of the projective space.

Example (the involution and the projective space). Let $G = \mathbb{Z}/2$ act antipodally on $S^n$, so that the action is free and $K_{\mathbb{Z}/2}^0(S^n)\cong K^0(\mathbb{RP}^n)$; the representation ring is $R(\mathbb{Z}/2) = \mathbb{Z}[\epsilon]/(\epsilon^2-1)$ with $\epsilon$ the sign representation, and the equivariant bundles on $S^n$ are the bundles on the projective space with the lifted involutions. The Atiyah–Segal theory of the real $K$-theory of a real space, in which the complexification of the real bundles and the involution are the data, is the $\mathbb{Z}/2$-equivariant refinement of the ordinary theory and is the historical origin of the equivariant $K$-theory of Atiyah and Segal.

Summary

Equivariant $K$-theory is the Grothendieck group of the $G$-vector bundles over a compact $G$-space, graded by the suspension, $K_G^{-n}(X)=K_G^0(\Sigma^nX_+)$, with the representation ring $R(G)=K_G^0(\mathrm{pt})$ as its value at a point and as the ring over which it is a module. Bott periodicity $K_G^0(X)\cong K_G^2(X)$ holds equivariantly, the theory has natural six-term exact sequences for pairs and an induction–restriction sequence for a subgroup, and it is computed on the orbits by $K_G^0(G/H)=R(H)$, on free actions by $K_G^0(X)\cong K^0(X/G)$ and on trivial actions by $K_G^0(X)\cong K^0(X)\otimes R(G)$. The Atiyah–Segal completion theorem identifies the completion of the equivariant theory at the augmentation ideal of $R(G)$ with the ordinary $K$-theory of the Borel construction, $K_G^*(X)^{\wedge}_I\cong K^*(X_G)$, which is the $K$-theoretic analogue of the Borel model of equivariant cohomology. The equivariant Chern character is a rational isomorphism from the equivariant $K$-theory to the Weyl-invariant Borel cohomology, and the Atiyah–Segal localization theorem computes the equivariant $K$-theory of a torus action from the fixed set after inverting the Euler classes. The ordinary theory is that of Topological K-Theory, the Borel model is that of Equivariant Cohomology, and the operator-algebraic identification $K_G^*(X)\cong K_*(C_0(X)\rtimes G)$ and the equivariant index theorem are those of K-Theory of Operator Algebras, KK-Theory and Part IV. Nothing analytic and nothing geometric was used beyond the named connections.

Summary of Notation

Symbol	Meaning
$G$, $X$	Compact Lie group and compact $G$-space
$G$-vector bundle	Complex bundle with a fibrewise linear $G$-action covering $X$
$\mathrm{Vect}_G(X)$	Monoid of $G$-bundles under direct sum
$K_G^0(X)$	Grothendieck group of the $G$-bundles
$R(G) = K_G^0(\mathrm{pt})$	Representation ring of $G$
$I = \ker(R(G)\to\mathbb{Z})$	Augmentation ideal; the ideal of the completion
$K_G^{-n}(X)$, $K_G^1(X)$	Higher groups; $K_G^0\cong K_G^2$ by Bott periodicity
$(X,Y)$, six-term sequence	Exact sequence of a $G$-pair; induction–restriction
$K_G^0(G/H)=R(H)$	Value on the orbit; representation ring of the isotropy
$K_G^0(X)\cong K^0(X/G)$	Free action; orbit-space $K$-theory
$K_G^0(X)\cong K^0(X)\otimes R(G)$	Trivial action
$K_G^(X)^{\wedge}_I\cong K^(X_G)$	Atiyah–Segal completion theorem
$\mathrm{ch}_G$	Equivariant Chern character; Weyl-invariant Borel cohomology
localization at $X^T$	Atiyah–Segal localization theorem for a torus