03.08.10 · modern-geometry / k-theory

Equivariant K-theory $K_{G} (X)$ and $R (G)$

shipped3 tiersLean: none

Anchor (Master): Atiyah-Segal *The index of elliptic operators II* (Ann. Math. 87, 1968); Segal *Equivariant K-theory* (Publ. IHES 34, 1968); Atiyah-Segal *Completion theorem* (J. Diff. Geom. 3, 1969); Atiyah *Bott periodicity and the index of elliptic operators* (Quart. J. Math. 19, 1968); Lawson-Michelsohn §I.9 + §III

Intuition [Beginner]

Equivariant K-theory is K-theory done in the presence of a symmetry group. If a compact group $G$ acts on a space $X$ , then vector bundles over $X$ acquire a richer structure when their fibres rotate compatibly with the group action. Equivariant K-theory $K_{G} (X)$ records the Grothendieck group of these symmetry-aware bundles, and it captures information that ordinary K-theory cannot see.

The most basic example is when $X$ is a single point. A vector bundle over a point is a vector space; a $G$ -equivariant vector bundle over a point is a vector space carrying a representation of $G$ . The Grothendieck group of representations is the representation ring $R (G)$ , and it sits at the foundation of everything that follows. So $K_{G} (pt) = R (G)$ is the coefficient ring of equivariant K-theory, just as the integers are the coefficient ring of ordinary K-theory.

The reason this matters is that many natural geometric problems carry symmetries. Elliptic operators on a manifold acted on by a group, fixed-point computations, induced representations — all of these are best organised by passing to the equivariant framework first and then specialising back at the end.

Visual [Beginner]

A schematic with three frames: on the left, an ordinary vector bundle over a base space, with a fibre at one point; in the middle, the same bundle equipped with a group action that rotates each fibre while moving its base point compatibly; on the right, the single special case where the base is a point, leaving only a representation of the group. The arrow connecting the middle to the right is labelled $X = pt$ and the right-hand frame is captioned $R (G) = K_{G} (pt)$ .

The picture captures the essential message: equivariant K-theory packages two pieces of data at once — the underlying topology of the bundle and the action of the group on its fibres — and the case of a point isolates the representation-theoretic half of that package.

Worked example [Beginner]

Compute the representation ring of the circle group $G = U (1)$ and then of the torus $T^{2} = U (1) \times U (1)$ .

Step 1. The circle group $U (1)$ has one irreducible complex representation in each integer degree: for each $n$ in the integers, the representation $χ_{n}$ sends $z$ in $U (1)$ to multiplication by $z^{n}$ on the complex line. So the irreducibles of $U (1)$ are parametrised by the integers.

Step 2. The Grothendieck group of finite-dimensional complex representations of $U (1)$ is the free abelian group on the irreducibles, which is the free abelian group on the integers. Calling the generator corresponding to $χ_{1}$ by the letter $t$ , the inverse $χ_{- 1}$ is given by $t^{- 1}$ , and the ring structure (tensor product) gives $χ_{m}$ tensor $χ_{n} = χ_{m + n}$ , which is $t^{m} \cdot t^{n} = t^{m + n}$ . The representation ring is $R (U (1)) = Z [t, t^{- 1}]$ , the ring of Laurent polynomials.

Step 3. For the torus $T^{2}$ , the irreducibles are products of irreducibles for the two factors, parametrised by pairs of integers $(m, n)$ . The representation ring is the tensor product of the two circle-rings: $R (T^{2}) = Z [t_{1}, t_{1}^{- 1}, t_{2}, t_{2}^{- 1}]$ , Laurent polynomials in two variables.

Step 4. The pattern extends: $R (T^{n}) = Z [t_{1}^{\pm}, \dots, t_{n}^{\pm}]$ for the $n$ -torus. Each variable corresponds to one of the standard $U (1)$ -eigenvalue characters.

What this tells us: the representation ring of a torus is a ring of Laurent polynomials in the eigenvalue coordinates, and this is the building block out of which the representation rings of more elaborate compact Lie groups are assembled. The full unitary group $U (n)$ has representation ring $Z [x_{1}, \dots, x_{n}]^{S_{n}}$ , the symmetric polynomials in the eigenvalues — the Weyl-group invariants inside the torus ring.

Check your understanding [Beginner]

Exercise (easy, true-false).

True or false: if the group $G$ acts on the space $X$ with only the identity action, then $K_{G} (X)$ agrees with ordinary $K (X)$ .

Hint

When every group element acts as the identity, a $G$ -equivariant vector bundle still requires a fibrewise representation of $G$ , even though no points of $X$ are moved.

Answer

False. Even when $G$ acts as the identity on $X$ , a $G$ -equivariant vector bundle over $X$ has fibres that are $G$ -representations, and the structure remembers which representations appear at each point. In fact, when $X$ has the identity action, $K_{G} (X)$ identifies with $K (X)$ paired with $R (G)$ in the simple case where $X$ is connected — the K-theory of $X$ paired with the representation ring. The representation-ring factor is the genuinely new data, so equivariant K-theory is strictly richer than ordinary K-theory whenever $R (G)$ is larger than $Z$ .

Formal definition [Intermediate+]

Let $G$ be a compact Lie group and let $X$ be a compact Hausdorff $G$ -space — a topological space equipped with a continuous action $G \times X \to X$ . A $G$ -equivariant complex vector bundle over $X$ is a complex vector bundle $π : E \to X$ together with a continuous $G$ -action on $E$ such that $π$ is $G$ -equivariant ( $π (g \cdot e) = g \cdot π (e)$ ) and the map $E_{x} \to E_{g \cdot x}$ induced by $g$ is complex-linear on each fibre.

The isomorphism classes of $G$ -equivariant complex vector bundles on $X$ , denoted $Vect_{G} (X)$ , form a commutative monoid under direct sum. The equivariant K-theory of $X$ is the Grothendieck group of this monoid: $$ K_G(X) := \mathrm{Groth}(\mathrm{Vect}_G(X)). $$ Tensor product of $G$ -bundles makes $K_{G} (X)$ a commutative ring with unit the class of the rank-one product line bundle carrying the identity $G$ -action. Pullback along an equivariant continuous map $f : X \to Y$ produces a ring homomorphism $f^{*} : K_{G} (Y) \to K_{G} (X)$ .

Definition (representation ring). Let $G$ be a compact Lie group. The representation ring $R (G)$ is the Grothendieck group of the commutative monoid of isomorphism classes of finite-dimensional continuous complex unitary representations of $G$ under direct sum, equipped with the ring structure from tensor product of representations. The unit is the class of the one-dimensional identity representation; the augmentation $ϵ : R (G) \to Z$ sends a virtual representation to its virtual dimension.

Proposition (identification of $R (G)$ with $K_{G} (pt)$ ). A $G$ -equivariant complex vector bundle over a one-point space $pt$ is precisely a finite-dimensional complex representation of $G$ . The Grothendieck-group construction matches on both sides, so $K_{G} (pt) = R (G)$ as commutative rings.

The unique map $X \to pt$ is $G$ -equivariant for the identity action on the target, and pullback equips every $K_{G} (X)$ with the canonical structure of an $R (G)$ -module. Module structure: a class $[V] \in R (G)$ acts on $[E] \in K_{G} (X)$ by $[V] \cdot [E] = [V \otimes E]$ , where $V \otimes E$ is the bundle whose fibre at $x$ is $V \otimes E_{x}$ and on which $G$ acts diagonally.

Counterexamples to common slips

$K_{G} (X)$ is not the $G$ -fixed-points of $K (X)$ in general. The construction tracks the full $G$ -action on bundles, not just bundles whose isomorphism class is fixed by $G$ . The two can disagree even when $G$ acts freely on $X$ .
The representation ring $R (G)$ is generally not a polynomial ring: $R (S U (2)) = Z [V]$ is a polynomial ring on the standard rep, but $R (S O (3)) = Z [V_{3}]$ where $V_{3}$ is the $3$ -dimensional adjoint, and $R (U (n)) = Z [x_{1}, \dots, x_{n}]^{S_{n}}$ is a ring of symmetric polynomials, not a polynomial ring on the irreducibles.
A complex representation is not the same as a real representation. The complex representation ring $R (G)$ is what appears in $K_{G}$ for complex equivariant K-theory; the real version $R O (G)$ corresponds to $K O_{G}$ and is generally different (e.g., for $G = Z /2$ , $R O (G) = Z [σ] / (σ^{2} - 1)$ but $R (G) = Z [σ] / (σ^{2} - 1)$ with the sign representation in both — the rings agree only because the relevant representations are real; for $G = S U (2)$ the rings differ).
Equivariant Bott periodicity is not the assertion that $K_{G}$ is two-periodic in suspension by an arbitrary two-cell. The Bott element lives in $K_{G} (V)$ for $V$ a complex representation of $G$ ; suspension by a real two-dimensional representation that is not complex (e.g., the sign representation summed with itself for $G = Z /2$ ) does not yield the same isomorphism.

Key theorem with proof [Intermediate+]

Theorem (equivariant Bott periodicity; Atiyah-Segal 1968). Let $G$ be a compact Lie group, $V$ a finite-dimensional complex representation of $G$ regarded as a $G$ -space, and $X$ a compact $G$ -space. Pullback along the zero-section produces a natural isomorphism $$ K_G(X) \xrightarrow{\sim} K_G(X \times V) $$ in the appropriate $Z /2$ -graded sense, given by multiplication with the equivariant Bott class $β_{V} \in K_{G} (V)$ .

The case $V = C$ with identity $G$ -action recovers ordinary Bott periodicity. The case of an arbitrary complex representation $V$ — twisted by $G$ -action — is the equivariant refinement and is the substantive new content.

Proof. The argument follows Atiyah-Segal's strategy of reducing to the non-equivariant Bott theorem via the Thom isomorphism. Step one: define the equivariant Thom class. For a complex $G$ -vector bundle $E \to X$ , the equivariant Koszul complex $$ \Lambda^\bullet E^* = \bigoplus_{k=0}^{\mathrm{rk}, E} \Lambda^k E^* $$ carries a $Z /2$ -grading by parity of $k$ and a differential given by interior product with the tautological section, acyclic off the zero section. The class $$ \lambda_E = \sum_{k=0}^{\mathrm{rk}, E} (-1)^k [\Lambda^k E^*] \in K_G(E) $$ is the equivariant Thom class.

Step two: prove the equivariant Thom isomorphism $$ \phi_E: K_G(X) \xrightarrow{\sim} \widetilde K_G(\mathrm{Th}(E)), \qquad x \mapsto \pi^* x \cdot \lambda_E, $$ where $Th (E)$ is the Thom space of $E$ (one-point compactification of $E$ rel. zero section) and $π : E \to X$ is the projection. Verification: the equivariant Koszul resolution makes $λ_{E}$ a class whose restriction to each fibre generates the reduced K-theory of the fibre, freely over $R (G)$ . The Mayer-Vietoris and induction arguments of the non-equivariant case transfer term by term because the construction is $G$ -equivariant at every step.

Step three: specialise. Take $E = V$ , a product bundle in the topological sense, equipped with the $G$ -action coming from a representation of $G$ on the fibre $V$ . The Thom space $Th (V) = S^{V}$ (the equivariant suspension $G$ -sphere) and the Thom isomorphism reads $$ K_G(X) \cong \widetilde K_G(X^+ \wedge S^V), $$ which is the asserted suspension isomorphism. In the special case $V = C$ this is two-fold suspension and the isomorphism is the classical Bott map.

Step four (compatibility). The Bott class $β_{V} \in K_{G} (V)$ is the image of the generator under the Thom isomorphism applied to $X = pt$ . The composition with the canonical map $K_{G} (X) \to K_{G} (X \times V)$ is multiplication by $β_{V}$ , as asserted. Naturality in $X$ and in $V$ follows from the functoriality of the Thom-isomorphism construction. $□$

Theorem (Atiyah-Segal completion theorem; Atiyah-Segal 1969). Let $G$ be a compact Lie group, $I \subset R (G)$ the augmentation ideal (kernel of $dim : R (G) \to Z$ ), and $E G$ a contractible $G$ -space with free $G$ -action. The natural map $K_{G} (pt) \to K_{G} (E G)$ extends to a ring isomorphism $$ \widehat{K_G(\mathrm{pt})}_I \cong K(BG), $$ where the left side is the $I$ -adic completion of $R (G)$ and $B G = E G / G$ is the classifying space of $G$ .

The proof is deferred to the Full proof set; the statement is the bridge between equivariant K-theory of contractible $G$ -spaces and ordinary K-theory of classifying spaces.

Bridge. The equivariant Bott theorem builds toward the full machinery of the equivariant index theorem and the Atiyah-Segal completion theorem, both of which appear again in 03.09.10 (Atiyah-Singer index theorem) in the family-equivariant form developed by Atiyah-Singer III. The foundational reason equivariant K-theory carries a representation-ring action is exactly that every $G$ -space admits a canonical equivariant map to the one-point space, and pullback along this map equips $K_{G} (X)$ with the canonical $R (G)$ -module structure. This is exactly the same mechanism that organises every equivariant cohomology theory: the coefficient ring lives over a point, and the module structure on a general space is pullback from the point. The central insight is that the representation ring $R (G)$ is the universal equivariant scalar ring, and the bridge is that the Atiyah-Segal completion theorem identifies the $I$ -adically completed $R (G)$ with the ordinary K-theory of the classifying space $B G$ — putting these together, equivariant computations over arbitrary $G$ -spaces and ordinary computations over classifying spaces are two facets of the same algebraic structure, separated only by an adic completion. This recurs as a pattern in 03.09.21 (family-equivariant index) where the index of an equivariant operator lives in $R (G)$ before specialising via characters; and it generalises the classical fact that the cohomology of $B G$ records the polynomial invariants of the representation ring of $G$ .

Exercises [Intermediate+]

Exercise 3 (medium, symbolic).

Show that $R (S U (2)) ≅ Z [V]$ as rings, where $V = V_{2}$ is the standard two-dimensional representation, and identify the irreducibles $V_{n}$ in terms of $V$ .

Hint

The symmetric powers $Sym^{n - 1} (V)$ are the irreducibles of $S U (2)$ . The Clebsch-Gordan rule $V \otimes V_{n} = V_{n + 1} \oplus V_{n - 1}$ gives a recurrence.

Answer

Each $V_{n}$ is the symmetric power $Sym^{n - 1} (V)$ of dimension $n$ . The Clebsch-Gordan rule $$ V \otimes V_n = V_{n+1} \oplus V_{n-1} $$ gives a recurrence: in the ring $R (S U (2))$ , writing $v = [V]$ , we have $v \cdot V_{n} = V_{n + 1} + V_{n - 1}$ , so $V_{n + 1} = v V_{n} - V_{n - 1}$ . Starting from $V_{1} = 1$ and $V_{2} = v$ , the recursion produces $V_{n} = U_{n - 1} (v /2)$ where $U_{k}$ is the Chebyshev polynomial of the second kind — a polynomial in $v$ of degree $n - 1$ with integer coefficients. So the irreducibles span the polynomial ring $Z [v]$ in $v$ as a free abelian group, and the multiplication is determined by the tensor-product rule, giving $R (S U (2)) ≅ Z [V]$ . Rubric: full credit for the Clebsch-Gordan derivation plus the Chebyshev identification.

Exercise 4 (medium, symbolic).

Compute the augmentation ideal $I \subset R (Z / p)$ for $p$ a prime and identify $I^{k}$ in the ring.

Hint

The augmentation is $dim : R (Z / p) \to Z$ . Use the identification $R (Z / p) = Z [t] / (t^{p} - 1)$ .

Answer

In $R (Z / p) = Z [t] / (t^{p} - 1)$ , the augmentation $ϵ : R \to Z$ sends $t \mapsto 1$ , so the augmentation ideal is $I = (t - 1)$ . Computing powers: $(t - 1)^{p} = \sum_{k = 0}^{p} (k p) t^{k} (- 1)^{p - k}$ . Modulo $t^{p} - 1$ , using $t^{p} = 1$ , we obtain $(t - 1)^{p} = - p + p \cdot (element with augmentation 0)$ , so $(t - 1)^{p} \in p \cdot R + I^{2} \cdot I^{p - 2}$ . The completed ring $R (Z / p)_{I}$ is isomorphic to the $p$ -adic integers ring $Z_{p} [[t - 1]] / ((t - 1)^{p} \cdot (unit) - p \cdot (unit))$ which collapses to $Z_{p} [[t - 1]]$ as a complete local ring with residue field $F_{p}$ . Rubric: full credit for identifying $I = (t - 1)$ and noting the $p$ -adic flavor of the completion.

Exercise 5 (medium, short-answer).

Show that for a finite group $G$ and a compact $G$ -space $X$ on which $G$ acts freely, $K_{G} (X) ≅ K (X / G)$ .

Hint

A $G$ -equivariant bundle on $X$ with free action descends to a bundle on the quotient $X / G$ , and conversely a bundle on $X / G$ pulls back to a $G$ -equivariant bundle on $X$ .

Answer

For free actions, the quotient map $q : X \to X / G$ is a principal $G$ -bundle. A $G$ -equivariant complex vector bundle $E \to X$ descends uniquely to a bundle $\overset{ˉ}{E} \to X / G$ with $E = q^{*} \overset{ˉ}{E}$ , where $\overset{ˉ}{E}_{y} = E_{x} / (identification by stabiliser)$ for $x$ in the fibre $q^{- 1} (y)$ ; since the stabiliser is the identity element by freeness, no identification is needed and $\overset{ˉ}{E}_{y} ≅ E_{x}$ canonically. Conversely, pullback $q^{*}$ sends bundles on $X / G$ to $G$ -equivariant bundles on $X$ . The two functors are mutually inverse on isomorphism classes, descending to a ring isomorphism $K_{G} (X) ≅ K (X / G)$ . Rubric: full credit for the descent argument and the inverse pullback.

Exercise 6 (medium, short-answer).

Identify the $R (G)$ -module structure on $K_{G} (X)$ when $G$ acts on $X$ with the identity action, and conclude $K_{G} (X) ≅ K (X) \otimes_{Z} R (G)$ for $X$ a finite CW complex.

Hint

When the action is the identity, a $G$ -equivariant bundle decomposes as a sum of isotypic components, one for each irreducible of $G$ .

Answer

When $G$ acts as the identity on $X$ , each $G$ -equivariant bundle $E$ decomposes fibrewise into isotypic pieces $E = ⨁_{[V]} E_{[V]}$ indexed by the irreducible representations $[V]$ of $G$ , where the isotypic component $E_{[V]}$ has fibre $Hom_{G} (V, E_{x}) \otimes V$ at $x$ . The Hom-piece $Hom_{G} (V, E_{x})$ depends only on $x$ — it is an ordinary complex vector space — and assembles into an ordinary vector bundle $E_{V}^{(0)}$ on $X$ . The map $E \mapsto \sum_{V} [E_{V}^{(0)}] \otimes [V]$ defines an isomorphism $K_{G} (X) \to K (X) \otimes R (G)$ . The $R (G)$ -module structure comes from the $R (G)$ -factor: $[W] \cdot ([F] \otimes [V]) = [F] \otimes [W \otimes V]$ . Rubric: full credit for the isotypic decomposition and the resulting tensor-product identification.

Exercise 7 (hard, short-answer).

Verify that $R (T) = Z [t_{1}^{\pm}, \dots, t_{n}^{\pm}]$ for the maximal torus $T = U (1)^{n}$ of $U (n)$ , and that the restriction homomorphism $R (U (n)) \to R (T)$ identifies $R (U (n))$ with the symmetric polynomials $Z [t_{1}^{\pm}, \dots, t_{n}^{\pm}]^{S_{n}}$ .

Hint

The torus is abelian, so its irreducibles are products of $U (1)$ -characters. For $U (n)$ , the Weyl group $S_{n}$ permutes the eigenvalue coordinates, and a representation of $U (n)$ restricts to a representation of $T$ whose character is $S_{n}$ -invariant by the orbit-eigenvalue argument.

Answer

For the $n$ -torus $T = U (1)^{n}$ , the irreducibles are the characters $χ_{(k_{1}, \dots, k_{n})} : (z_{1}, \dots, z_{n}) \mapsto z_{1}^{k_{1}} \dots z_{n}^{k_{n}}$ for any tuple of integers $(k_{1}, \dots, k_{n})$ in $Z^{n}$ . Tensoring multiplies exponents, so $R (T) = Z [t_{1}^{\pm}, \dots, t_{n}^{\pm}]$ with $t_{i}$ corresponding to the $i$ -th coordinate character. For $U (n)$ , any representation $ρ$ when restricted to $T$ decomposes into eigencharacters, and its restricted character is the formal sum $ch_{T} (ρ ∣_{T}) = \sum_{μ} m_{μ} \cdot t^{μ}$ where $μ$ ranges over weights and $m_{μ}$ are multiplicities. Conjugation by elements of $U (n)$ that permute the diagonal coordinates produces the Weyl group $S_{n}$ , which acts on $T$ by permuting the $t_{i}$ and which fixes the character of $ρ$ since $ρ ∣_{T}$ is determined by $ρ$ . So the image of $R (U (n)) \to R (T)$ lies in the symmetric polynomials $R (T)^{S_{n}}$ . Conversely, every symmetric Laurent polynomial in $t_{1}, \dots, t_{n}$ is the character of a virtual representation of $U (n)$ (Schur polynomials are the characters of irreducible polynomial representations, plus a determinant twist for the $- 1$ powers), so the map is surjective. Injectivity follows from the maximal-torus theorem: a representation is determined by its restriction to a maximal torus. Therefore $R (U (n)) ≅ R (T)^{S_{n}} = Z [t_{1}^{\pm}, \dots, t_{n}^{\pm}]^{S_{n}}$ . Rubric: full credit for the maximal-torus / Weyl-invariance argument with the Schur-polynomial surjectivity remark.

Exercise 8 (hard, short-answer).

State the Atiyah-Segal completion theorem and use it to compute $K^{*} (B Z / p)$ for $p$ a prime as an $I$ -adically completed ring.

Hint

The theorem says $K^{*} (B G) = R (G)_{I}$ . For $G = Z / p$ , the representation ring is $Z [t] / (t^{p} - 1)$ and the augmentation ideal is $(t - 1)$ .

Answer

The Atiyah-Segal completion theorem states that for a compact Lie group $G$ , the natural map $K_{G} (E G) \to K (B G)$ is the $I$ -adic completion: $K (B G) ≅ K_{G} (pt)_{I} = R (G)_{I}$ , where $I = ker (ϵ : R (G) \to Z)$ is the augmentation ideal. For $G = Z / p$ , $R (G) = Z [t] / (t^{p} - 1)$ and $I = (t - 1)$ . The polynomial $t^{p} - 1$ factors as $(t - 1) (t^{p - 1} + t^{p - 2} + \dots + 1)$ , so $t^{p} - 1 \equiv 0$ in $R (Z / p)$ means $(t - 1) \cdot ((t^{p} - 1) / (t - 1)) = 0$ . The completion at $I = (t - 1)$ inverts the second factor (which equals $p$ modulo $(t - 1)$ , hence becomes a unit only after $p$ -adic completion). The completion is $R (Z / p)_{(t - 1)} = Z_{p} [[t - 1]] / ((t - 1)^{p - 1} (t - 1) - p \cdot (unit))$ which simplifies to $K^{*} (B Z / p) ≅ Z_{p} [[y]]$ with $y = t - 1$ , equipped with the implicit relation coming from $t^{p} = 1$ . In K-theory grading, the result lives in degree zero (Bott-periodicity), and the answer matches Atiyah's 1961 direct computation of $K^{*} (B Z / p)$ via the Atiyah-Hirzebruch spectral sequence. Rubric: full credit for stating the theorem and identifying the $p$ -adic completion of $Z [t] / (t^{p} - 1)$ at $(t - 1)$ .

Lean formalization [Intermediate+]

Mathlib does not currently package equivariant K-theory; the formalisation is one of the largest open targets in the K-theory roadmap. A schematic of the intended statements:

import Mathlib.Topology.Algebra.Group.Basic
import Mathlib.RepresentationTheory.Maschke
import Mathlib.CategoryTheory.Monoidal.Category
import Mathlib.Algebra.GroupRingAction

namespace EquivariantKTheory

variable (G : Type*) [TopologicalGroup G] [CompactSpace G]
variable (X : Type*) [TopologicalSpace X] [MulAction G X] [ContinuousSMul G X]

/-- A G-equivariant complex vector bundle over a G-space. -/
structure GEquivariantBundle where
  total      : Type*
  proj       : total → X
  fibreSpace : X → Type*
  -- continuous G-action on total commuting with proj, linear on fibres
  smul       : G → total → total
  -- ...all the bundle and equivariance axioms
  sorry

/-- Equivariant K-theory: Grothendieck group of equivariant bundles. -/
noncomputable def KG : Type* := sorry -- Grothendieck of (Vect_G X / iso)

instance : CommRing (KG G X) := sorry -- ring from direct sum + tensor

/-- Representation ring R(G) = K_G(pt). -/
noncomputable def RepRing : Type* := KG G PUnit

/-- R(G)-module structure on K_G(X) via pullback along X → pt. -/
instance : Module (RepRing G) (KG G X) := sorry

/-- Equivariant Bott periodicity for complex G-representations. -/
theorem equivariant_bott
    (V : Type*) [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V]
    [Representation G V] :
    KG G X ≃+* KG G (X × V) := sorry

/-- Atiyah-Segal completion theorem (statement schematic). -/
theorem atiyah_segal_completion
    (G : Type*) [TopologicalGroup G] [CompactSpace G] :
    True := sorry -- K_G(EG)^∧_I ≅ K(BG); needs EG, BG, I-adic completion

end EquivariantKTheory

The formalisation gap is substantial: Mathlib needs equivariant continuous-map categories, the equivariant bundle category, the Grothendieck-completion functor on a symmetric monoidal category with direct sums, the identification of $K_{G} (pt)$ with the representation Grothendieck ring, the $R (G)$ -module structure assembled from a base-change functor, and the equivariant Thom-isomorphism / Bott-periodicity package. The completion theorem additionally requires the $I$ -adic completion functor in commutative algebra wired to the K-theory output. None of these are present in the form needed; the entire pipeline is a roadmap of contributions.

Advanced results [Master]

Theorem (induction and restriction; Segal 1968). Let $H \subset G$ be a closed subgroup of a compact Lie group $G$ . The forgetful functor from $G$ -bundles to $H$ -bundles induces a ring homomorphism $Res_{H}^{G} : K_{G} (X) \to K_{H} (X)$ for every $G$ -space $X$ . The induced-representation functor produces an $R (H)$ -linear map $Ind_{H}^{G} : K_{H} (X) \to K_{G} (X)$ when $X$ is a $G$ -space regarded as an $H$ -space by restriction, satisfying the Frobenius reciprocity formula $Ind_{H}^{G} (Res_{H}^{G} (a) \cdot b) = a \cdot Ind_{H}^{G} (b)$ for $a \in K_{G} (X)$ , $b \in K_{H} (X)$ .

The proof is functorial. Restriction is forgetful on the bundle level, hence a ring map. Induction is constructed by associated-bundle: given an $H$ -equivariant bundle $E \to X$ , form $G \times_{H} E$ as a $G$ -bundle over $G \times_{H} X = X$ (since $X$ is already a $G$ -space, $G \times_{H} X$ has a canonical $G$ -equivariant collapse to $X$ ). The Frobenius identity follows from the standard adjunction $G \times_{H} (E \otimes F ∣_{H}) = (G \times_{H} E) \otimes F$ .

Theorem (equivariant index theorem; Atiyah-Segal 1968). Let $M$ be a compact $G$ -manifold, $D$ an elliptic $G$ -equivariant differential operator with symbol class $σ (D) \in K_{G} (T M)$ . The equivariant analytic index $$ \mathrm{ind}_G(D) = [\ker D] - [\mathrm{coker}, D] \in R(G) $$ coincides with the equivariant topological index $t - ind_{G} (σ (D)) \in K_{G} (pt) = R (G)$ , where $t - ind_{G} : K_{G} (T M) \to R (G)$ is constructed by the equivariant Thom isomorphism and Atiyah-Segal embedding into a $G$ -representation.

Specialising the equivariant index to the trace at $g \in G$ recovers the Atiyah-Bott Lefschetz fixed-point formula: for $g$ acting on $M$ with isolated non-degenerate fixed points ${p_{i}}$ , $$ \mathrm{tr}(g, \mathrm{ind}(D)) = \sum_{p_i} \frac{\mathrm{tr}(g, \sigma_{p_i})}{\det(1 - dg_{p_i})}. $$

Theorem (Atiyah-Segal completion theorem; Atiyah-Segal 1969). For a compact Lie group $G$ and $E G$ a contractible $G$ -space with free action, the natural homomorphism $R (G) = K_{G} (pt) \to K_{G} (E G)$ identifies $K_{G} (E G)$ with the $I$ -adic completion $R (G)_{I}$ , where $I = ker (dim : R (G) \to Z)$ is the augmentation ideal. Furthermore, the projection $E G \to B G$ induces an isomorphism $K_{G} (E G) ≅ K (B G)$ , so $K (B G) ≅ R (G)_{I}$ .

The theorem connects equivariant K-theory of a contractible space (which encodes only the $R (G)$ structure) with ordinary K-theory of the corresponding classifying space, via $I$ -adic completion. Special cases recover Atiyah's 1961 computation of $K^{*} (B G)$ for various $G$ as completions of representation rings: $K^{*} (B U (n)) = Z [[c_{1}, \dots, c_{n}]]$ via the completion of $R (U (n)) = Z [x_{1}, \dots, x_{n}]^{S_{n}}$ at the augmentation, $K^{*} (B Z / p) = Z_{p} [[y]]$ , and Atiyah's original 1961 computation of $K^{*} (B S^{1}) = Z [[t - 1]]$ as the completion of $Z [t, t^{- 1}]$ .

Theorem (localisation at conjugacy class; Atiyah-Segal localisation). Let $G$ be a compact Lie group, $g \in G$ an element generating a closed subgroup $⟨ g ⟩$ , and $X^{g} \subset X$ the fixed-point set. Localising $R (G)$ at the prime ideal $p_{g} = {χ : χ (g) = 0}$ , the restriction map $$ K_G(X)_{\mathfrak{p}g} \xrightarrow{\sim} K_G(X^g){\mathfrak{p}_g} $$ is an isomorphism.

The localisation theorem reduces equivariant computations on a $G$ -space $X$ to computations on the fixed-point set $X^{g}$ , after inverting the conjugacy-class-of- $g$ data in $R (G)$ . This is the K-theoretic precursor of Borel-localisation in equivariant cohomology and underlies the Atiyah-Bott-Berline-Vergne integration formula for equivariant characteristic classes.

Theorem ( $R (G)$ as a $λ$ -ring). The representation ring $R (G)$ of any compact Lie group $G$ is a $λ$ -ring, with $λ^{k}$ acting by exterior power on representations: $λ^{k} ([V]) = [Λ^{k} V]$ . The Adams operations $ψ^{k}$ defined by the Newton formula coincide on irreducible characters with the action of the $k$ -th power map on $G$ : $ψ^{k} (χ) (g) = χ (g^{k})$ .

The $λ$ -ring structure on $R (G)$ is the prototype that motivated the abstract $λ$ -ring axioms of Atiyah-Tall 1969. The Adams operations on $R (G)$ encode the action of the power maps $g \mapsto g^{k}$ on conjugacy classes, dual to the character-theoretic side. This duality is the algebraic shadow of the Adams-Riemann-Roch theorem for the K-theory pushforward $π_{!} : K_{G} (M) \to K_{G} (pt) = R (G)$ from a $G$ -manifold $M$ to a point.

Synthesis. Equivariant K-theory is the foundational reason that representation theory of compact groups and K-theory of spaces share so much structural DNA. The central insight is that $R (G)$ , the representation ring of $G$ , is the canonical coefficient ring of equivariant K-theory — the value of the functor on the one-point $G$ -space — and every equivariant K-theory $K_{G} (X)$ is an $R (G)$ -algebra through pullback from this universal point. Putting these together, the four pillars of equivariant K-theory — equivariant Bott periodicity, the equivariant index theorem, the Atiyah-Segal completion theorem, and the localisation theorem — each appear again in 03.09.10 (Atiyah-Singer index theorem) and in 03.09.21 (family-equivariant index) as the technical engine driving fixed-point formulas, character-valued indices, and the connection between equivariant cohomology and equivariant K-theory through the Chern character.

The bridge is that every classical question about a $G$ -action — fixed points, induced representations, characters of representations on cohomology of moduli, equivariant cobordism — admits a K-theoretic refinement whose value is an element of $R (G)$ , generalising the integer-valued classical statement to a representation-valued statement that records the action of $G$ on the answer. This is exactly the same organising idea that recurs in equivariant cohomology, in equivariant cobordism, and in derived geometry through the loop-space construction: passing from $X$ to its inertia stack $Λ X = {(x, g) : g x = x}$ converts $G$ -equivariant data on $X$ into ordinary data on $Λ X$ , and the K-theory of the inertia stack with its $R (G)$ -module structure is precisely $K_{G} (X)$ . Putting these together, equivariant K-theory builds toward delocalised equivariant K-theory (Baum-Brylinski-MacPherson, Adem-Ruan), the K-theoretic Lefschetz fixed-point theorem of Atiyah-Bott, and ultimately the family-equivariant index theorem appears again in 03.09.21 (family-equivariant index theorem) where index classes live in the K-theory of the parameter space and refine to virtual representations of $G$ when an action is present. The same pattern generalises to twisted equivariant K-theory and to equivariant elliptic cohomology, both of which take the representation-ring structure of $R (G)$ as the seed and assemble higher-categorical machinery around it.

Full proof set [Master]

Proposition ( $R (G)$ -module structure on $K_{G} (X)$ ), proof. The unique map $p : X \to pt$ is $G$ -equivariant for the identity action on $pt$ . Functoriality of equivariant K-theory produces a ring map $p^{*} : K_{G} (pt) \to K_{G} (X)$ . The image of $[V] \in R (G)$ is the product bundle $X \times V$ equipped with the diagonal $G$ -action ( $g \cdot (x, v) = (g x, g v)$ ). For any $G$ -equivariant bundle $E \to X$ , the product $p^{*} [V] \cdot [E] = [(X \times V) \otimes_{X} E] = [V \otimes E]$ as a $G$ -bundle, recovering the asserted formula. The compatibility with addition follows from the additivity of pullback; multiplicativity of pullback gives the ring-homomorphism property. The $R (G)$ -module structure is thus the canonical structure pulled back from the point. $□$

Theorem (equivariant Bott periodicity), proof. Detailed argument. Let $V$ be a complex $G$ -representation. Construct the equivariant Thom class as follows. The Koszul complex $Λ^{∙} V^{*} = Λ^{0} V^{*} \oplus Λ^{1} V^{*} \oplus \dots$ carries a $Z /2$ -grading by parity and a differential $d : Λ^{k} V^{*} \to Λ^{k - 1} V^{*}$ given by interior product with the tautological section $σ (v) = v \in V \to V^{*}$ via the metric. The differential squares to zero, and the cohomology of $(Λ^{∙} V^{*}, d)$ is supported on the zero section: away from zero, the Koszul complex is acyclic by the standard Koszul resolution argument (an explicit chain homotopy is given by interior product with $v /∣ v ∣^{2}$ when $v \neq = 0$ ). The class $λ_{V} = \sum_{k} (- 1)^{k} [Λ^{k} V^{*}]$ is a virtual $G$ -bundle on $V$ supported at the origin in K-theory, i.e., representing a class in $K_{G} (V) = K_{G} (V, V ∖ 0)$ .

For a $G$ -space $X$ , the external product $$ K_G(X) \otimes K_G(V) \to K_G(X \times V), \qquad x \otimes y \mapsto \pi_X^* x \cdot \pi_V^* y, $$ sends $x \otimes λ_{V} \mapsto π_{X}^{*} x \cdot π_{V}^{*} λ_{V}$ . The Bott map $$ \beta: K_G(X) \to K_G(X \times V), \qquad x \mapsto x \cdot \lambda_V $$ is the candidate isomorphism. To verify it is bijective, use the Mayer-Vietoris and the explicit Thom-isomorphism construction: cover $X \times V$ by $X \times {∣ v ∣ < 2}$ and $X \times {∣ v ∣ > 1}$ , on each piece either the Koszul complex is contractible to the zero section or the bundle splits as a product. The compatible local isomorphisms glue to a global isomorphism by the Mayer-Vietoris five-lemma applied to the equivariant K-theory long exact sequences. This is the equivariant generalisation of the Atiyah-Bott clutching-function proof of ordinary Bott periodicity; full details in Atiyah-Segal 1968 §2.

The case $V = C$ with the standard $U (1)$ -action recovers ordinary Bott periodicity: $K (pt) \to K (S^{2}) = K (C P^{1})$ is the multiplication by $[H] - 1$ where $H$ is the tautological line bundle, and $λ_{V}$ for $V$ the standard $U (1)$ -rep equals exactly this Bott element in $K_{U (1)} (C) = R (U (1)) \cdot β$ . $□$

Theorem (Atiyah-Segal completion theorem), proof sketch. The proof proceeds by reduction to the case $G$ finite via the Borel construction, then to the case of cyclic groups via the structure theory of finite abelian groups, and finally to the case $G = Z / p$ via direct computation. For the cyclic case, $R (Z / p) = Z [t] / (t^{p} - 1)$ with augmentation ideal $I = (t - 1)$ ; the completion is $Z_{p} [[t - 1]]$ . Direct comparison with Atiyah's 1961 computation of $K^{*} (B Z / p)$ via the Atiyah-Hirzebruch spectral sequence verifies the isomorphism in this base case. For general compact Lie $G$ , the Borel construction $E G \times_{G} X$ and an inductive argument over the skeleta of $B G$ produces the completion theorem in full generality; the inductive step uses that each cell of $B G$ is finite-dimensional and that the Borel construction commutes with $I$ -adic completion in the appropriate sense. Full proof in Atiyah-Segal 1969 Equivariant K-theory and completion (J. Diff. Geom. 3, 1-18). $□$

Theorem (Frobenius reciprocity for induction and restriction), proof. Let $H \subset G$ be a closed subgroup, $E$ an $H$ -equivariant bundle on a $G$ -space $X$ , and $F$ a $G$ -equivariant bundle on $X$ . The induced bundle $Ind_{H}^{G} E = G \times_{H} E$ is the quotient of $G \times E$ by the relation $(g h, e) \sim (g, h \cdot e)$ . The associated-bundle formula $$ G \times_H (E \otimes (F|_H)) = (G \times_H E) \otimes F $$ follows from the canonical isomorphism $G \times_{H} (V \otimes W) = (G \times_{H} V) \otimes W$ for any $G$ -rep $W$ and $H$ -rep $V$ , applied fibrewise. Passing to K-theory classes gives the Frobenius formula $Ind_{H}^{G} (Res_{H}^{G} (a) \cdot b) = a \cdot Ind_{H}^{G} (b)$ . $□$

Theorem ( $R (G)$ as $λ$ -ring), proof. Define $λ^{k} : R (G) \to R (G)$ on a representation $V$ by $λ^{k} (V) = Λ^{k} V$ , the $k$ -th exterior power with induced $G$ -action $g \cdot (v_{1} \land \dots \land v_{k}) = g v_{1} \land \dots \land g v_{k}$ . Check the $λ$ -ring axioms: $λ^{0} = 1$ , $λ^{1} = id$ , and $λ^{k} (V \oplus W) = \sum_{i + j = k} λ^{i} V \otimes λ^{j} W$ (the exterior power of a direct sum). The Newton-polynomial formula then defines $ψ^{k} = N_{k} (λ^{1}, \dots, λ^{k})$ , and on a representation of dimension one $V = χ$ , $ψ^{k} (χ) = χ^{k}$ . By the splitting principle for compact Lie group representations (every rep is virtually a sum of one-dimensional reps after restriction to a maximal torus), this extends: $ψ^{k} (ρ)$ has character $χ_{ρ} (g^{k})$ , identifying $ψ^{k}$ with the $k$ -th power-map pullback on characters. $□$

Connections [Master]

Topological K-theory 03.08.01. Equivariant K-theory $K_{G} (X)$ specialises to ordinary K-theory $K (X)$ when $G$ is the one-element group, in which case $R (G) = Z$ and every $G$ -bundle is a plain bundle. More substantively, the augmentation map $K_{G} (X) \to K_{G} (X) \otimes_{R (G)} Z = K_{G} (X) / I \cdot K_{G} (X)$ partly recovers $K (X)$ — fully so in good cases like $X$ a free $G$ -space. The unit 03.08.01 provides the underlying K-theory infrastructure on which the equivariant refinement is built.
Adams operations 03.08.02. The representation ring $R (G)$ is the prototype $λ$ -ring: Adams operations $ψ^{k}$ on $R (G)$ act on irreducible characters by the rule $ψ^{k} (χ) (g) = χ (g^{k})$ , the pullback along the $k$ -th power map. This is the canonical example that motivated the abstract $λ$ -ring formalism of Atiyah-Tall 1969 and the systematic study of $ψ^{k}$ on topological K-theory. The equivariant generalisation of Adams operations to $K_{G} (X)$ is immediate by exterior power on $G$ -bundles, and the eigenvalue computation $ψ^{k} (β_{V}) = k^{d i m V} β_{V}$ on the Bott class governs the K-theoretic computations underlying the equivariant index theorem.
Bott periodicity 03.08.07. The equivariant Bott periodicity theorem $K_{G} (X) ≅ K_{G} (X \times V)$ for $V$ a complex $G$ -representation is the equivariant refinement of ordinary Bott periodicity, recovered when $G$ is the one-element group. The proof factors through the equivariant Thom isomorphism, which itself reduces to the equivariant Koszul resolution of a complex representation; ordinary Bott periodicity is the case of an identity action by $G$ on a one-dimensional complex space. The unit 03.08.07 provides the non-equivariant base case from which the equivariant version is bootstrapped.
Atiyah-Singer index theorem 03.09.10. The Atiyah-Singer index theorem becomes the equivariant index theorem in the presence of a compact group action: the analytic index $[ker D] - [coker D] \in R (G)$ records not just the integer dimension difference but the virtual character difference, and the topological index $t - ind_{G} : K_{G} (T M) \to R (G)$ computes this character through the equivariant Thom isomorphism. Specialising the equivariant index to fixed-point data via the Atiyah-Segal localisation theorem recovers the Atiyah-Bott Lefschetz fixed-point formula. The unit 03.09.10 is the natural downstream consumer where the equivariant machinery developed here is deployed.
Family-equivariant index theorem 03.09.21. The family-equivariant index of a $G$ -equivariant family of elliptic operators parametrised by a space $Y$ lives in $K_{G} (Y)$ , an $R (G)$ -module. When the family is parametrised by a point, this reduces to a single virtual representation in $R (G)$ ; for general $Y$ , the family index in $K_{G} (Y)$ encodes the $G$ -action on the family of kernel and cokernel bundles. The unit 03.09.21 is the principal application of equivariant K-theory in the index-theory framework and a direct consumer of the $R (G)$ -module structure developed here.
Bismut superconnection 03.09.23. The family Chern character of an equivariant family-index in $K_{G} (B)$ is computed at the level of differential forms by the equivariant Bismut superconnection of 03.09.23, with the $G$ -invariant connection on the fibration replacing the ordinary horizontal connection used in the non-equivariant case. Specialising the base to a point recovers the equivariant Chern character $K_{G}^{0} (pt) = R (G) \to H_{G}^{*} (pt; R)$ developed here, and the Bismut-Cheeger eta-form is the equivariant analogue of the family transgression form. The unit 03.09.23 is therefore the differential-geometric refinement that computes the Chern character of the equivariant K-theory classes constructed here, and the agreement of the two pictures — algebraic in $R (G)$ , analytic in heat kernels — is the family-index theorem in its equivariant form.
Atiyah-Hirzebruch spectral sequence 03.13.04. The Atiyah-Segal completion theorem $K (B G) ≅ R (G)_{I}$ is the algebraic explanation for the structure of the Atiyah-Hirzebruch spectral sequence for $K^{*} (B G)$ : the $E_{2}$ page is $H^{*} (B G; Z)$ which encodes characteristic classes, and convergence to the $I$ -adic completion of $R (G)$ records the representation-ring data as a topological refinement. Atiyah's 1961 computations of $K^{*} (B G)$ for classical groups were the original motivation and the first verification of what would become the completion theorem.
Kirillov character formula via the equivariant index 03.09.25. The Kirillov formula is the explicit geometric computation of the character map $ch_{G} : K_{G} (G / T) \to R (G)$ developed here. Concretely: the class $[L_{λ}] \in K_{T} (pt)$ pushed forward to $K_{G} (pt) = R (G)$ along the homogeneous fibration $G / T \to pt$ produces $[V_{λ}]$ via Borel-Weil, and Kirillov's formula expresses the equivariant Chern character of that pushforward — evaluated at a regular element $g = e^{X}$ — as the Fourier transform of the Liouville measure on the coadjoint orbit $O_{λ} \subset g^{*}$ corrected by the Duflo Jacobian $j (X)^{1/2}$ . The Atiyah-Bott Lefschetz formula (an equivariant K-theory computation in $R (T)$ then induced up to $R (G)$ ) gives the Weyl character formula on the same input; Kirillov is the orbit-integral reformulation of the same equivariant index. The downstream unit 03.09.25 is therefore the canonical worked example of the equivariant Chern character / induction-restriction / Atiyah-Segal localisation machinery developed here applied to homogeneous spaces.

Historical & philosophical context [Master]

The foundations of equivariant K-theory were laid by Atiyah and Segal in two foundational papers of 1968. Graeme Segal's Equivariant K-theory appeared in Publications mathématiques de l'IHÉS 34 (1968) pp. 129-151 ^{[Segal 1968 Equivariant K-theory]}; this is the definitive originator paper, giving the full axiomatic framework: $G$ -equivariant vector bundles on $G$ -spaces, the Grothendieck-completion ring $K_{G} (X)$ , the canonical $R (G)$ -module structure through pullback from the one-point space, the long exact sequences for $G$ -CW pairs, the equivariant Bott periodicity, the induction-restriction adjunction $Ind_{H}^{G} ⊣ Res_{H}^{G}$ for closed subgroups, and the change-of-group spectral sequence. Segal's exposition is still the cleanest axiomatic treatment, and the subsequent literature builds on its conventions.

The same year, Atiyah and Segal published The index of elliptic operators II in Annals of Mathematics (2) 87 (1968) pp. 531-545 ^{[Atiyah-Segal 1968 Index II]}, establishing the equivariant index theorem: the analytic index of a $G$ -equivariant elliptic operator $D$ on a compact $G$ -manifold lives in the representation ring $R (G)$ and equals the equivariant topological index $t - ind_{G} (σ (D))$ computed from the K-theoretic symbol class through the equivariant Thom isomorphism. The equivariant index theorem is the natural common generalisation of the Atiyah-Singer index theorem (Atiyah-Singer 1968 I, the same volume) and the Atiyah-Bott Lefschetz fixed-point formula for elliptic complexes (Ann. of Math. (2) 86, 1967, pp. 374-407) ^{[Atiyah-Bott 1967 Lefschetz I]}; specialising the equivariant index to a single group element $g \in G$ yields the trace-formula version of the Lefschetz theorem.

The Atiyah-Segal completion theorem appeared in Equivariant K-theory and completion in Journal of Differential Geometry 3 (1969) pp. 1-18 ^{[Atiyah-Segal 1969 Completion]}. The theorem identifies the ordinary K-theory $K (B G)$ of the classifying space with the $I$ -adic completion of the representation ring $R (G)$ at the augmentation ideal. This bridges two different worlds: the world of compact-group representation theory, captured by $R (G)$ , and the world of stable homotopy theory of classifying spaces, captured by $K (B G)$ and the Atiyah-Hirzebruch spectral sequence. Atiyah's 1961 paper Characters and cohomology of finite groups (Publ. Math. IHÉS 9, pp. 23-64) ^{[Atiyah 1961 Characters and cohomology]} had already computed $K^{*} (B G)$ for finite groups by direct spectral-sequence arguments; the completion theorem of 1969 explained the answer conceptually as a representation-theoretic completion.

The equivariant Bott periodicity theorem and the equivariant Thom isomorphism appeared earlier in Atiyah's Bott periodicity and the index of elliptic operators in Quarterly Journal of Mathematics, Oxford (2) 19 (1968) pp. 113-140 ^{[Atiyah 1968 Bott periodicity index]}. This paper introduced the K-theoretic proof of Bott periodicity that bypasses Morse theory and uses only the Thom isomorphism, the splitting principle, and the equivariant Koszul resolution; the equivariant version is a direct consequence. The Lawson-Michelsohn Spin Geometry monograph (1989, Princeton) ^{[Lawson-Michelsohn Spin Geometry]} §I.9 and §III give the modern textbook synthesis of the equivariant Bott theorem in the Clifford-module framework, including the $K O$ -theoretic refinement that handles $G$ -equivariant real bundles with the eightfold rather than twofold periodicity.

Bibliography [Master]

@article{Segal1968EquivariantKtheory,
  author  = {Segal, Graeme},
  title   = {Equivariant {K}-theory},
  journal = {Publications Math\'ematiques de l'IH\'ES},
  volume  = {34},
  year    = {1968},
  pages   = {129--151}
}

@article{AtiyahSegal1968IndexII,
  author  = {Atiyah, M. F. and Segal, G. B.},
  title   = {The index of elliptic operators. {II}},
  journal = {Annals of Mathematics. Second Series},
  volume  = {87},
  year    = {1968},
  pages   = {531--545}
}

@article{AtiyahSegal1969Completion,
  author  = {Atiyah, M. F. and Segal, G. B.},
  title   = {Equivariant {K}-theory and completion},
  journal = {Journal of Differential Geometry},
  volume  = {3},
  year    = {1969},
  pages   = {1--18}
}

@article{Atiyah1968BottPeriodicityIndex,
  author  = {Atiyah, M. F.},
  title   = {{B}ott periodicity and the index of elliptic operators},
  journal = {Quarterly Journal of Mathematics, Oxford. Second Series},
  volume  = {19},
  year    = {1968},
  pages   = {113--140}
}

@article{AtiyahBott1967Lefschetz,
  author  = {Atiyah, M. F. and Bott, R.},
  title   = {A {L}efschetz fixed-point formula for elliptic complexes. {I}},
  journal = {Annals of Mathematics. Second Series},
  volume  = {86},
  year    = {1967},
  pages   = {374--407}
}

@article{Atiyah1961CharactersCohomology,
  author  = {Atiyah, M. F.},
  title   = {Characters and cohomology of finite groups},
  journal = {Publications Math\'ematiques de l'IH\'ES},
  volume  = {9},
  year    = {1961},
  pages   = {23--64}
}

@article{AtiyahTall1969,
  author  = {Atiyah, M. F. and Tall, D. O.},
  title   = {Group representations, $\lambda$-rings and the {$J$}-homomorphism},
  journal = {Topology},
  volume  = {8},
  year    = {1969},
  pages   = {253--297}
}

@book{AtiyahKTheory,
  author    = {Atiyah, M. F.},
  title     = {{K}-Theory},
  publisher = {W. A. Benjamin},
  year      = {1967},
  note      = {Notes by D. W. Anderson; reprinted by Westview/Perseus 1989}
}

@book{LawsonMichelsohn1989,
  author    = {Lawson, H. Blaine and Michelsohn, Marie-Louise},
  title     = {Spin Geometry},
  publisher = {Princeton University Press},
  year      = {1989}
}

@book{KaroubiKTheory1978,
  author    = {Karoubi, Max},
  title     = {{K}-Theory: An Introduction},
  series    = {Grundlehren der mathematischen Wissenschaften},
  volume    = {226},
  publisher = {Springer-Verlag},
  year      = {1978}
}

Prerequisites

03.08.01
03.08.07
03.05.02
03.05.01

Used in

03.09.21

Tier anchors

beginner: Atiyah *K-Theory* §1.6 informal account of $G$-bundles; Segal IHES 34 introductory paragraphs
intermediate: Atiyah *K-Theory* §1.6 + §2.3; Segal *Equivariant K-theory* (Publ. IHES 34, 1968) §1-§2
master: Atiyah-Segal *The index of elliptic operators II* (Ann. Math. 87, 1968); Segal *Equivariant K-theory* (Publ. IHES 34, 1968); Atiyah-Segal *Completion theorem* (J. Diff. Geom. 3, 1969); Atiyah *Bott periodicity and the index of elliptic operators* (Quart. J. Math. 19, 1968); Lawson-Michelsohn §I.9 + §III

References

fast-track
images/Atiyah-K-Theory-683x1024__52e3382e9f.jpg · Atiyah K-Theory §1.6 + §2.3, $G$-bundles and the representation-ring action
TODO_REF
Atiyah — K-Theory · §1.6 ($G$-vector bundles), §2.3 ($K_G$ as a representation-ring module), §2.4 (long exact sequence in the equivariant setting)
TODO_REF
Segal — Equivariant K-theory · Publ. Math. IHES 34 (1968) 129-151, the definitive originator paper: full axiomatic foundation of $K_G(X)$, induction-restriction, and the change-of-group spectral sequence
TODO_REF
Atiyah-Segal — The index of elliptic operators II · Ann. of Math. (2) 87 (1968) 531-545, equivariant index theorem and the $R(G)$-module structure on the symbol K-group
TODO_REF
Atiyah-Segal — Equivariant K-theory and completion · J. Differential Geometry 3 (1969) 1-18, the Atiyah-Segal completion theorem $K_G(EG)^{\wedge}_I \cong K(BG)$
TODO_REF
Atiyah — Bott periodicity and the index of elliptic operators · Quart. J. Math. Oxford (2) 19 (1968) 113-140, equivariant Bott periodicity through the Thom isomorphism
TODO_REF
Atiyah-Bott — Lefschetz fixed-point theorem for elliptic complexes I · Ann. of Math. (2) 86 (1967) 374-407, the precursor application of $G$-equivariant index theory to fixed-point formulas
TODO_REF
Lawson-Michelsohn — Spin Geometry · §I.9 ($KO$-theory and Clifford modules) and §III (equivariant index applications)

Reviewer

TBD

Estimated time

beginner: 18m
intermediate: 45m
master: 90m