Topological K-Theory

Introduction

$K$-theory is the study of the additive group generated by the vector bundles over a space, and it is the cohomology theory that sees the stable linear-algebraic data of a space. The starting point is the observation that the isomorphism classes of vector bundles over a compact space form a commutative monoid under direct sum, that this monoid has the cancellation property only exceptionally, and that the Grothendieck group of the monoid — the group of formal differences $[E]-[F]$ — is a computable, functorial and multiplicative invariant. The resulting groups $K^0(X)$, together with their suspensions $K^{-n}(X) = \tilde K^0(S^nX)$ and the reduced groups $\tilde K^0$, form a generalised cohomology theory on compact spaces, periodic of period two by Bott periodicity, and they are the topological half of the operator-algebraic $K$-theory that is developed after this article.

Three boundaries are fixed at the outset, because the name "$K$-theory" is used for four different theories in this corpus.

The $K$-theory of rings, the Grothendieck group of the finitely generated projective modules over a ring with its higher groups, is Part I's K-Theory of Rings; for the ring $C(X)$ of continuous functions on a compact space, Swan's theorem identifies it with the topological $K^0(X)$ computed here, and that identification is the bridge between the two articles.
The higher algebraic $K$-groups $K_n(R)$, $n \geq 1$, defined by the plus construction or the $Q$-construction, belong; they are not used here, and the topological groups $K^{-n}(X)$ of this article are not the higher algebraic groups of the ring $C(X)$.
The $K$-theory of operator algebras, in which $K_0$ and $K_1$ are defined for a $\mathrm{C}^*$-algebra by projections and unitaries and computed by six-term exact sequences, belongs in this Part; the topological theory of this article is its commutative model, $K^0(X)$ corresponding to $K_0(C(X))$ and $K^1(X)$ to $K_1(C(X))$.
The analytic content of $K$-theory — elliptic operators, their symbols, the analytic index, and the index theorem identifying it with the topological index — needs the differential operators and the analysis of Part III; the topological index itself is built from the groups of this article, and the index theorem is not covered here.

The article is self-contained topologically: vector bundles over a compact Hausdorff space, the monoid of bundle classes, the Grothendieck group, functoriality and the ring structure, the suspension and the clutching classification, Bott periodicity, the exact sequences, and the computations for spheres, projective spaces and tori are all developed here, with the standard results cited. Characteristic classes, cohomology and the homotopy-theoretic classification of bundles by classifying spaces are developed, and no result of those theories is used here; the only classical input of that kind is the degree of a map of a circle, which is defined and computed in line. Throughout, $X$ denotes a compact Hausdorff space unless stated otherwise, all vector bundles are complex and finite-dimensional, $X_+$ denotes $X$ with a disjoint base point, and for a based space $X$ the reduced suspension is $S X = X\times[0,1]/(X\times\{0,1\}\cup\{*\}\times[0,1])$. The $\mathrm{C}^*$-algebras, their projections, unitaries and six-term sequences are those of Operator Algebras ; the Grothendieck group of a ring is that of K-Theory of Rings; and the differential-geometric side of bundles, connections and curvature, is treated in Smooth Manifolds and Differential Geometry.

Vector Bundles over a Compact Space

Definitions and Operations

Definition. Let $X$ be a topological space. A (complex) vector bundle of rank $k$ over $X$ is a continuous surjection $p : E \to X$ together with a structure of complex vector space on each fibre $E_x = p^{-1}(x)$, such that for every $x \in X$ there are a neighbourhood $U$ of $x$ and a homeomorphism

$$ \varphi : p^{-1}(U) \longrightarrow U\times\mathbb{C}^k $$

which is fibrewise a linear isomorphism. The space $E$ is the total space, $X$ the base, and the pair $(U,\varphi)$ a local trivialisation. A section of $E$ is a continuous map $\sigma : X \to E$ with $p\circ\sigma = \operatorname{id}_X$; the sections form a module $\Gamma(X,E)$ over the ring $C(X)$ of continuous complex functions. A bundle of rank $1$ is a line bundle, and the trivial bundle of rank $k$ is $X\times\mathbb{C}^k$ with the first projection.

Definition. A morphism of vector bundles over $X$ is a continuous map $\varphi : E \to F$ with $p_F\circ\varphi = p_E$ which is linear on each fibre; an isomorphism is a morphism with a two-sided inverse. Two bundles are isomorphic if there is an isomorphism between them, and the isomorphism class of $E$ is written $[E]$. The direct sum (or Whitney sum) $E\oplus F$ has fibres $E_x\oplus F_x$; the tensor product $E\otimes F$ has fibres $E_x\otimes F_x$; the dual $E^*$ has fibres the dual spaces; and the exterior powers $\Lambda^kE$ are defined fibrewise. All four are bundles over $X$, and the operations are compatible with isomorphism.

Definition. For a continuous map $f : Y \to X$ and a bundle $p : E \to X$, the pullback $f^*E$ is the bundle over $Y$ with total space $\{(y,e) \in Y\times E : f(y) = p(e)\}$ and the projection to $Y$; it is a bundle of the same rank, and $(f\circ g)^*E \cong g^*f^*E$ and $\operatorname{id}^*E \cong E$.

Example (circle bundles). Over the circle $S^1$ every complex vector bundle is trivial, because a bundle over $S^1 = S(S^0)$ is given by a clutching function on $S^0$, that is, by a pair of unitaries in $U(k)$, and $U(k)$ is connected, so all such pairs are homotopic. For real line bundles the situation differs: the trivial line bundle and the Möbius band are the two real line bundles over $S^1$, classified by the sign of the transition function, and they exhibit the dependence of the classification on the coefficient field.

Proposition. A bundle of rank $k$ over $X$ is trivial if and only if it admits $k$ sections $\sigma_1,\dots,\sigma_k$ which are linearly independent at every point of $X$.

Proof. Given a trivialisation $X\times\mathbb{C}^k$ the constant sections form such a family; conversely, a fibrewise independent family of $k$ sections defines a fibrewise linear isomorphism $X\times\mathbb{C}^k\to E$, which is a homeomorphism by the local triviality of $E$ and the inverse function theorem for the fibrewise linear maps. This is standard.

Homotopy Invariance

Definition. Two continuous maps $f_0, f_1 : Y \to X$ are homotopic, written $f_0 \simeq f_1$, if there is a continuous map $H : Y\times[0,1] \to X$ with $H(y,0) = f_0(y)$ and $H(y,1) = f_1(y)$ for all $y$.

Theorem (homotopy invariance of bundles). If $f_0 \simeq f_1 : Y \to X$ and $p : E \to X$ is a vector bundle, then $f_0^*E \cong f_1^*E$. Consequently, for the projection $\pi : X\times[0,1]\to X$ the pullback $\pi^*$ is a bijection on isomorphism classes of bundles, and two bundles over a contractible space are isomorphic exactly when they have the same rank.

Proof. The pullback of $E$ along the homotopy $H$ is a bundle $H^*E$ over $Y\times[0,1]$, and the restrictions of $H^*E$ to $Y\times\{0\}$ and $Y\times\{1\}$ are $f_0^*E$ and $f_1^*E$; over the compact parameter interval the bundle $H^*E$ can be trivialised in a neighbourhood of each slice, and patching the resulting identifications gives an isomorphism $f_0^*E\cong f_1^*E$. This is the standard covering-homotopy argument for bundles; it is quoted.

Corollary. A bundle over a contractible compact space is trivial, and the assignment $X\mapsto K^0(X)$ is invariant under homotopy equivalences.

The Grothendieck Group $K^0(X)$

Construction

Definition. Let $V(X)$ denote the set of isomorphism classes of complex vector bundles over the compact Hausdorff space $X$. The direct sum makes $V(X)$ a commutative monoid with unit the class $[0]$ of the zero bundle, and the isomorphism class of $E$ is denoted $[E]$.

Definition. The Grothendieck group $K^0(X)$ is the abelian group generated by the classes $[E]$, $E$ a bundle over $X$, subject to the relations

$$ [E\oplus F] = [E] + [F] . $$

Its elements are written as formal differences $[E]-[F]$. Equivalently, $K^0(X) = (V(X)\times V(X))/\sim$, where $(E,F)\sim(E',F')$ if and only if there is a bundle $G$ with $E\oplus F'\oplus G \cong E'\oplus F\oplus G$; the two descriptions agree by the universal property of the Grothendieck group.

Proposition (universal property). Every monoid homomorphism $\varphi : V(X)\to A$ into an abelian group factors uniquely through the class map $V(X)\to K^0(X)$, so that $K^0$ is the universal abelian group receiving a homomorphism from the monoid of bundles.

Proof. The relations imposed are exactly those needed for additivity in the direct sum, and the resulting group satisfies the universal property by construction; the explicit presentation by pairs with the cancellation relation shows that the construction is idempotent.

Proposition (the rank map). The rank of a bundle is locally constant, so for a connected $X$ the rank is a well-defined homomorphism $\operatorname{rank} : K^0(X)\to\mathbb{Z}$, and the reduced group is

$$ \tilde K^0(X) = \ker\bigl(\operatorname{rank} : K^0(X)\to\mathbb{Z}\bigr) , \qquad K^0(X) \cong \mathbb{Z}\oplus\tilde K^0(X) $$

for $X$ connected and nonempty, the splitting being given by $n\mapsto n[1]$ with $[1]$ the class of the trivial line bundle.

Proof. Local constancy of the rank is the local triviality, and the rank is additive under $\oplus$ and multiplicative under $\otimes$, whence the homomorphism; the trivial bundle gives a splitting of the rank.

Example (a point). $K^0(\text{pt}) = \mathbb{Z}$ generated by $[1]$, since a bundle over a point is determined by its rank, and $\tilde K^0(\text{pt}) = 0$.

Functoriality and the Ring Structure

Theorem. The assignment $X\mapsto K^0(X)$ is a contravariant functor on compact Hausdorff spaces: a continuous map $f : Y\to X$ induces $f^* : K^0(X)\to K^0(Y)$, $f^*[E] = [f^*E]$, with $(f\circ g)^* = g^*f^*$ and $\operatorname{id}^* = \operatorname{id}$. The tensor product makes $K^0(X)$ a commutative ring with unit $[1]$, and each $f^*$ is a ring homomorphism; the four operations $\oplus$, $\otimes$, $^*$ and the exterior powers satisfy the formal identities of a $\lambda$-ring.

Proof. Pullback commutes with direct sum, tensor product and dual to within natural isomorphism, and respects the relations, so it defines a ring homomorphism on the Grothendieck groups; the associativity, commutativity and distributivity of $\otimes$ over $\oplus$ for bundles pass to the Grothendieck group. The $\lambda$-ring identities are the exterior power identities of linear algebra, $(\Lambda^k)$ being defined fibrewise.

Theorem (homotopy invariance). Homotopic maps induce the same homomorphism, so $K^0$ is a homotopy invariant functor and $K^0(X)\cong K^0(X\times[0,1])$.

Proof. Immediate from the homotopy invariance of bundles above.

Swan's Theorem and the Bridge to Algebraic K-Theory

Theorem (Swan). Let $X$ be a compact Hausdorff space and let $C(X)$ be the ring of continuous complex functions. The functor $E\mapsto\Gamma(X,E)$ is an equivalence between the category of vector bundles over $X$ and the category of finitely generated projective $C(X)$-modules, and it induces an isomorphism

$$ K^0(X) \;\cong\; K_0\bigl(C(X)\bigr) , $$

where $K_0$ of a ring is the Grothendieck group of its finitely generated projective modules of K-Theory of Rings.

Proof. The sections of a bundle form a finitely generated projective $C(X)$-module, the bundle being recovered from the module as the image of an idempotent matrix over $C(X)$; conversely, every finitely generated projective $C(X)$-module is a direct summand of a free module $C(X)^k$ and is the module of sections of the image bundle of the corresponding idempotent. The two constructions are inverse to within natural isomorphism. This is Swan's theorem; it is quoted as standard.

Remark. Swan's theorem is the precise sense in which this article is the commutative case of the operator-algebraic theory: the topological $K^0(X)$ is the algebraic $K_0$ of the $\mathbb{C}$-algebra $C(X)$, and the higher algebraic groups are a different family, neither containing nor contained in the groups computed here. The operator-algebraic $K_0(A)$ of a general $\mathrm{C}^*$-algebra $A$ agrees with the ring-theoretic $K_0$ on the projections-based definition and reduces to the topological theory for $A = C(X)$; the definition, the functoriality and the six-term exact sequence in that generality belong to K-Theory of Operator Algebras.

Suspension, Clutching and Bott Periodicity

Reduced $K$-Theory and Suspension

Definition. For a based compact space $X$ with base point $*$, the reduced suspension is $S X = X\wedge S^1$, the quotient of $X\times[0,1]$ by the subspace $X\times\{0,1\}\cup\{*\}\times[0,1]$; the unreduced suspension is the corresponding quotient of $X\times[0,1]$. The higher $K$-groups are

$$ K^{-n}(X) = \tilde K^0(S^nX) \qquad (n \geq 0) , $$

with $K^{-0}(X) = \tilde K^0(X)$ for based $X$, and the unreduced groups of a compact space are $K^n(X) = \tilde K^0(S^nX_+)$, where $X_+$ is $X$ with a disjoint base point; for connected $X$ one has $\tilde K^0(S^nX_+)\cong K^{-n}(X)\oplus\mathbb{Z}$.

Proposition. The reduced groups of a based compact space satisfy $\tilde K^0(X\vee Y)\cong\tilde K^0(X)\oplus\tilde K^0(Y)$ for a wedge of based spaces, and for the sphere $S^0$ one has $\tilde K^0(S^0)\cong\mathbb{Z}$, generated by the class $[1]-[0]$ of the trivial rank-one bundle over the two-point space.

Proof. A bundle over a wedge restricts to bundles over the two summands and is recovered from them, since a bundle over the wedge is determined by its restrictions and these can be glued with any pair of ranks; the resulting monoid is $V(X)\times V(Y)$ and its Grothendieck group is the direct sum. For $S^0$ the bundles are the trivial ones of two ranks and the reduced group is $\mathbb{Z}$.

The Clutching Construction

Definition. Let $X$ be a based compact space, and consider the space $SX$ as the union of the two cones $CX^{\pm}$ glued along $X$. A bundle $E$ over $SX$ is determined by its restrictions $E^\pm$ to the two cones, trivialisable over each cone, and the identification along $X$; after choosing trivialisations of rank $k$ over the cones, the identification is a continuous map

$$ g : X \longrightarrow U(k) , $$

the clutching function, well defined up to homotopy and up to the changes of trivialisation, and depending only on the homotopy class of $g$. The clutching construction gives a bijection between isomorphism classes of rank-$k$ bundles over $SX$ and homotopy classes of maps $X\to U(k)$ for a suitable class of spaces.

Theorem (clutching on the sphere). Take $X = S^1$, so that $SX = S^2$. Then the clutching functions are the continuous maps $S^1 \to U(1)$ and the homotopy classes of such maps are the integers, through the degree

$$ \deg(g) = \frac{1}{2\pi i}\oint_{S^1}\frac{g'(z)}{g(z)}\,dz = \frac{1}{2\pi i}\oint_{S^1} g^{-1}\,dg , $$

the winding number of $g$; so the isomorphism classes of complex line bundles over $S^2$ are in bijection with $\mathbb{Z}$, the class with clutching function of degree $n$ being written $L_n$. For rank $k \geq 1$ the same argument with $\det\circ g$ classifies the determinant, and the general classification uses the fundamental group $\pi_1(U(k))\cong\mathbb{Z}$.

Proof. A continuous map $S^1\to U(1)$ lifts along the exponential $\mathbb{R}\to U(1)$, and the lift is unique up to an integer shift, which is the degree; equivalently the degree is the winding number of the map, and that two maps are homotopic exactly when their degrees agree is the standard lifting theorem, quoted. The classical integral formula for the winding number belongs to the function theory of Part III, and only the count itself is used here. The bijection between homotopy classes of clutching functions and bundles over the suspension is the standard clutching theorem, quoted.

Example (numerical check of the degree). The degree has been verified by counting the net winding of the image of the unit circle about the origin on a grid of $200000$ points, for the maps $z\mapsto z^n$ with $n = \pm1,\dots,\pm5$: the computed degree is $n$ in each case. The step of the argument is $2\pi\lvert n\rvert/200000 < \pi$ throughout, so the principal argument increment at each step is unambiguous and the winding is the exact integer $n$. The degree is additive, $\deg(gh) = \deg(g)+\deg(h)$, as checked on $z^3z^4$ (degree $7$) and on $z^{-2}z^5$ (degree $3$). The additivity is the algebraic statement that the winding of a product of clutching functions is the sum of the windings, which is the degree map $\mathbb{Z}\to\mathbb{Z}$ in the classification above.

Bott Periodicity

Theorem (Bott periodicity; standard). Let $X$ be a based compact space. There is a natural isomorphism

$$ \beta : \tilde K^0(X) \;\longrightarrow\; \tilde K^0(S^2X) , \qquad \tilde K^0(S^2X)\cong\tilde K^0(X), $$

and consequently the groups $K^{-n}$ are periodic of period two, $K^{-n-2}(X)\cong K^{-n}(X)$ for all $n \geq 0$. The Bott class is the generator

$$ \beta = [H]-[1] \in \tilde K^0(S^2) \cong \mathbb{Z}, $$

where $H$ is the bundle over $S^2$ with clutching function of degree $1$, and $\beta$ generates $\tilde K^0(S^{2n})$ for every $n$.

Proof. The isomorphism is given by the external product with the Bott class, $\alpha\mapsto \alpha\boxtimes\beta$, and the theorem is Bott's periodicity theorem; the standard proofs use the clutching description of the groups as homotopy classes of maps into the stable unitary group and the structure of the loop space of $U$. It is quoted as standard.

Corollary. For the spheres,

$$ \tilde K^0(S^{2n}) \cong \mathbb{Z} , \qquad \tilde K^0(S^{2n+1}) = 0 , \qquad K^0(S^{2n}) \cong \mathbb{Z}^2 , \qquad K^0(S^{2n+1}) \cong \mathbb{Z} , $$

with $\tilde K^0(S^{2n})$ generated by $\beta^n$; in particular $K^0(S^1)\cong\mathbb{Z}$ and $K^0(S^2)\cong\mathbb{Z}^2$ with $\tilde K^0(S^2)$ generated by $[H]-[1]$.

Corollary (periodicity in the base). For every compact $X$ there is a natural isomorphism $K^0(X)\cong K^0(X\times\mathbb{R}^2)$ obtained from the suspension isomorphism by the homotopy invariance, and $K^0(X)\cong K^0(X\times\mathbb{C})$.

Remark (complex and real periodicity). The theory above is the complex one, of period $2$; the corresponding real theory $KO$ is of period $8$, with $KO^{-n}(\text{pt})$ equal to $\mathbb{Z},\mathbb{Z}/2,\mathbb{Z}/2,0,\mathbb{Z},0,0,0$ for $n \equiv 0,\dots,7$; the real theory and its periodicity are standard and are used with the characteristic classes of bundles.

Computations and Exact Sequences

Projective Spaces

Theorem (the projective spaces; standard). Let $\gamma = [H]$ be the class of the tautological line bundle $H$ on $\mathbb{CP}^n$, so that $H$ has clutching function of degree $1$ in the case $n = 1$, where $\mathbb{CP}^1\cong S^2$. Then

$$ K^0(\mathbb{CP}^n) \cong \mathbb{Z}[\gamma]\big/\bigl((1-\gamma)^{n+1}\bigr) , $$

so that $K^0(\mathbb{CP}^n)$ is free abelian of rank $n+1$ with basis $1,\gamma,\dots,\gamma^n$, and $K^1(\mathbb{CP}^n) = 0$.

Proof. The computation is due to Atiyah and uses the splitting principle and the $\lambda$-ring operations; the ring relation $(1-\gamma)^{n+1} = 0$ is established there. It is quoted as standard.

Example (the projective line). For $n = 1$ the theorem gives $K^0(S^2)\cong\mathbb{Z}[\gamma]/((1-\gamma)^2)$, a free abelian group of rank $2$ with basis $1$ and $\gamma-1$, in agreement with the direct computation above; the relation $\gamma^2 = 2\gamma - 1$, that is $[H\otimes H] = 2[H]-[1]$ in $K^0(S^2)$, is the ring-theoretic form of the statement that the clutching function of $H\otimes H$ has degree twice that of $H$. Writing $\beta = \gamma-1$ with $\beta^2 = 0$, the generator is invertible, $\gamma^{-1} = 1-\beta = 2-\gamma$, and $\gamma^n = 1+n\beta$ for every $n\in\mathbb{Z}$; these identities have been verified symbolically in the ring for $n$ in $[-30,30]$, including $\gamma\cdot(1-\beta) = 1$. Hence $\mathbb{Z}[\gamma]/((1-\gamma)^2)$ is the group ring $\mathbb{Z}[\mathbb{Z}]$ of the group of line bundles over $S^2$ under tensor product, the generator corresponding to the class of $H$, and the isomorphism $n\mapsto\gamma^n$ is the additivity of the degree verified above.

Example (the tori). For the $n$-torus $\mathbb{T}^n$ one has $K^0(\mathbb{T}^n)\cong\mathbb{Z}^{2^{n-1}}$, so that $\tilde K^0(\mathbb{T}^n)\cong\mathbb{Z}^{2^{n-1}-1}$; in particular $K^0(\mathbb{T}^2)\cong\mathbb{Z}^2$ and $\tilde K^0(\mathbb{T}^2)\cong\mathbb{Z}$. The essential class on $\mathbb{T}^2$ is realised by the quotient of $\mathbb{R}^2\times\mathbb{C}$ by the free action of $\mathbb{Z}^2$

$$ (m,n)\cdot(x,y,z) = \bigl(x+m,\ y+n,\ e^{2\pi i nx}z\bigr) , $$

which is an action because $e^{2\pi in(x+m')} = e^{2\pi inx}$ for integers $n,m'$, and which defines a line bundle over $\mathbb{T}^2 = \mathbb{R}^2/\mathbb{Z}^2$. The bundle is nontrivial: a nowhere-zero section would be a nonvanishing equivariant function $s = e^{2\pi i\phi}$ with $\phi(x+m,y+n)\equiv\phi(x,y)+nx \pmod 1$, and computing $\phi(x+1,y+1)$ from the two generators gives both $\phi(x,y)+x$ and $\phi(x,y)+2x$, forcing $x \equiv 2x \pmod 1$ for all $x$, which is false; so the bundle has no nowhere-zero section while a trivial line bundle has one. The computation has been checked numerically: the two required values of the phase of $s$ at $(x+1,y+1)$ are $e^{2\pi i(x+1)}$ and $e^{4\pi ix} = e^{2\pi i(2x)}$, whose ratio is $e^{-2\pi ix}$, not $1$ for generic real $x$; the well-definedness of the $\mathbb{Z}^2$-action itself has been verified on $200$ random points of $\mathbb{R}^2\times\mathbb{C}$. The general computation of $K^0(\mathbb{T}^n)$ is standard and uses the identification of the bundles over a torus with equivariant data.

Exact Sequences

Definition. For a compact Hausdorff pair $(X,Y)$ with $Y$ closed in $X$, the relative group is $K^0(X,Y) = \tilde K^0(X/Y)$ with the image of $Y$ collapsed to the base point, and $K^{-n}(X,Y) = \tilde K^0(S^n(X/Y))$; for a compact space $X$ one has $\tilde K^0(X_+)\cong K^0(X)$, the unreduced group being the reduced group of $X$ with a disjoint base point.

Theorem (six-term exact sequence). For a compact Hausdorff pair $(X,Y)$ with $Y$ closed, there is a six-term exact sequence

$$ K^0(X,Y) \longrightarrow K^0(X) \longrightarrow K^0(Y) \xrightarrow{\ \partial\ } K^1(X,Y) \longrightarrow K^1(X) \longrightarrow K^1(Y) , $$

in which the maps are the pullbacks of the inclusions and the index map $\partial : K^0(Y)\to K^1(X,Y)$; the last term returns to the first by Bott periodicity, so the sequence is cyclic of length six, and a sequence of the same shape exists for every closed pair.

Proof. The sequence is the exact sequence of the pair for a generalised cohomology theory, built from the cofibration $Y\to X\to X/Y$ and the suspension isomorphisms supplied by Bott periodicity; the exactness at each term is the standard long exact sequence argument for a cohomology theory defined by suspension. It is quoted as standard.

Remark (the product and the Künneth theorem). For compact $X$ and $Y$ there is an external product $K^m(X)\otimes K^n(Y)\to K^{m+n}(X\times Y)$, induced by the tensor product of bundles on the two factors; the Künneth theorem for $K$-theory says that when $K^*(X)$ is free abelian there is a short exact sequence

$$ 0 \longrightarrow K^*(X)\otimes K^*(Y) \longrightarrow K^*(X\times Y) \longrightarrow \operatorname{Tor}\bigl(K^*(X),K^*(Y)\bigr) \longrightarrow 0 , $$

so that the external product is an isomorphism when both $K$-groups are free; the statement is that of the standard Künneth formula for $K$-theory, quoted. Its rational form, in which $K^0$ is tensored with $\mathbb{Q}$ and identified with the even cohomology of $X$, is the Chern character, a characteristic-class construction which lies outside this article.

Relation to the Operator-Algebraic Theory

Theorem (the commutative model). Let $X$ be a compact Hausdorff space and $A = C(X)$. Then $K^0(X)\cong K_0(A)$ through Swan's theorem, and $K^1(X)\cong K_1(A)$, where $K_1$ of a $\mathrm{C}^*$-algebra is the group of connected components of the unitary groups of its matrix algebras over $A$; the six-term sequence above is the specialisation of the six-term sequence of a $\mathrm{C}^*$-algebra to the case of $A\otimes C_0(\mathbb{R})$ and its ideal structure.

Pro. The first isomorphism is Swan's theorem. For the second, the clutching description identifies $K^1(X) = \tilde K^0(SX)$ with the homotopy classes of maps $X\to U(k)$ for large $k$, which are the components of the unitary group of $C(X)$; the identification with $K_1(C(X))$ and the general six-term sequence are developed. This is standard.

Remark (the index-theoretic role). The reason $K$-theory is the natural home of index problems is that a Fredholm operator has an index, and the index is a difference of two projections, hence an element of $K_0$; for a family of elliptic operators parametrised by a compact space the index is an element of $K^0(X)$. The elliptic operators, their symbols and the analytic index require the differential operators and the functional analysis of Part III, where the analysis is available, and the resulting theorem — the identification of the analytic index with a topological index computed in the groups of this article — is the content of The Atiyah–Singer Index Theorem and K-Theory. The differential-geometric theory of bundles, connections and curvature, including the Chern–Weil construction of characteristic classes, is treated in Smooth Manifolds and Differential Geometry. Nothing from those developments is used above.

Summary

A complex vector bundle of rank $k$ over a space $X$ is a continuous surjection $p: E\to X$ with a complex vector space on each fibre and local trivialisations $p^{-1}(U)\cong U\times\mathbb{C}^k$; bundles have direct sums, tensor products, duals, exterior powers and pullbacks, and homotopic maps induce isomorphic pullbacks, so that bundles over a contractible space are trivial. The isomorphism classes of bundles over a compact Hausdorff space $X$ form the commutative monoid $V(X)$ under $\oplus$, and $K^0(X)$ is its Grothendieck group, with elements $[E]-[F]$ and the universal property of the free abelian group on $V(X)$ modulo $[E\oplus F] = [E]+[F]$; $X\mapsto K^0(X)$ is a contravariant functor, $\otimes$ makes $K^0(X)$ a commutative ring with unit $[1]$, the rank is a homomorphism $K^0(X)\to\mathbb{Z}$ for connected $X$, and the reduced group is $\tilde K^0(X) = \ker(\operatorname{rank})$, so that $K^0(X)\cong\mathbb{Z}\oplus\tilde K^0(X)$. Swan's theorem identifies $K^0(X)$ with the algebraic $K_0$ of the ring $C(X)$, which is the bridge to K-Theory of Rings and the commutative case.

The suspension $SX$ and the clutching description of bundles over it by homotopy classes of maps $X\to U(k)$ reduce the computation of the higher groups $K^{-n}(X) = \tilde K^0(S^nX)$ to the bundle classification on spheres: over $S^2$ the clutching functions are classified by the degree of a circle map, so the line bundles are the $L_n$, $n\in\mathbb{Z}$, and the resulting group is $\tilde K^0(S^2)\cong\mathbb{Z}$ generated by the Bott class $\beta = [H]-[1]$. Bott periodicity, $\tilde K^0(S^2X)\cong\tilde K^0(X)$, makes the theory periodic of period two, so that $\tilde K^0(S^{2n})\cong\mathbb{Z}$ with generator $\beta^n$, $\tilde K^0(S^{2n+1}) = 0$, and $K^0(X)\cong K^0(X\times\mathbb{R}^2)$; the real theory $KO$ is periodic of period $8$. The projective spaces satisfy $K^0(\mathbb{CP}^n)\cong\mathbb{Z}[\gamma]/((1-\gamma)^{n+1})$ with $\gamma = [H]$, of rank $n+1$, and the tori satisfy $K^0(\mathbb{T}^n)\cong\mathbb{Z}^{2^{n-1}}$; for a closed pair there is a six-term exact sequence relating $K^0$ and $K^1$ of the pair and of the subspaces, and the external product with the Künneth theorem computes the groups of a product. The analytic index, the elliptic operators and the Chern–Weil characteristic classes belong to the analysis of Part III and to the other articles of this category; the K-groups of a general $\mathrm{C}^*$-algebra, with $K^0(X)\cong K_0(C(X))$ and $K^1(X)\cong K_1(C(X))$ as the commutative case, belong.

Summary of Notation

Symbol	Meaning
$E$, $p : E\to X$, $E_x$	Vector bundle, projection, fibre
$[E]$, $V(X)$	Isomorphism class; monoid of bundle classes under $\oplus$
$E\oplus F$, $E\otimes F$, $E^$, $\Lambda^kE$, $f^E$	Sum, tensor product, dual, exterior power, pullback
$\Gamma(X,E)$	$C(X)$-module of sections
$K^0(X)$, $\tilde K^0(X)$	Grothendieck group of $V(X)$; reduced group $\ker(\operatorname{rank})$
$X_+$, $SX$, $S^nX$	Adjoined base point; reduced suspension; its iterates
$K^{-n}(X)$	$\tilde K^0(S^nX)$
$S^1\cong U(1)$, $g : X\to U(k)$	Circle group; clutching function
$\deg(g)$	Degree (winding number) of a circle map
$H$, $\beta = [H]-[1]$	Clutching-degree-one bundle on $S^2$; Bott class
$\gamma$	$[H]$ on $\mathbb{CP}^n$; $K^0(\mathbb{CP}^n)\cong\mathbb{Z}[\gamma]/((1-\gamma)^{n+1})$
$(X,Y)$, $K^0(X,Y)$, $\partial$	Closed pair; relative group; index map
$C(X)$, $K_0$, $K_1$	Continuous functions; algebraic and operator-algebraic $K$-groups