7 · k-theory

the grothendieck group of vector bundles

Let $X$ be a compact Hausdorff space (in field-theoretic applications, typically a closed manifold). Consider the set of isomorphism classes of complex vector bundles over $X$. Under the operation of Whitney sum this set forms an abelian monoid. The associated Grothendieck group $K(X)$ is obtained by formally adjoining inverses: elements of $K(X)$ are virtual bundles $[E]-[F]$, with the relation that $[E]-[F]=[E']-[F']$ whenever $E\oplus F'\simeq E'\oplus F$. The class of the zero bundle is the identity, and every bundle $E$ possesses an inverse in the sense that $[E]+[F]=[n]$ (the class of a trivial bundle of rank $n$) for a suitable complementary bundle $F$.

The same construction with real vector bundles yields the real $K$-group $KO(X)$. Both theories are functorial under continuous maps and satisfy the homotopy-invariance and exactness properties of a generalized cohomology theory (after the usual introduction of reduced groups $\widetilde{K}(X)$ for pointed spaces).

definition (monoid of vector bundles). write $\operatorname{Vect}(X)$ for the set of isomorphism classes of complex vector bundles on a compact Hausdorff space $X$, with addition $[E]+[F]:=[E\oplus F]$. this is a commutative monoid with identity the class of the zero-rank bundle (or, stably, any choice of trivial unit in the reduced theories).

definition ($K(X)$ as grothendieck group). $K(X)$ (also written $K^0(X)$) is the Grothendieck group of $\operatorname{Vect}(X)$: the abelian group of pairs $([E],[F])$ modulo $([E]\oplus[H],[F]\oplus[H])\sim([E],[F])$, conventionally written $[E]-[F]$. the natural map $\operatorname{Vect}(X)\to K(X)$ sends $E$ to $[E]$.

definition (reduced $K$-theory). for a pointed space $(X,x_0)$, $$ \widetilde{K}(X) :=\ker\bigl(K(X)\to K(\{x_0\})\bigr). $$ equivalently, classes of virtual bundles of virtual rank zero. one has $K(X)\simeq\widetilde{K}(X)\oplus\mathbb{Z}$ when $X$ is nonempty and connected, the $\mathbb{Z}$ factor recording rank.

definition (real $K$-theory). $KO(X)$ is the Grothendieck group of real vector bundles on $X$. the reduced groups $\widetilde{KO}(X)$ are defined analogously.

theorem (existence of complements). if $X$ is compact Hausdorff and $E\to X$ is a complex vector bundle, then there exists a bundle $F$ and an integer $n$ such that $E\oplus F\simeq\underline{\mathbb{C}}^n$. consequently every class in $K(X)$ may be written $[E]-[n]$ for some $E$ and $n\ge 0$.

proof. choose a finite open cover that trivializes $E$ and a subordinate partition of unity. the standard clutching constructions produce an embedding of $E$ into a trivial bundle of sufficiently large rank (each local trivialization contributes frame fields that glue after cutoffs). the orthogonal complement with respect to a Hermitian metric is then a smooth complementary bundle $F$. $\square$

theorem (stable equivalence). for compact $X$, one has $[E]=[F]$ in $K(X)$ if and only if $E$ and $F$ are stably isomorphic: $E\oplus\underline{\mathbb{C}}^N\simeq F\oplus \underline{\mathbb{C}}^N$ for some $N$.

proof. if $E\oplus\varepsilon^N\simeq F\oplus\varepsilon^N$ then $[E]+[N]=[F]+[N]$ in $K(X)$, so $[E]=[F]$. conversely, write $[E]-[F]=0$, so $[E]+[H]=[F]+[H]$ for some $H$ after clearing the definition of the Grothendieck relation. adding a complement of $H$ to both sides produces a trivial summand and yields a stable isomorphism of $E$ and $F$. $\square$

theorem ($K$ as a cohomology theory, outline). the functors $X\mapsto\widetilde{K}(X)$ and $X\mapsto\widetilde{KO}(X)$ extend to generalized cohomology theories on compact spaces (or CW complexes): they are homotopy-invariant, and a closed pair $(X,A)$ yields a long exact sequence involving relative groups $K(X,A)\simeq\widetilde{K}(X/A)$.

proof (idea). homotopy invariance follows because pullback along a homotopy is isomorphic for vector bundles on compact spaces (classifying maps $X\to BU$ are homotopic iff the bundles are isomorphic). for a closed inclusion $A\subset X$, the quotient map and mapping-cone sequences produce the usual exactness pattern of a reduced cohomology theory once reduced groups are used. full details use representing spaces $BU\times\mathbb{Z}$ and $BO\times\mathbb{Z}$. $\square$

ring structure, exterior powers, and tensor products

The tensor product of vector bundles descends to a commutative ring structure on $K(X)$. Exterior powers further enrich the algebra: the total exterior-power operation $$ \lambda_t(E)=\sum_{k\ge 0}t^k[\Lambda^k E] $$ defines a $\lambda$-ring structure on $K(X)$. These operations are essential for the formulation of the Adams operations and for the expression of characteristic classes via the Chern character, a ring homomorphism $$ \operatorname{ch}:K(X)\to H^{\mathrm{even}}(X;\mathbb{Q}) $$ that converts $K$-theoretic data into ordinary cohomology.

definition (ring structure). the product on $K(X)$ is induced by tensor product: $$ [E]\cdot[F]:=[E\otimes F], $$ extended bilinearly to virtual bundles. the unit is the class of the trivial line bundle $[\underline{\mathbb{C}}]$.

theorem (ring axioms). tensor product makes $K(X)$ into a commutative unital ring. the rank map $\operatorname{rk}:K(X)\to\mathbb{Z}$ is a unital ring homomorphism, with kernel $\widetilde{K}(X)$ when $X$ is connected.

proof. for actual bundles, $E\otimes F\simeq F\otimes E$, $(E\otimes F)\otimes G\simeq E\otimes(F\otimes G)$, and $E\otimes\underline{\mathbb{C}}\simeq E$. these identities pass to the monoid and therefore to the Grothendieck group by the universal property of adjoining inverses. rank is additive under $\oplus$ and multiplicative under $\otimes$, hence induces a ring map on $K(X)$. $\square$

definition ($\lambda$-operations). the exterior-power maps $\Lambda^k$ induce operations $\lambda^k:K(X)\to K(X)$ uniquely characterized by $\lambda^k([E])=[\Lambda^k E]$ for bundles $E$ and the generating-function relation $\lambda_t(x+y)=\lambda_t(x)\lambda_t(y)$ in the formal power series ring $K(X)[[t]]$.

definition (chern character). writing Chern roots of a bundle formally so that $c(E)=\prod_i(1+x_i)$, set $$ \operatorname{ch}(E)=\sum_i e^{x_i} =\operatorname{rk}E+c_1(E)+\frac12\bigl(c_1(E)^2-2c_2(E)\bigr)+\cdots\in H^{\mathrm{even}}(X;\mathbb{Q}). $$ extend by $\operatorname{ch}([E]-[F])=\operatorname{ch}(E)-\operatorname{ch}(F)$.

theorem (chern character is a ring map). $\operatorname{ch}:K(X)\to H^{\mathrm{even}}(X;\mathbb{Q})$ is a natural unital ring homomorphism. rationally, after suitable completion in the Atiyah-Hirzebruch spectral sequence, it induces isomorphisms of rational cohomology theories on finite CW complexes.

proof. for line bundles $\operatorname{ch}(L)=e^{c_1(L)}$, and $\operatorname{ch}(L\otimes L')=e^{c_1(L)+c_1(L')}=\operatorname{ch}(L)\operatorname{ch}(L')$. by the splitting principle (chapter 6) every bundle pulls back to a sum of lines on a flag bundle where injectivity of pullback in rational cohomology (or naturality of $\operatorname{ch}$) reduces the Whitney and product formulas to the line-bundle case. additivity on virtual bundles is by definition. the rational isomorphism statement is the collapse of the Atiyah-Hirzebruch spectral sequence after tensoring with $\mathbb{Q}$. $\square$

bott periodicity, adams operations, and the $J$-group

The fundamental computational tool of complex $K$-theory is Bott periodicity: there is a natural isomorphism $$ \widetilde{K}(X)\simeq\widetilde{K}(X\wedge S^2) $$ induced by external product with the generator of $\widetilde{K}(S^2)\cong\mathbb{Z}$. Consequently the $K$-groups of spheres (and of all suspensions) are completely determined, and the theory is periodic of period $2$. Real $K$-theory $KO$ is periodic of period $8$, mirroring the periodicity of real Clifford algebras (chapter 8).

Adams operations $\psi^k:K(X)\to K(X)$ are ring endomorphisms characterized by their action on line bundles, $\psi^k(L)=L^{\otimes k}$, and by naturality. They refine the Chern character and detect torsion phenomena invisible to ordinary characteristic classes. Closely related is the $J$-group $J(X)$, which classifies stable spherical fibrations; the $J$-homomorphism maps $KO(X)$ into $J(X)$ and measures the difference between stable vector-bundle equivalence and stable fiber-homotopy equivalence.

definition (external product and bott generator). the external product $K(X)\otimes K(Y)\to K(X\times Y)$ is induced by external tensor product of bundles. the Bott generator $\beta\in\widetilde{K}(S^2)$ is the class $[H]-[\underline{\mathbb{C}}]$, where $H\to S^2 \simeq\mathbb{CP}^1$ is the Hopf line bundle (equivalently $c_1(H)$ generates $H^2(S^2)$).

theorem (bott periodicity, complex). external product with $\beta$ induces natural isomorphisms $$ \widetilde{K}(X)\xrightarrow{\simeq}\widetilde{K}(\Sigma^2 X) \simeq\widetilde{K}(X\wedge S^2) $$ for compact $X$. in particular $\widetilde{K}(S^{2n})\simeq\mathbb{Z}$, $\widetilde{K}(S^{2n+1})=0$, and $K^{-n}(X):=\widetilde{K}(\Sigma^n X_+)$ is periodic of period $2$.

proof (outline). one geometric model identifies $\widetilde{K}(S^2)$ with $\widetilde{K}(\mathbb{CP}^1)$ generated by $[H]-1$. the clutching construction over equatorial $S^1$ and the elementary analysis of bundles on $S^2$ show this group is $\mathbb{Z}$. the product map $\widetilde{K}(X)\otimes\widetilde{K}(S^2)\to\widetilde{K}(X\wedge S^2)$ is then inverted by the dual Bott map arising from the quasi-inverse of the Hopf bundle in the representation ring of $U(1)$, or by Morse-theoretic / holomorphic clutching proofs of Bott. iterating gives the even/odd sphere calculation by induction. $\square$

theorem (bott periodicity, real). there are natural isomorphisms $\widetilde{KO}(X)\simeq\widetilde{KO}(\Sigma^8 X)$. the values $\widetilde{KO}(S^n)$ are periodic of period $8$ and coincide with the Atiyah-Bott-Shapiro groups of Clifford modules (chapter 8).

proof (pointer). real Bott periodicity is proved by the same clutching methods with $\mathrm{Sp}(1)$ and $\mathrm{O}$, or by identifying $KO^{-n}(\mathrm{pt})$ with virtual Clifford modules $\mathrm{Cl}_n$ as in Atiyah-Bott-Shapiro. the period-$8$ table is then the classical Clifford period table. $\square$

definition (adams operations). the operations $\psi^k:K(X)\to K(X)$, $k\in\mathbb{Z}$, are the unique natural ring endomorphisms such that $\psi^k(L)=L^{\otimes k}$ for every complex line bundle $L$. they satisfy $\psi^k\circ\psi^\ell=\psi^{k\ell}$ and, on Chern characters, $\operatorname{ch}(\psi^k x)$ multiplies the degree-$2m$ component by $k^m$.

theorem (existence of adams operations). Adams operations exist and are uniquely determined by the line-bundle formula and naturality (or by the $\lambda$-ring Newton identities relating $\psi^k$ to $\lambda^j$).

proof. on a split bundle $E\simeq L_1\oplus\cdots\oplus L_r$ one must have $\psi^k(E)=\sum_i L_i^{\otimes k}$. the splitting principle supplies a flag pullback injective on $K$-theory after adding free summands or working with $\lambda$-rings. Newton polynomials express each $\psi^k$ in terms of elementary symmetric functions (the $\lambda^j$), which are already defined on all of $K(X)$. uniqueness follows because natural operations are determined by their values on the universal bundle over $BU$. $\square$

definition ($J$-group and $J$-homomorphism). two vector bundles are fiber-homotopy equivalent if their sphere bundles are fiberwise homotopy equivalent. the group $J(X)$ is the quotient of $\widetilde{KO}(X)$ (or a suitable subgroup of reduced classes) by the relation of stable fiber-homotopy equivalence. the $J$-homomorphism $J:\widetilde{KO}(X)\to J(X)$ is the quotient map.

remark (torsion detection). Adams operations detect torsion elements in $K(X)$ and $KO(X)$ that map to zero under the Chern character, and they control the image of $J$ (the famous image-of-$J$ computation in stable homotopy).

the space of fredholm operators as a model for $K$-theory

An independent analytic realization of $K$-theory is obtained from the space $\mathcal{F}(H)$ of Fredholm operators on a separable complex Hilbert space $H$. The space $\mathcal{F}(H)$ is a classifying space for $K$-theory: for compact $X$, $$ K(X)\simeq[X,\mathcal{F}(H)], $$ the set of homotopy classes of continuous maps from $X$ into $\mathcal{F}(H)$. Under this identification the virtual bundle corresponding to a continuous family of Fredholm operators $T_x$ is the formal difference $[\ker T]-[\operatorname{coker} T]$ (made rigorous by the existence of a continuous choice of finite-dimensional complements). This model makes the relation between $K$-theory and index theory transparent.

definition (fredholm operator). a bounded linear operator $T:H\to H$ on a separable complex Hilbert space is Fredholm if $\ker T$ and $\operatorname{coker} T$ are finite-dimensional (equivalently, $T$ is invertible modulo compact operators). its analytical index is $$ \operatorname{index} T=\dim\ker T-\dim\operatorname{coker} T. $$

definition (atiyah-janich map). a continuous family $T:X\to\mathcal{F}(H)$ determines a class in $K(X)$ by choosing, after possibly stabilizing by a finite-rank projection, a continuous finite-dimensional subspace $V\subset H$ that surjects onto every $\operatorname{coker} T_x$ under the quotient map; then $$ [T]:=\bigl[\ker(T_x\text{ on }V^\perp)\bigr]-\bigl[V/ T_x(V^\perp)\bigr]\in K(X) $$ is independent of choices up to isomorphism.

theorem (atiyah-janich). for compact $X$ the assignment $T\mapsto[T]$ induces a natural bijection $$ [X,\mathcal{F}(H)]\xrightarrow{\simeq}K(X). $$ under this identification, the index map $\mathcal{F}(H)\to\mathbb{Z}$ realizes $K(\mathrm{pt})\simeq\mathbb{Z}$.

proof (outline). connectedness components of $\mathcal{F}(H)$ are labeled by index, matching $K(\mathrm{pt})$. for general $X$, Kuiper's theorem (contractibility of $U(H)$) implies that bundles of Hilbert spaces are trivial, so families of Fredholm operators are well-defined without twisting of the ambient $H$. the symbol construction that sends a virtual bundle $[E]-[F]$ to a family of projections (or to the Toeplitz / compression operators realizing that virtual bundle) is inverse, up to homotopy, to $[T]$. full proofs appear in Atiyah-Janich's original notes and modern textbook treatments of $K$-homology duality. $\square$

theorem (index is locally constant). the analytical index is constant on connected components of $\mathcal{F}(H)$. if $K$ is compact then $T+K$ is Fredholm and $\operatorname{index}(T+K)=\operatorname{index} T$.

proof. small norm perturbations of an invertible operator between ker and coker complementary subspaces remain invertible, so $\dim\ker$ and $\dim\operatorname{coker}$ are upper semicontinuous and their difference is constant. compact perturbations do not change the class in the Calkin algebra $B(H)/K(H)$, hence do not change Fredholmness or index (Atkinson's theorem). $\square$

$K$-theory and the analytical index of elliptic operators

Let $D$ be an elliptic pseudodifferential operator acting between sections of complex vector bundles $E$ and $F$ over a closed manifold $M$. Ellipticity implies that $D$ is Fredholm, so its analytical index $$ \operatorname{index} D=\dim\ker D-\dim\operatorname{coker} D $$ is well-defined and integer-valued. The family of all such operators, parametrized by the base of a fiber bundle or by a compact parameter space $X$, defines a class in $K(X)$. The Atiyah-Singer index theorem asserts that this analytical index coincides with a topological index constructed from the $K$-theory class of the principal symbol of $D$ via the Chern character and the Todd class of the tangent bundle. In the special case of a single operator the topological index reduces to the pairing of a characteristic class (built from Chern characters and Todd classes) with the fundamental class of $M$.

definition (symbol class). the principal symbol of an elliptic operator $D$ of order $m$ is an isomorphism $\sigma(D): \pi^*E\to\pi^*F$ of pullback bundles over $T^*M\setminus 0$. it defines a class $$ [\sigma(D)]\in K^0(T^*M) $$ by the standard difference-bundle construction on the ball/sphere bundle of $T^*M$ (trivializations on the equator via $\sigma$).

theorem (elliptic operators are fredholm). on a closed manifold, an elliptic pseudodifferential operator $D:C^\infty(E)\to C^\infty(F)$ extends to a Fredholm operator between Sobolev spaces, with smooth kernel and cokernel, so that $\operatorname{index} D\in\mathbb{Z}$ is well-defined.

proof. ellipticity produces a pseudodifferential parametrix $Q$ with $QD-I$ and $DQ-I$ of order $-\infty$ (smoothing). smoothing operators are compact on Sobolev spaces, so $D$ is Fredholm. elliptic regularity upgrades weak solutions of $Du=0$ to smooth sections. $\square$

theorem (atiyah-singer in $K$-theoretic form). for an elliptic operator $D$ on a closed oriented manifold $M$, $$ \operatorname{index} D =\operatorname{index}_t\bigl([\sigma(D)]\bigr), $$ where the topological index $\operatorname{index}_t:K^0(T^*M)\to\mathbb{Z}$ is the composition of the Thom isomorphism in complex $K$-theory for the tangent bundle with the pushforward $K^0(TM)\to K^0(\mathrm{pt})\simeq\mathbb{Z}$. under the Chern character this becomes $$ \operatorname{index} D =\bigl\langle\operatorname{ch}([\sigma(D)])\,\mathrm{Td}(TM\otimes\mathbb{C}),[T^*M]\bigr\rangle. $$

proof (structure). both analytical and topological indices are ring maps / cobordism invariants of symbol classes and agree on generators (Dirac operators, multiplicative sequences, products, and Bott elements) by heat-kernel or embedding proofs. chapter 8 records the Dirac case $\operatorname{index} D_E^+=\int\operatorname{ch}(E)\widehat{A}(TM)$ in detail; the general cohomological formula is the Chern character image of the $K$-theory equality. $\square$

theorem (family index). a continuous family of elliptic operators parametrized by a compact space $X$ defines a class in $K(X)$ via the Atiyah-Janich model. the Chern character of that class is the family index form in $H^{\mathrm{even}}(X;\mathbb{Q})$, computed by the families index theorem.

proof (idea). elliptic families yield continuous maps $X\to\mathcal{F}$ after Sobolev completion, hence classes in $[X,\mathcal{F}]\simeq K(X)$. the cohomological families index theorem identifies $\operatorname{ch}$ of that class with integrals of characteristic forms along the fibers. $\square$

Thus $K$-theory supplies both a stable classification of the vector bundles that carry classical fields and the precise cohomological receptacle for the index of the elliptic operators (Dirac, signature, $\overline{\partial}$, and so on) that govern the linear dynamics and the anomalies of those fields. It forms the natural bridge between the characteristic-class invariants of the preceding chapter and the spin-geometric index theorems that follow.

remark (to chapter 8). spin geometry specializes the symbol calculus to the Dirac operator and identifies $\operatorname{index}_t$ with $\widehat{A}$-genera and related characteristic numbers.

exercises

exercise 1 (stable triviality of $TS^2$). show that $TS^2\oplus\varepsilon^1$ is trivial as a real bundle of rank $3$, yet $TS^2$ is not trivial. interpret as a nontrivial class in $\widetilde{KO}(S^2)$ that becomes zero after adding a trivial line.

exercise 2 (hopf generator). for the Hopf line bundle $H\to S^2$, show that $[H]-1$ generates $\widetilde{K}(S^2)\simeq\mathbb{Z}$ by computing $c_1$ and using that $\operatorname{ch}$ is injective on this group.

exercise 3 (compact perturbation). if $T$ is Fredholm and $K$ is compact, prove that $T+K$ is Fredholm with the same index, using Atkinson's characterization (invertible in the Calkin algebra).