Topological K-theory ($K^{\ast }(X)$)

Complex topological K-theory is the generalized cohomology theory built from stable equivalence classes of complex vector bundles. Its degree-zero group is the Grothendieck group $K^0(X)$, and its reduced theory is denoted ${\tilde{K}}^{\ast }(X)$.

For compact metric $X$, odd topological K-theory is the homotopy set of norm-continuous maps into unitaries equal to the identity modulo compact operators on a fixed separable infinite-dimensional Hilbert space. Uniform finite-rank approximation over $X$ and over homotopies gives the equivalent stable model ${\mathrm{colim}}_N[X,G{L}_N(\mathbb{C})]$. Block sum gives an abelian group: a finite-dimensional unitary path swaps blocks, and a two-block rotation contracts $\mathrm{diag}(u,u^{-1})$ to the identity. On the circle the determinant winding identifies $K^1(S^1)$ with $\mathbb{Z}$; Bott periodicity gives $K^1(S^n)=\mathbb{Z}$ in odd dimensions and zero in even dimensions.

If $L$ is the complexification of a real vector bundle of the tautological bundle and $\mu =[L]-1$, thenRealizing Real projective space as the circle bundle of the square of the tautological bundle on Complex projective space gives this through the K-theory Gysin sequence of a sphere bundle.

For a finite covering map $p:Y\to X$, define the fiber of the transferred vector bundle by $(p_{!}V{)}_x={⨁}_{y\in p^{-1}(x)}{V}_y$. Local trivializations of the covering map make this a vector bundle, and compatibility with direct sums extends it to Topological K-theory.

The K-theory transfer of a finite covering distributes over a pulled-back tensor product of vector bundles:For an $n$-sheeted covering map, $p_{!}(1)$ is the permutation vector bundle of rank $n$. Its difference from $n$ is a nilpotent element by nilpotence of rank-zero K-theory classes, so it is a unit after inverting $n$. Thus pullback on Topological K-theory becomes injective after this localization of a ring.

The rank of a vector bundle is locally constant on the base. Taking the difference of the ranks extends to the Grothendieck group of vector bundles, giving the rank map in Topological K-theory. For a finite CW complex, its kernel consists of classes of rank zero on every connected component.

For a finite CW complex, every element in the kernel of the rank map in topological K-theory is a nilpotent element. One proof adjoins one positive-dimensional cell at a time: the restriction kernel is square-zero by the relative product in topological K-theory. For connected spaces the rank-zero kernel is Reduced topological K-theory; for disconnected spaces the kernel of restriction to just one basepoint can contain nonzero idempotents.

For a nonempty subcomplex $A\subset X$, relative Topological K-theory is $K^i(X,A)={\tilde{K}}^i(X/A)$, where the collapsed subcomplex is the basepoint. It fits into the K-theory six-term exact sequence.

The relative topological K-theory productis induced by the reduced diagonal $X/A\to (X/A)\wedge (X/A)$. If $X/A$ is a positive-dimensional sphere, this diagonal is null-homotopic, so the kernel of $K^0(X)\to K^0(A)$ is square-zero.

For a pair $(X,A)$ of CW complexes, relative topological K-theory and Bott periodicity give the cyclic exact sequence

The Grothendieck group of a commutative monoid is its universal abelian-group completion. For vector bundles under direct sum, its elements are formal differences $[E]-[F]$ modulo stabilization.

For a based compact space, reduced K-theory is the kernel of restriction to the basepoint. It satisfies

Multiplication by the Bott element $\beta \in {\tilde{K}}^0(S^2)$ gives the Bott isomorphism

The Bott element is the reduced class of the tautological complex line bundle over $S^2\cong {\mathbb{CP}}^1$, up to the choice of sign. Its exterior powers generate the reduced K-theory of even-dimensional spheres.

Complex Bott periodicity gives

If a finite CW-complex $Y$ has only even-dimensional cells, thenand $K^0(Y)$ is free abelian, with one generator for each cell. This follows by induction from the six-term exact sequence for adjoining a wedge of even-dimensional cells.

If $Y$ is a finite even-cell complex, the exterior product is an isomorphismCellular induction proves this because $K^0(Y)$ is free and $K^{-1}(Y)=0$.

If $t=1-[\overline{\gamma }]$ for the tautological complex line bundle, then

Let $E_0,E_1$ be complex vector bundles of rank $d$ over a finite CW complex of dimension at most $2d$. If $E_0\oplus {\mathbb{C}}^r\cong E_1\oplus {\mathbb{C}}^r$ for some $r$, then $E_0\cong E_1$. This follows because the stabilization map $BU(d)\to BU$ is $2d$-connected, so stable equivalence is injective on bundles over such a complex.

Two rank-$d$ complex vector bundles over ${\mathbb{CP}}^d$ are isomorphic exactly when their Chern classes agree. Equality of Chern classes gives equality of their Chern characters; the Chern character is injective here because $K^0({\mathbb{CP}}^d)$ is torsion-free. The bundles are therefore stably isomorphic, and stable cancellation for complex vector bundles applies because ${\mathbb{CP}}^d$ has real dimension $2d$.

For a self-map $f:Z\to Z$, the mapping torus has an exact sequence containingIf $K^{-1}(Z)=0$, then $K^0(T_f)\cong \ker (1-f^{\ast }:K^0(Z)\to K^0(Z))$.

For the factor swap on ${\mathbb{CP}}^2\times {\mathbb{CP}}^2$, the invariant subgroup ofhas basis $1,xy,x^2{y}^2,x+y,x^2+y^2,x{y}^2+x^{2}y$. Hence the mapping torus has $K^0\cong {\mathbb{Z}}^6$.

For a complex vector bundle $E\to X$, there is a map $p:F(E)\to X$ such that $p^{\ast }$ is injective on cohomology and $p^{\ast }E$ splits as a sum of line bundles. Symmetric identities in the Chern roots can therefore be proved after this pullback.

If a pulled-back complex vector bundle splits as $L_1\oplus \cdots \oplus L_r$, its formal Chern roots are $x_i=c_1(L_i)$. The Chern classes are the elementary symmetric polynomials in these roots.

If the formal Chern roots of $E$ are $x_1,\dots ,x_r$, thenIt extends to virtual bundles and is a ring homomorphism because direct sum joins root lists while tensor product replaces them by all sums $x_i+y_j$.

Under the natural identificationthe Chern character maps ${\tilde{K}}^0(S^{2n})$ into integral cohomology. A Bott generator is an exterior product of $n$ degree-two Bott elements, whose Chern characters multiply to an integral top-dimensional generator.

For a complex vector bundle $E\to S^{2n}$,is divisible by $(n-1)!$. Since the lower Chern classes vanish, the Newton identity givesand the Chern character on an even-dimensional sphere is integral.

For a complex vector bundle $E\to X$, the K-theory Thom classgenerates the Thom isomorphism $K^i(X)\cong {\tilde{K}}^i(\mathrm{Th}(E))$.

Pulling the K-theory Thom class back along the zero section gives

Let $\gamma$ be the tautological line bundle and put $t=1-[\gamma ]$. The Euler sequence on complex projective space givesHenceand evaluating at $s=-1$ in $K^0({\mathbb{CP}}^n)=\mathbb{Z}[t]/(t^{n+1})$ gives

The cofibration $S(E)\to D(E)\to \mathrm{Th}(E)$ and the Thom isomorphism give

Let $L$ be the complexification of a real vector bundle of the tautological bundle over Real projective space ${\mathbb{RP}}^{2n+1}$. For $k\ge 1$, the K-theory Euler class of $L^{\oplus k}$ is $-2^{k-1}([L]-1)$, andThe K-theory Gysin sequence of a sphere bundle gives an extension by $\mathbb{Z}$, which splits as an exact sequence of abelian groups.

The K-theory Gysin sequence of a sphere bundle and $e^K(T{\mathbb{CP}}^n)=(n+1)t^n$ giveand

For $E=\gamma \oplus \gamma$ over ${\mathbb{CP}}^n$ and $t=1-[\overline{\gamma }]$, the K-theory Euler class is $t^2$. Hencefor $n\ge 1$.

For $n=0$, the base is a point, the sphere bundle is $S^3$, and its odd K-theory is $\mathbb{Z}$.

The Adams operations are natural ring endomorphisms of complex K-theory characterized on line bundles by

If $u\in {\tilde{K}}^0(S^{2m})$ is a Bott element generator, then

Let ${\mathbb{CP}}_k^{k+n}={\mathbb{CP}}^{k+n}/{\mathbb{CP}}^{k-1}$. If the bottom-cell inclusion $S^{2k}={\mathbb{CP}}_k^k\to {\mathbb{CP}}_k^{k+2}$ admits a retraction, then $24$ divides $k$. Writing $x=[\overline{\gamma }]-1$, a retracted Bott generator has the form $x^k+a{x}^{k+1}+b{x}^{k+2}$. Comparing it with its image under ${\psi }^2(x)=2x+x^2$ gives $a=-k/2$ and $b=k(3k+5)/24$; integrality forces both $3$ and $8$ to divide $k$.

The cannibalistic class is defined byIt satisfies ${\rho }^k(E\oplus F)={\rho }^k(E){\rho }^k(F)$. For a line bundle,

For $p_{!}:K^{-1}(S(E))\to K^0(X)$ in the K-theory Gysin sequence,This follows by applying ${\psi }^k$ to $\delta x={\lambda }_E{p}_{!}(x)$.

Choose $a,b$ with $p_{!}(a)=t^{n-1}$ and $p_{!}(b)=t^n$, where $t=1-[\overline{\gamma }]$. Then