Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 142 (Characteristic Classes and K-Theory)

Let $p:\mathbb{P}(E)\to X$ be the projectivization of a real vector bundle, let $L_E$ be its tautological bundle, and putThe mod-two projective bundle formula says that $H^{\ast }(\mathbb{P}(E);{\mathbb{F}}_2)$ is a free $H^{\ast }(X;{\mathbb{F}}_2)$-module on $1,u,\dots ,u^{d-1}$. The Projective bundle definition of Stiefel–Whitney classes is the unique relation

Apply the splitting principle for real vector bundles. After an injective pullback, writeIf $x_i=w_1(L_i)$, the projective-bundle relation factors assoThe line summands of $E\oplus E^{\prime }$ are the union of the two lists, henceComparing the degree-$k$ components proves the Whitney product formula for Stiefel–Whitney classesInjectivity of the splitting pullback returns the identity to $X$.

For real line bundles, the transition functions take values in $O(1)=\{\pm 1\}$. Tensor product multiplies these signs, while the identification $\{\pm 1\}\cong \mathbb{Z}/2$ turns multiplication into addition. The corresponding degree-one characteristic classes therefore satisfy the First Stiefel–Whitney class of a tensor product of real line bundles formulaEquivalently, this follows from the classification of real line bundles by $H^1(X;{\mathbb{F}}_2)$.

Now take $M={\mathbb{RP}}^n$ and $E=k{\gamma }_{\mathbb{R}}^{1,n+1}$. Sincea line in $E_x$ is the fixed line ${\gamma }_x$ tensored with a line in ${\mathbb{R}}^k$. Thus the projectivization of copies of the real tautological line bundle isLet $x$ and $v$ be the degree-one generators pulled back from the first and second factors. The mod-two cohomology ring of real projective space and the Künneth theorem giveThe tautological line $L_E$ is the tensor product of the two tautological lines, so $w_1(L_E)=x+v$. In the alternative generator $u=w_1(L_E)$, the same ring is

The stable tangent-bundle identitygivesFor the vertical part of the tangent bundle of a projectivized real vector bundle,Each of the $k$ line summands on the right has first Stiefel–Whitney classThe Whitney product formula for Stiefel–Whitney classes therefore yieldsthe Total Stiefel–Whitney class of the projectivization of copies of the tautological line.

The Bott isomorphism is multiplication by the Bott element $\beta \in {\tilde{K}}^0(S^2)$:Together with the suspension isomorphism andit gives the Complex K-theory of a sphere

Letbe a CW filtration in which each quotient $Y_r/Y_{r-1}$ is a wedge of even-dimensional spheres. The six-term exact sequence in Topological K-theory, the sphere calculation, and induction giveand a short exact sequence whose new summand in $K^0(Y_r)$ is free abelian on the newly attached cells. Every such extension splits as an extension of free abelian groups, so $K^0(Y)$ is free, with one generator for each cell. This proves the Complex K-theory of an even-cell complex result.

The exterior product definesFor a point it is the identity. Attaching one layer of even cells gives corresponding exact sequences on the source and target; the sphere case is the suspension isomorphism, and induction with the Five lemma proves that the product map remains an isomorphism. This is the Künneth theorem for complex K-theory with an even-cell factor.

For a mapping torus $T_f$, the K-theory Wang sequence of a mapping torus containsWhen $K^{-1}(Z)=0$, exactness gives

For $Z={\mathbb{CP}}^2\times {\mathbb{CP}}^2$, the Complex K-theory of complex projective space and the K-theory Künneth isomorphism giveThe factor swap interchanges $x$ and $y$. Its invariant subgroup has the basisIt follows that the K-theory of the mapping torus of the factor swap on two complex projective planes is

The splitting principle for complex vector bundles says that for every complex vector bundle $E\to X$ there is a map $p:F(E)\to X$ such that $p^{\ast }$ is injective on cohomology andsplits into complex line bundles. Write $x_i=c_1(L_i)$ for the formal Chern roots.

Define the Chern character after this injective pullback byEach homogeneous component is a symmetric polynomial in the $x_i$ with rational coefficients, hence a polynomial in the elementary symmetric functions $c_j(E)$. It therefore descends uniquely to $H^{\mathrm{ev}}(X;\mathbb{Q})$ and depends only on $E$. Seton the Grothendieck group $K^0(X)$; additivity under direct sums makes this well defined.

If $E$ has roots $x_i$ and $F$ has roots $y_j$, then $E\otimes F$ has roots $x_i+y_j$. ConsequentlyIt also sends the trivial line to $1$, so it is a unital ring homomorphism.

For $S^{2n}$, a generator of ${\tilde{K}}^0(S^{2n})$ is the $n$-fold exterior product of the degree-two Bott element. The Chern character respects exterior products, and the degree-two Bott element has Chern character equal, up to sign, to the integral generator of ${\tilde{H}}^2(S^2;\mathbb{Z})$. Its $n$-fold product maps to the integral top-dimensional generator. Hence the Chern character on an even-dimensional sphere is integral.

Let the formal Chern roots of $E\to S^{2n}$ be $x_1,\dots ,x_r$ and write $p_n={\sum }_i{x}_i^n$. Sinceall lower Chern classes $c_1(E),\dots ,c_{n-1}(E)$ vanish. The Newton identities then reduce toThe degree-$2n$ term of the Chern character is thereforeIts evaluation on the fundamental class is an integer by integrality of the reduced Chern character. Thusis divisible by $(n-1)!$, proving the Divisibility of the top Chern number on an even-dimensional sphere.

The cofibrationgives the long exact sequence of a pair in Topological K-theory. Identify $D(E)$ with $X$ by deformation retraction and use multiplication by the K-theory Thom classto identify the relative term with $K^{\ast }(X)$. Pullback along the zero section sends ${\lambda }_E$ to the K-theory Euler classThe map from the relative term to $K^{\ast }(D(E))$ is therefore multiplication by $e^K(E)$, giving the K-theory Gysin sequence of a sphere bundle

Forover ${\mathbb{CP}}^n$, put $t=1-[\overline{\gamma }]$. The Complex K-theory of complex projective space isandThe Gysin sequence consequently identifiesfor $n\ge 1$. This is the Odd K-theory of the sphere bundle of two tautological lines.

If $n=0$, then the base is a point and $Y=S^3$, so $K^{-1}(Y)\cong \mathbb{Z}$ by Bott periodicity.

The cannibalistic class is defined by the identityfor the Adams operation ${\psi }^k$. The Thom class of a direct sum is the product of the pulled-back Thom classes. Applying the ring homomorphism ${\psi }^k$ givesIf $L$ is a line bundle, restriction along the zero section givesso

Let $\delta :K^{-1}(S(E))\to {\tilde{K}}^0(\mathrm{Th}(E))$ be the boundary map. By definition of $p_{!}$,The natural operation ${\psi }^k$ commutes with $\delta$, and thereforeCancelling the Thom class proves the Adams operation and the boundary pushforward of a sphere bundle formula

Choose the basis $a,b$ of $K^{-1}(Y)$ characterized byFor $E=\gamma \oplus \gamma$,Modulo $t^{n+1}$, this givesSince $p_{!}$ identifies $K^{-1}(Y)$ with this kernel, the Second Adams operation on the odd K-theory of the sphere bundle of two tautological lines isFor $n=0$, the single generator of $K^{-1}(S^3)$ is multiplied by $4$.