Chern character of a complex vector bundle

Assume AC. Let $E\to X$ be a numerable complex vector bundle over a finite CW complex $X$. Write $X={⨆}_{\alpha }{X}_{\alpha }$ for its finitely many path components and $n_{\alpha }$ for the constant rank of $E{\mathrm{\mid }}_{X_{\alpha }}$. On a component of positive rank, let $c_i(E{\mathrm{\mid }}_{X_{\alpha }})$ and the Chern roots $t_1,\dots ,t_{n_{\alpha }}$ have the meanings fixed by Chern classes from the projective-bundle relation and Complex flag bundle and Chern roots. On a rank-zero component all positive Chern classes and all positive-degree Chern-character components are defined to be zero.

For each $n\ge 1$ and $k\ge 0$ let $N_{n,k}$ be the Newton polynomial expressing the $k$-th power sum in $n$ variables: the unique polynomial with $N_{n,k}(e_1(x),\dots ,e_n(x))={\sum }_{i=1}^n{x}_i^k$ for all $x_1,\dots ,x_n$, which exists by Fundamental theorem of symmetric polynomials: unique expression as a polynomial in $e_1,\dots ,e_n$ applied to the symmetric polynomial ${\sum }_i{x}_i^k$. Here and below integral Chern classes and roots are sent to rational cohomology by the coefficient map $\mathbb{Z}\to \mathbb{Q}$ before evaluating a rational expression. Give the $i$-th polynomial variable weight $i$. The uniqueness in the symmetric-polynomial theorem shows that $N_{n,k}$ is weighted homogeneous of weight $k$: decompose it by weight and substitute the homogeneous elementary symmetric polynomials; injectivity of that substitution forces every weight other than $k$ to vanish. On a positive-rank component define ${\mathrm{ch}}_k(E){\mathrm{\mid }}_{X_{\alpha }}:=\frac{1}{k!}{N}_{n_{\alpha },k}(c_1(E{\mathrm{\mid }}_{X_{\alpha }}),\dots ,c_{n_{\alpha }}(E{\mathrm{\mid }}_{X_{\alpha }}))\mathrm{.}$ On a rank-zero component define every ${\mathrm{ch}}_k$ to be zero. These componentwise classes determine an element of $H^{2k}(X;\mathbb{Q})$. The Chern character is $\mathrm{ch}(E):={\sum }_{k\ge 0}{\mathrm{ch}}_k(E)\in H^{\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{n}}(X;\mathbb{Q})\mathrm{.}$

The polynomial definition is intrinsic to the Chern classes. The following argument justifies its equivalent description by roots, including uniqueness in rational cohomology.

On a positive-rank component write $q:Y\to X_{\alpha }$ for its flag bundle. It has CW homotopy type and $q^{\ast }E={⨁}_i{L}_i$ by Complex flag bundle and Chern roots and Complex splitting principle with integral injective pullback. Choose a homotopy equivalence $h:W\to Y$ from a path-connected CW complex. The pulled-back line bundles and their sum are numerable. Apply Chern naturality, normalization and Whitney sum on the actual CW base $W$, using Naturality, normalization, and Whitney sum for Chern classes, to obtain $(qh{)}^{\ast }{c}_i(E)=e_i(c_1(h^{\ast }{L}_1),\dots ,c_1(h^{\ast }{L}_n))$. Line normalization and Euler naturality Naturality, orientation sign, and Whitney product for Euler classes identify $c_1(h^{\ast }{L}_j)=h^{\ast }{t}_j$. Since $h^{\ast }$ is an isomorphism by Homotopic maps induce equal maps in singular cohomology, this proves $q^{\ast }{c}_i(E)=e_i(t_1,\dots ,t_n)$ integrally on $Y$.

Here is a direct proof that $q^{\ast }$ is injective with rational coefficients. Each projective stage $p:T\to B$ in the flag tower has path-connected paracompact Hausdorff CW-type base and global tautological Euler class $x$. Choose a homotopy equivalence $w:V\to B$ from a path-connected CW complex and pull back the bundle. The projective bundle $p_V:T_V\to V$ is numerable, hence Serre by Numerable fiber bundles are hurewicz fibrations. The pulled-back classes $1,x,\dots ,x^{r-1}$ restrict to an integral basis on each fiber by Euler naturality and The complex tautological Euler class restricts to the projective-fiber generator applied to the trivial rank-$r$ bundle over a point. The Schubert CW structure of this fiber ${\mathbb{C}\mathbb{P}}^{r-1}={\mathrm{Gr}}_1({\mathbb{C}}^r)$ has one cell in each dimension $0,2,\dots ,2r-2$ and no other cells, by Schubert cells in real and complex Grassmannians and Schubert cells give the stable Grassmannian CW structure. Thus its cellular cochain complex has one copy of the coefficient ring in each indicated even degree and zero in odd degrees, so every differential vanishes. The coefficient-natural cellular comparison Cellular cochains compute cohomology with local coefficients shows that the images over $\mathbb{Q}$ of the integral basis $1,x,\dots ,x^{r-1}$ are a rational basis: in the corresponding cellular coordinates each integral generator is $\pm 1$, which remains nonzero and generating after $\mathbb{Z}\to \mathbb{Q}$. Coefficient change preserves cup products directly by the face formula in Singular cup product on cochains. Leray--Hirsch over $\mathbb{Q}$ Leray–Hirsch module isomorphism now makes $p_V^{\ast }$ injective, since its module basis includes $1$. If $p^{\ast }a=0$, pulling back to $T_V$ gives $p_V^{\ast }{w}^{\ast }a=0$, whence $w^{\ast }a=0$ and $a=0$ because $w$ is a homotopy equivalence. Thus every stage is rationally injective, and so is their finite composite $q$. Rank-one stages are identities and obey the same argument with the single basis element $1$.

Consequently, in rational cohomology, $q^{\ast }{\mathrm{ch}}_k(E)=\frac{1}{k!}{\sum }_{i=1}^{n_{\alpha }}{t}_i^k,$ and ${\mathrm{ch}}_k(E)$ is the unique class with that pullback. The rank-zero prescription is canonical. The series is a finite sum: for $X=\mathrm{\varnothing }$ every group is zero; otherwise $H^{2k}(X;\mathbb{Q})=0$ for $2k>\dim X$ by Cohomology of a finite CW complex vanishes above its dimension, so only finitely many terms are nonzero. The coefficient $1\mathrm{/}k!$ lies in $\mathbb{Q}$, and no division by zero occurs. Concretely ${\mathrm{ch}}_0(E)$ is the locally constant rank function, ${\mathrm{ch}}_1(E)=c_1(E)$, ${\mathrm{ch}}_2(E)=\frac{1}{2}(c_1^2-2c_2)$, and so on; the class depends only on the isomorphism class of $E$ (as do its Chern classes), and $\mathrm{ch}(0)=0$.