Splitting principle for complex vector bundles

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.