Chern character isomorphism

Remarks

The Chern character, extended to all degrees, is a natural transformation of generalized cohomology theories from $K^*(-) \otimes \Q$ to $\prod_k H^{*+2k}(-; \Q)$. Both are cohomology theories on finite CW complexes, because tensoring with $\Q$ preserves exact sequences. The map is an isomorphism on a point, where both sides are $\Q$ in even degrees and $0$ in odd degrees. Induction on the cells, using the five lemma and the long exact sequences, gives the isomorphism for all finite CW complexes. Equivalently, the Atiyah–Hirzebruch spectral sequence for K-theory collapses rationally.

Example: for $X = S^2$, $K^0(S^2)$ is free abelian with basis $1$ and $[H]$, where $H$ is the tautological line bundle, and $\operatorname{ch}(H) = 1 + c_1(H)$, where $c_1(H)$ generates $H^2(S^2; \Z)$. So $\operatorname{ch}$ maps $K^0(S^2)$ isomorphically onto the integral lattice $H^0(S^2; \Z) \oplus H^2(S^2; \Z)$ in this case.

Tensoring with $\Q$ is necessary. For the real projective plane, $\tilde{K}(\R P^2) \cong \Z/2$, while $\tilde{H}^{\text{even}}(\R P^2; \Q) = 0$, so torsion in K-theory is lost. Also, the image of $\operatorname{ch}$ need not lie in integral cohomology: $\operatorname{ch}(E) = \operatorname{rank} E + c_1 + \frac{1}{2}(c_1^2 - 2 c_2) + \cdots$ has denominators.

Going deeper

For a student

The rational isomorphism is used constantly to compute with K-theory: classes in K-theory are tested through their Chern characters. It is the bridge between the K-theoretic form of index theorems and their cohomological form, where the Chern character of the symbol is integrated against the Todd class. Integrality questions, that is, which rational classes are Chern characters of actual bundles, lead to deep constraints such as those used by Adams in the Hopf invariant one problem.