TITLE: Explanation for the Chern character
QUESTION [51 upvotes]: The Chern character is often seen as just being a convenient way to get a ring homomorphism from K-theory to (ordinary) cohomology.
The most usual definition in that case seems to just be to define the Chern character on a line bundle as $\mathrm{ch}(L) = \exp(c_1(L))$ and then extend this; then for example $\mathrm{ch}(L_1 \otimes L_2) = \exp(c_1(L_1 \otimes L_2)) = \exp(c_1(L_1) + c_2(L_2)) = \mathrm{ch}(L_1) \mathrm{ch}(L_2)$; then we can use this to define a Chern character on general vector bundles.
This all seems a bit ad-hoc, and it doesn't give much insight as to why such a thing exists anyway.
An explanation I like a lot better comes from even complex oriented cohomology theories. Given any complex oriented periodic cohomology theory, such as K-theory or periodic (ordinary) cohomology, we have $H(\mathbb{CP}^\infty) \cong H(P)[[t]]$ for $P$ a point. Seeing as $\mathbb{CP}^\infty$ is the classifying space for line bundles, this gives us a way of having "generalised Chern classes" for any line bundle corresponding to any cohomology theory, and even for any vector bundle.
We have a link between complex oriented periodic cohomology theories and formal group laws, corresponding to what corresponds to $c_1(L_1 \otimes L_2)$ in $\mathbb{CP}^\infty$: for ordinary cohomology, as above, we get that $c_1(L_1 \otimes L_2) = c_1(L_1) + c_1(L_2)$ which gives the additive formal group law, and for K-theory we get $c_1(L_1 \otimes L_2) = c_1(L_1) + c_1(L_2) + c_1(L_1) c_2(L_2)$ which is the multiplicative formal group law. The fact that over $\mathbb{Q}$ (but not over $\mathbb{Z}$) there is an isomorphism between the formal group laws given by the exponential map, and this reflects in the cohomology, giving the Chern character $K(X) \otimes_\mathbb{Z} \mathbb{Q} \to \prod_n H^{2n}(X,\mathbb{Q})$.
I'm not too sure what the exact formulation in that second case is, but more importantly I was wondering if there are any other, cleaner interpretations of the Chern character (I've been hearing about generalised Chern characters, and I have no idea where they would come from in this case). It seems like there should be a way to link the Chern character to things like the genus of a multiplicative sequence, and tie it in with other similar ideas for example the Todd genus or the L-genus given by similar formal power series. I guess the trouble is that I don't see how these related ideas all fit in together.

REPLY [3 votes]: I can give you my interpretation of the Chern character in terms of morphism of formal group laws. Denote with $\gamma$ the universal line bundle over $\mathbb{P}^\infty$ and denote its first Chern class with $c_1(\gamma) = x_H \in H^2(\mathbb{P}^\infty)$. Also denote with $k = \gamma-1 \in K(\mathbb{P}^\infty)$ and with $k_i : = p_i^*(\gamma)-1 \in K(\mathbb{P}^\infty \times \mathbb{P}^\infty)$. Then by definition the Chern character of $k \in K(\mathbb{P}^\infty)$ is equal to $e^{x_H}-1 \in HP^0(\mathbb{P}^\infty, \mathbb{Q}):=\prod_{i\geq0}H^{2i}(\mathbb{P^\infty},\mathbb{Q})$. The naturality of the Chern character and the isomorphism $HP^0(\mathbb{P}^\infty,\mathbb{Q}) \cong \mathbb{Q}[[x_H]], HP^0(\mathbb{P}^\infty \times \mathbb{P}^\infty,\mathbb{Q}) \cong \mathbb{Q}[[x_1,x_2]]$ gives the following commutative square
$\require{AMScd}$
\begin{CD}
    K(\mathbb{P}^\infty) @>ch>> \mathbb{Q}[[x_H]]\\
    @V \mu^* V V @VV \mu^* V\\
    K(\mathbb{P}^\infty \times \mathbb{P}^\infty) @>>ch> \mathbb{Q}[[x_1,x_2]]
\end{CD}
where $\mu$ is the multiplication in the H-space $\mathbb{P}^\infty$. If we follow the element $1+k \in K(\mathbb{P}^\infty)$ we get
$\require{AMScd}$
\begin{CD}
    1+k @>ch>> e^{x_H}\\
    @V \mu^* V V @VV \mu^* V\\
    1+k_1+k_2+k_1k_2 @>>ch> e^{x_1+x_2}
\end{CD}
Now let $m$ be te multiplicative formal group law and $a$ be te additive one. Then, by the previous diagram we get that
$
m(ch(k_1),ch(k_2)) = m(e^{x_1}-1,e^{x_2}-1) = e^{a(x_1,x_2)}-1 
$
and the Chern character is identified to a morphism (actually an isomorphism) of fgl.