Explanation for the Chern character

Question

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.

Answer 1

There is a nice discussion about multiplicative sequences, &c., in Lawson and Michelsohn's book "Spin Geometry". It discusses things like the Todd genus, the A-hat genus, and so on, but also the Chern character and the ring homomorphism from K-theory to ordinary cohomology. It is a readable exposition and perhaps "connects the dots" in a way that would be helpful to you.

Answer 2

For a general complex oriented cohomology theory represented by a ring spectrum $R$, there is a "Hurewicz map" from $R$ to its smash product $H\mathbb{Z}\wedge R$ with the Eilenberg-Mac Lane object for the integers. $R$ has a formal group law associated to it as you stated. So does $H\mathbb{Z}\wedge R$; in fact, it carries the formal group law from $H\mathbb{Z}$ (the additive group), the one from $R$, and an isomorphism between them. You can think of this isomorphism of as a "logarithm" for the formal group law of $R$.

For certain complex oriented cohomology theories $R$ (the so-called "Landweber exact" theories) you can say more. Complex K-theory, which is represented in the stable homotopy category by a spectrum called $KU$, is one such example. In Landweber exact cases, the Hurewicz map of graded rings from $\pi_*R$ to $H_*R$ is the universal map from $\pi_*R$ (with its formal group law) to a ring where this formal group law has a choice of logarithm.

In the case of K-theory (and in some other cases), this universal ring is the rationalization. So you can think of the Chern character as simply the Hurewicz homomorphism, or as the universal way to adjoin a logarithm to the formal group law of K-theory.

Answer 3

There is a beautiful explanation of the Chern character that my student Fei Han proved in his thesis: The Chern character is given by the map that "crosses with the circle". The hard part is to explain why domain and range of this map are K-theory respectively de Rham cohomology. This uses isomorphisms

$K^0(X) \cong 1|1-EFT[X]$ and $H^{ev}_{dR}(X) \cong 0|1-EFT[X]$

where $d|1-EFT[X]$ are concordance classes of $d|1$-dimensional Euclidean field theories over the manifold $X$. Since the circle of length one is a Euclidean 1-manifold, it is not hard to believe, modulo the precise definitions, that crossing with it gives a map as required.

In fact, his result works even before taking concordance classes, where the left hand side is replace by vector bundles with connection and the right hand side becomes (even closed) differential forms.

Answer 4

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.