How to write K-theory Conner-Floyd Chern classes in terms of Adams operations?

Question

From Adams, we know that the algebra of (unstable, degree-zero) cohomology operations $K^0(BU)$ can be written as formal infinite linear combinations of canonical generators

$$\mu_n := \sum_{i=0}^{n} (-1)^{n-k}\binom{n}{k} \psi^k$$

However, from the collapse of the Atiyah-Hirzebruch spectral sequence for BU, we also know that $K^0(BU) \cong \mathbb{Z}[[c_1^K,c_2^K,\ldots ]]$ where $c_i^K$ are the Conner-Floyd Chern classes (where I'm renormalizing them to degree zero by an appropriate power of $t\in \pi_2KU$).

Thus, it should be possible to write the Chern classes in terms of the Adams operations. How can I find these expressions?

Doing the reverse is not so bad, using Hirzebruch's theory of genera: I get that $\psi^k$ is $(1+c_1^K+c_2^K+\ldots)^k$. But unfortunately I'm completing lacking in the power series wizardry that would allow me to invert this.

This was wrong, since the multiplication of characteristic series of genera does not correspond to anything over on the cohomology operations side, coinciding neither with cup product nor composition (which are themselves distinct).

Answer 1

I am not sure about the facts you mention, and I don't think I'll quite answer your question, but here are some facts I do know.

First, it is not the case that all $KU$-operations can be written as (even infinite) sums of Adams operations; Adams operations are additive, and general $KU$-operations need not be. So I won't say how to write the Chern classes in terms of Adams operations, but I'll try to say something about how they relate; I apologize if I am just repeating facts you are already familiar with.

One technical remark is that $KU$-operations aren't governed by $KU(BU)$, but really by $KU(\mathbb{Z}\times BU)$. I'll talk about $ku({-}) = [{-},BU]$ instead, which is valued in nonunital rings and has operations governed by $ku(BU) = \mathbb{Z}[[c_1,c_2,\ldots]]^+$, the augmentation ideal of $KU(BU)$. From now on, all my rings will be nonunital, but you can feel free to add a unit and think of them as augmented rings instead. I think my $c_i$'s are the same as yours; they are so that if we write $ku(BU(1)) = \mathbb{Z}[[u]]^+$ with $u=l-1$ with $l$ the tautological line bundle, then pullback under summation $ ku(BU)\rightarrow ku(BU(1)^{\times n})$ sends $c_i$ to the $i$'th elementary symmetric polynomial in $u_1,\ldots,u_n$.

Adams operations can be defined in any $\lambda$-ring, and $ku({-})$ is valued in $\lambda$-rings in the usual way. So you can write $\psi^n$ as a polynomial in $\lambda^1,\ldots,\lambda^n$, but not conversely. I think the relation is $$ t\frac{d}{dt}\log(1+\sum_{n\geq 1}\lambda^n(x) t^n) = \sum_{n\geq 1}(-1)^{n+1}\psi^n(x)t^n, $$ but I would not swear on the signs. Moreover, you can identify sums of Adams operations as exactly the additive operations acting on all $\lambda$-rings. Since this relates $\lambda$-operations and Adams operations, I'll just say how $\lambda$-operations relate to Chern classes.

For $\lambda$-rings like $ku({-})$ that are comprised of things like degree zero virtual bundles, it's useful to introduce the $\gamma$-operations. If I set $\lambda_t(x) = 1+\sum_{n\geq 1}\lambda^n(x)t^n$ for a formal variable $t$, and similarly define $\gamma_t$, then these are uniquely determined by asking first that $\gamma_t(x+y) = \gamma_t(x)\gamma_t(y)$, and second that if $\lambda_t(l)=1+lt$ then $\gamma_t(l-1)=1+(l-1)t$. You can explicitly relate these with the $\lambda$-operations via $\gamma_t = \lambda_{t/(1-t)}$, and a $\lambda$-ring structure is equivalent to a $\gamma$-ring structure.

Finally, a splitting principle argument lets you show that the class $c_n\in ku(BU)$ corresponds to the operation $\gamma^n$.