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).

  • Why does not your last statement imply that $\psi^k$ is $(\psi_1)^k$? – მამუკა ჯიბლაძე Oct 11 at 19:46
  • it does, in the usual cup-product multiplication on $K^0(BU)$. but $K^0(BU)$ has the "extra" multiplicative structure of composition, which gives it a second, different algebra structure, and it's via this algebra structure that it acts on other K-theory rings. – xir Oct 11 at 21:01
  • actually seeing the answer now i think perhaps one wants to restrict to additive operations to make things like "algebra action" make sense. – xir Oct 11 at 21:06
  • But does not $\psi^k$ equal $(\psi_1)^k$ only on line bundles? – მამუკა ჯიბლაძე Oct 11 at 21:17
  • i'm not sure what you mean by that. how are you proposing to "act" by $\psi^k$ on a line bundle? the only answer i can think of is by acting on $K^0$ of the relevant space via the action of the (additive) K-theory operations algebra, and that action is via the compositional algebra structure, not via the cup-product one, like i said. – xir Oct 11 at 21:19
up vote 6 down vote accepted

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 $\gamma^n$.

  • oh i never even thought of considering non-additivity but of course you're right a class in [BU, BU] doesn't have to be an H-space map for the addition. so i guess i'm really just talking about additive operations; i should go back and find if this is the case in my references as well. it's slightly confusing i guess since the additivity is a statement about having this second algebra structure under composition, a nuance i hadn't considered – xir Oct 11 at 21:08
  • ah, okay i think these $\gamma_t$ operations are the ingredient that i was not aware of. this formally looks like it corresponds to what i want. i'll think about how this relates to what i'm trying to do – xir Oct 11 at 21:24

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