Question

Let $U^* \rightarrow H^* $ be the homomorphism describing the complex orientation of $H^* $ from complex cobordism. Let $t_1, t_2, ...$ be indeterminates.

My question is: Does there exist an integral cohomology operation $H^* (X) \rightarrow H^*(X)[\mathbf{t}]$ that makes the following diagram commute for any $X$? If so can we describe it in a way that doesn't involve complex cobordism?

Here $s_{\mathbf{t}}$ is the total Landweber-Novikov operation.

In particular, since the Landweber-Novikov operation satisfies the Riemann-Roch type formula for proper, complex-oriented maps $$ s_{\mathbf{t}} f_{*}x = f_{*}(c_{\mathbf{t}}(\nu_{f}) \cdot s_{\mathbf{t}}x)$$ (where $\nu_{f}$ is the virtual class $1-\nu_i$ and $\nu_i$ is the stable normal bundle of the proper, complex-oriented map $f: Z \rightarrow X$) we would expect ths operations to satisfy something similar.

EDIT: Whoops, original title didn't have to do with the question :) But it did have to do with the motivation behind asking it! Maybe another time...

Answer 1

This is more of a comment than an answer. I just wanted to point out that such an operation, if it were to exist, cannot be stable.

This follows from the answer to http://mathoverflow.net/questions/50519/integral-cohomology-stable-operations, which asserts that the set of stable cohomology operations in integral cohomology of a given degree $2k>0$ is finite, and the observation that $s_{(k)}(cf_1) = cf_1^{k+1}$, where $cf_1\in U^2(BU)$ is the first universal Conner-Floyd-Chern class.

To see this, suppose that there is a stable operation $\sigma_{(k)}:H^\ast(-)\to H^{\ast+2k}(-)$ which extends the stable operation $s_{(k)}:U^\ast(-)\to U^{\ast+2k}(-)$. Then we would have $$ \sigma_{(k)}(c_1)= c_1^{k+1}, $$ where $c_1\in H^2(BU)$ is the first universal Chern class. But since the sum of stable operations is stable, we would have $$ n\sigma_{(k)}(c_1)=0 $$ for $n$ sufficiently large, contradicting the fact that $nc_1^{k+1}\neq 0$.