Vrftjkry

Let $MU$ be the complex bordism spectrum and let $Hmathbb{Z}$ be the Eilenberg-Maclane spectrum.

Is it know what the structure of the complex cobordism cohomology $MU^{*}(Hmathbb{Z})$ is?

EDIT: What if instead $Hmathbb{Z}$, one consider $Hmathbb{Z}/(p)$ for a prime $p$?

One can prove that $mathrm{Map}(Hmathbf{F}_p,MU)$ is contractible. We know that $Hmathbf{F}_p$ is dissonant (Theorem 4.7 of Ravenel's "Localization with Respect to Certain Periodic Homology Theories"), but $MU$ is harmonic (Theorem 4.2 of that paper). Since dissonant spectra (resp. harmonic spectra) are by definition $E$-acyclic (resp. local) for the spectrum $E = bigvee_p E_p$, where $E_p = bigvee_{0leq n<infty} K(n)$, the claim follows. (I just realized that this is Corollary 4.10 of Ravenel's paper.)

It is, however, not the case that $mathrm{Map}(Hmathbf{Z},MU)$ is contractible. (What I wrote previously was incorrect.) The spectrum $Hmathbf{Z}$ is not dissonant, so we cannot immediately apply the above argument. Since $MU$ is harmonic, there is, however, an equivalence between $mathrm{Map}(Hmathbf{Z},MU)$ and $mathrm{Map}(L_E Hmathbf{Z},MU)$. We therefore need to understand $L_E Hmathbf{Z}$. By the discussion at this question, we can conclude that $L_{E_p} Hmathbf{Z} simeq Hmathbf{Q}_p$. It therefore suffices to understand $MU^ast(Hmathbf{Q})$. But $Hmathbf{Q}$ is the colimit of multiplication by $2,3,5,7,cdots$ on the sphere, so $MU^ast(Hmathbf{Q})$ admits a description in terms of $lim^0$ and $lim^1$ of multiplication by $2,3,5,7,cdots$ on $pi_ast MU$. In particular, the $lim^1$ term is $mathrm{Ext}^1_mathbf{Z}(mathbf{Q,Z}) cong widehat{mathbf{Z}}/mathbf{Z}$.