For a prime number $p$ and the Brown-Peterson spectrum $BP$, let $T:BP\to H\mathbb{Z}\_{(p)}$ be the Thom map, and $T':BP\to H\mathbb{Z}\_p$ be the mop $p$ reduction of $T$. Tamanoi ([1](https://www.ams.org/journals/tran/1997-349-03/S0002-9947-97-01826-6/S0002-9947-97-01826-6.pdf)) determined the image of
$$T'\_\*:BP^\*(K(\mathbb{Z}\_{(p)},n+2))\to H\mathbb{Z}\_p^\*(K(\mathbb{Z}\_{(p)},n+2))$$
for $n\geq 1$.
My question is, whether the same has been considered, or follows easily from the above, for
$$T\_\*:BP^\*(K(\mathbb{Z}\_{(p)},n+2))\to H\mathbb{Z}\_{(p)}^\*(K(\mathbb{Z}\_{(p)},n+2)).$$
In particular, I would like to know if there is any nontrivial class of dimension $n+2$ in the image.


Some computation seems to suggest that for n=1, we have $p\iota\_{n+2}\in \operatorname{Im}T\_\*$ where $\iota\_{n+2}\in H\mathbb{Z}\_{(p)}^\*(K(\mathbb{Z}\_{(p)},n+2))$ is the fundamental class, but of course I could be missing something.



The Thom map for the Brown-Peterson cohomology
Here is an "answer" which may be or not be good enough for your purpose, but which is easy to prove.


Let's start with [Ravenel-Wilson-Yagita](https://hopf.math.purdue.edu/Ravenel-Wilson-Yagita/rav-wil-yag.pdf) Theorem 1.20. Applied to Eilenberg-Maclane spaces, it implies that their $BP$ cohomology is generated by the elements detected by mod $p$ cohomology, in fact the elements maps to the set of elements described by Tamanoi.


The information above suffices to conclude that as a graded abelian groups, the image of $BP^\*(K(Z\_{(p)},n)\to H^\*(K(Z\_{(p)},n),Z\_{(p)})$ is generated by some lift of Tamanoi's generator. In particular we can already conclude that $p\iota \_{n+2}$ is not in the image if $n>0$.


I am not completely sure but probably these elements are torsion-free, which should give the additive structure of the image. If not, just study the the Bockstein spectral sequence.


If you need to determine concretely which elements of $H^\*(K(Z\_{(p)},n),Z\_{(p)})$
are actually in the image, basically you will need to follow what Tamanoi did (fortunately not all). Step 1) find a description of what he calls $BP$-fundamental class in $H^\*(K(Z\_{(p)},n),Z\_{(p)})$. Step 2) Find out which cohomology operations in $HZ\_{(p)}^\*HZ\_{(p)}$ are covered by Landweber-Novikov operations. Step 3) find what the operations in the step 2 do to the element in the step 1.


