The Thom map for the Brown-Peterson cohomology

Question

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) 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.

Answer 1

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 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.