TITLE: Chromatic orientability of manifolds
QUESTION [13 upvotes]: If a compact manifold $M$ with empty boundary is oriented with respect to all the connective Morava $K$-theories $k(n)_*$, localized at a prime $p$, can one conclude that $M$ is orientable with respect to $p$-local Brown-Peterson theory $BP_*$?
[As an example, Chris Lloyd and I know that the hypothesis holds for the real Grassmanians $Gr_2(\mathbb R^m)$ with $m$ even (and $p=2$).]
Added later: Chris and I have been having fun studying the Morava $K$-theory of $Gr_d(\mathbb R^m)$ (at $p=2$), and among other things, it seems that when $m$ is even, all of the spaces $Gr_d(\mathbb R^m)$ are $k(n)$-orientable for all $n$ and $d$.   So I was just idly pondering what this means.

REPLY [7 votes]: If $p$ is odd then this is easy but dull. Using the truncation $k(n) \to HF_p$, a $k(n)$-orientable vector bundle is orientable in the usual sense. On the other hand, $p$-locally the Thom spectrum $MSO$ has a cell structure with only even cells, so is orientable with respect to any even ring spectrum.