Let $p$ be a prime and $n \in \mathbb N$. Does there exist a closed manifold which is of type $n$ after $p$-localization?

When $n= 0$ the answer is yes. When $p = 2$ and $n = 1$ we can take $\mathbb R \mathbb P ^2$.

Other than that, I'm not sure.

Notes:

• Recall that a finite CW complex $X$ is said to be of type $n$ if $\widetilde{K(n)}_\ast X \neq 0$ but $\widetilde{K(m)}_\ast X = 0$ for $m < n$, where $K(n)$ is the $n$th Morava $K$-theory at the prime $p$.

• Recall the thick subcategory theorem tells us that for every $n$ (and every $p$) there exists a finite CW complex which is of type $n$ after $p$-localization, and conversely that every finite CW complex is of type $n$ for some $n$ after $p$-localization. The question is whether "finite CW complex" can be upgraded to "closed manifold".

• When $n \geq 1$, a closed manifold of type $n$ can't be orientable (since if it's orientable, then by Poincare duality its reduced rational homology is nonvanishing, and $K(0) = H\mathbb Q$).