Question: Let $p$ be an odd prime. Does there exist a closed manifold $M$ with $\widetilde H^\ast(M; \mathbb Q) = 0$ but $\widetilde H^\ast(M; \mathbb F_p) \neq 0$?
When $p = 2$, an example is given by $\mathbb R \mathbb P^2$.
After discussion, this question turns out to be equivalent to this other question. That is, over at that question Saal Hardali explained that if $M$ is a closed manifold, then the chromatic type of $M$ at a prime $p$ is either 0 or 1. Both possibilities are realized at $p=2$; the question is whether chromatic type 1 is realized at odd primes. Chromatic type 0 just means having nonvanishing rational (co)homology. So the question is whether there exists a closed manifold $M$ which is rationally contractible but whose $p$-localization is nontrivial for an odd $p$. This turned out to be mistaken, thanks to Ben Wieland for pointing this out at the other question.
Side Question: When $p=2$, what are some other examples of $M$ with $\widetilde H^\ast(M;\mathbb Q) = 0$ but $\widetilde H^\ast(M;\mathbb F_2) \neq 0$ besides $M = \mathbb R \mathbb P^{2n}$ and products thereof?
Best Answer
As mme noted in the comments, such examples cannot exist in odd dimensions, for Euler characteristic reasons. They can't exist in dimension 2 either, by classification. I claim that in all other dimensions $2n > 2$ we have (plenty of) examples.
Let $N$ be a rational homology $2n$-sphere, that is a $2n$-manifold with $H_*(N; \mathbb{Q}) = H_*(S^{2n}; \mathbb{Q})$. For every prime $p$ there exists a rational homology $2n$-sphere with $\dim_{\mathbb{F}_p} H_*(N;\mathbb{F}_p) > 2$. For instance, you can take a spun lens space (any spun rational homology $2n-1$-sphere would do).
Now, the integral homology of $M = \mathbb{RP}^{2n} \# N$ splits as a direct sum of that of the two summands in all dimensions strictly between 0 and 2n, and it vanishes in dimension 2n (because $M$ is non-orientable) and it is $\mathbb{Z}$ in dimension 0 (because $M$ is connected). (This is Exercise 6 in Section 3.3 of Hatcher's Algebraic topology.) That is, $M$ is a rational homology ball, and its homology has as much $p$-torsion as that of $N$.
For the side question, if we choose $N$ to have no 2-torsion in its homology (e.g. spinning an odd lens space should do the trick), this gives plenty of examples.