(This is a re-post of [1](https://math.stackexchange.com/questions/1446061/non-realizability-of-mathbbq-as-a-cohomology-group))


In the paper "On the realizability of singular cohomology groups" by Kan and Whitehead, it is shown that there is no space $X$ and integer $n\geq 1$ such that $H^{n−1}(X)=0$ and $H^n(X)=\mathbb{Q}$ (cohomology with integral coefficients).


At the very end of the article there is a remark where it is stated that, at the time of writing (around 1960, I suppose), it was not known whether $\mathbb{Q}$ could be a (integral) singular cohomology group at all.


My question is: is this still not known?



Non-realizability of $\mathbb{Q}$ as a cohomology group
This may depend on your axioms, see


S. Shelah "The consistency of Ext(G,Z)=Q", Israel J. Math. 39 (1981), no. 1-2, 74–82. 


There it is shown that it is consistent with the generalised continuum hypothesis that there exists a group $G$ having $Ext(G, \mathbb{Z})=\mathbb{Q}$. Then a Moore space $M(G,n-1)$ has the required property.


