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Let $X$ be a compact smooth manifold. Given a $\mathbb{Q}$-basis$e_1,\dots,e_n$ of the vector space $H^k(X,\mathbb{Q})$. Is it true that there exist rational numbers $a_1,\dots,a_n$ such that $$ a_1e_1,\dots, a_ne_n $$ are $\mathbb{Z}$-basis of $H^k(X,\mathbb{Z})/torsion$? I think this is true but it seems not obvious to me either.

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Sorry, this is obviously false. Consider $\mathbb{Q}^2=\mathbb{Z}^2\otimes \mathbb{Q}$, then $e=(1,1),f=(-1,1)$ form a basis of $\mathbb{Q}^2$ but do not descend to a basis of $\mathbb{Z}^2$. –  M. K. Oct 5 '12 at 23:22
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