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Cohomology of toric blowup

Especially, I would like to prove that $H^{2k}(\widetilde{\mathbb{C}^{n}/G},\mathbb{Z})$ is torsion free for $k\leq n-1$. Do someone know an efficient method to deal with this kind of problems? … I managed to prove that $H^2(\widetilde{\mathbb{C}^{n}/G},\mathbb{Z})$ and $H^{2n-2}(\widetilde{\mathbb{C}^{n}/G},\mathbb{Z})$ are torsion free using Danilov combinatorial result. …
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