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Suppose that $a,b,c$ are positive integers such that $gcd(a,b,c) = 1$. Let $X$ be the complex weighted projective space $\mathbb{C} \mathbb{P}(1,a,b,c)$. How to compute the group $T(H^{3}(X,\mathbb{Z}))$?

Notation: If $A$ is an abelian group then $T(A)$ denotes the torsion subgroup.

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Additively, the integral cohomology is the same as for the ordinary projective space (multiplication in cohomology is different though), so there is no torsion. This is Theorem 1 in the paper of Kawasaki Cohomology of twisted projective spaces and lens complexes.

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