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.



Cohomology of weighted projective spaces
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](https://link.springer.com/article/10.1007/BF01429212).


