Cycles in oribifolds Suppose we have a compact, orientable, $n$-dimensional orbifold $X$, where $n \geq 3$. Suppose that there is a single isolated orbifold point $p_{0} \in X$, with a neighbourhood homeomorphic to $\mathbb{R}^{n} /\{\pm 1\}$, i.e. the cone over $\mathbb{RP}^{n-1}$. In particular $X$ has a single orbifold point with order $2$.
In what follows when I refer to homology, I mean just the singular homology of the underlying topological space of $X$. I think that the following statement should be true and I would like to ask for a proof or a reference for it.
Question: Suppose we have a integral cycle $C \in C_{d}(X,\mathbb{Z})$ in $X$ and $0 < d < \dim(X)$. Then $[2C]$ is represented by a cycle which is (set-theoretically) disjoint from $p_{0} \in X$.
 A: Here is the details of Moishe Kohan's nice solution. We will need the the following fact:
Fact: Let $m>0$ be an integer, for $0<i<m$ for any $\alpha \in H_{i}(\mathbb{RP}^{m},\mathbb{Z})$ satisfies $2 \alpha =0$.
Let $U$ be a small neighbourhood of the orbifold point, homeomorphic the cone on $\mathbb{RP}^{n-1}$, in particular $U$ is contractible. Let $V$ be a small neighbourhood of $X \setminus U$, so that $U \cap V$ is homeomorphic to $(0,1) \times \mathbb{RP}^{n-1}$, in particular it is homotopy equivelant to $\mathbb{RP}^{n-1}$. Let fix $0<d<\dim(X)$ and consider the following part of the MV sequence ascosiated to $(X,U,V)$. All homology group are taken with integral coeeffients, we use reduced homology to ensure the case $d=1$ works out.
$$\ldots \rightarrow H_{d}(U) \oplus H_{d}(V) \rightarrow H_{d}(X) \rightarrow H_{d-1}(U \cap V) \rightarrow \ldots $$ 
Consider $\beta \in H_{d}(X)$, then by the Fact, the image of $2\beta$ in $H_{d-1}(U \cap V)$ is zero, hence $2\beta$ is in the image of $H_{d}(U) \oplus H_{d}(V)$. The inclusion of any cycle in $V$ is clearly disjoint from the orbifold point. Any cycle contained in $U$ can also be made disjoint because $U$ is contractible.  Hence, the class $2\beta$ is represented by a cycle disjoint from the orbifold point.
