Let $X$ be a smooth, compact, orbifold of dimension $4$, where the stabilisers are only allowed to be cyclic groups. Let $p \in X$ be an isolated orbifold point (i.e. the orbifold chart about $p$ consists of a finite (cyclic) group acting linearly on $\mathbb{R^{4}}$, and the action is free on $\mathbb{R}^{4} \setminus \{0\}$).


Let $S$ be a compact, orientable surface and $f:S \rightarrow X$ be a continuous map and $[f(S)] \in H\_{2}(X,\mathbb{R})$ the real homology class it represents (in the homology of the underlying topological space of $X$). Suppose that $p$ is in the image of $f$, and this is the only orbifold point in the image of $f$. 


**Question:** Is it possible to find a $2$-cycle in $X$, disjoint from the orbifold locus, which represents the class $[f(S)]$?


If not, I would like to know an understandable counter-example. I am not sure whether the assumption of cyclic stabilisers is important here, but I am mainly interested in this case.



Can we perturb a surface away from an orbifold point?
The answer to the question is no. In general the intersection $[f(S)] \cdot [f(S)]$ can lie in $\mathbb{Q} \setminus \mathbb{Z}$, then of course in this case there is no such two cycle.


