Suppose $X^n$ is an orientable compact orbifold (without boundary) with stabilisers in codimension 2, and $\bar X^n$ is the underlying topological space. We can assume, moreover, that $X^n$ is a quotient of a manifold $X'^n$ by an action of finite group $G$.


Is it true that, for simplicial homologies of $\bar X^n$, we have $H\_{n-k}(\bar X^n, \mathbb R)$ is dual to $H\_k(\bar X^n,\mathbb R)$? 


If not, what is a simplest counterexample, and what is the *correct* statement? If yes, what would be a reference?


**PS.** It seems to me that this should be true in the case when $X^n$ is a global quotient of a manifold by a finite group, because I guess in this case the simplicial homology of $X^n$ should be equal to the invariants of the action of $G$ on $H\_k(X'^n)$. At the same time, actions on $H\_k(X'^n)$ and $H\_{n-k}(X'^n)$ are dual. Is this reasoning correct? 



A simple minded Poincare duality for orbifolds?
Your proof for global quotients is correct, although the homology is more naturally the co-invariants, though these are isomorphic to the invariants by the transfer. The dual of the co-invariants is naturally the invariants of the dual. So to say that the co-invariants of two dual spaces are again dual, you need to adjust the pairing by the the order of the group, though people don't usually worry about such factors.


For the general case, a useful tool is [homology manifolds](http://en.wikipedia.org/wiki/Homology_manifold). This is a local property which the underlying (or "coarse") space of an orbifold has with $\mathbb Q$ or $\mathbb R$ coefficients. The same argument you applied to the homology in the global quotient case applies to the local homology. Homology manifolds have Poincare duality. Unfortunately, the definition I linked in wikipedia is not quite correct. Your user name suggests that you might like sheaf theory. The correct definition is that a certain sheaf is constant; the incorrect definition checks only the stalks. In fact, this sheaf is the dualizing sheaf, though sheaves are overkill to proving PD for homology manifolds.


