# A simple minded Poincare duality for orbifolds?

Suppose $X^n$ is an orientable compact orbifold (without boundary) with stabilisers in codimesnion 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 contretemps, 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?

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For the general case, a useful tool is homology manifolds. 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.
@Geordie an example of what can go wrong is the mapping torus of a degree 2 map of a sphere. That is, glue the ends of $S\times I$ by a degree 2 map. The local cohomology is at every point the cohomology of a sphere. But the sheaf of local cohomology is not the constant sheaf $\mathbb Z$. The right thing to do is to check whether the restriction from a small open set to a point is an isomorphism, but it's actually multiplication by 2. Maybe these groups associated with points aren't actually stalks of everything. I think the relevant sheaf is loc sys $\mathbb Z[1/2]$ with monodromy 2. –  Ben Wieland Nov 18 '13 at 20:26