Orbifold

This is a variation of topological manifold

Definition

An orbifold chart on a manifold $M$ is an open set $U\subseteq M$ and an identification of $U$ with a quotient space $V/\Gamma$ where $V$ is open in $\mathbb {R} ^{n}$ and $\Gamma$ acts on $V$ by linear transformations.

An orbifold is the following data:

• A topological space $M$ that is Hausdoff and second-countable
• An orbifold atlas of $M$ viz an open cover of $M$ by orbifold charts

satisfying the following compatibility condition. If $x\in V_{1}$ and $y\in V_{2}$ for two orbifold charts $V_{1},V_{2}$ such that they correspond to the same point in $M$, then there is a homeomorphism between a neighbourhood of $x$ and a neighbourhood of $y$ that descends to the identity map in the corresponding open set fo $M$.