Orbifold

An orbifold is locally modeled on a quotient $U/G$ of an open subset of Euclidean space by a finite group action, with compatible quotient charts. The chart retains the local group action, not merely the underlying quotient topological space. For example, the plane modulo a rotation of order $m$ has a cone point with total angle $2\pi /m$. Its underlying space is homeomorphic to a plane, but its orbifold structure records the order-$m$ stabilizer. If every stabilizer is trivial, the local models reduce to those of a manifold.

A teardrop orbifold is an orbifold whose underlying topological space is a two-sphere and which has exactly one cone point of an orbifold of order $m>1$. A neighborhood of that point is modeled on a disk modulo rotations by multiples of $2\pi /m$; every other point has trivial local group.

Topologically, a disk modulo this finite rotation group is still a disk. The local stabilizer is therefore extra orbifold information invisible in the underlying sphere. The singularity order distinguishes teardrops that have the same underlying topological space.

The teardrop is a standard example of a bad orbifold: it has no manifold orbifold covering. Cutting it into an ordinary disk and a disk with one order-$m$ cone point exposes incompatible nontrivial covering behavior along their common boundary. Thus being locally a finite quotient does not imply being globally covered by a manifold.^[ref]

A cone point of an oriented two-dimensional orbifold is a point with a chart modeled on a disk modulo a nontrivial finite cyclic rotation group. For a group of order $m$, its local quotient metric has total angle $2\pi /m$, and its stabilizer is cyclic of order $m$.

For example, identifying the edges of a planar sector of angle $2\pi /3$ gives the local order-three model. The underlying quotient neighborhood is homeomorphic to a disk, so the cone order is not detectable from topology alone. A point with order one is an ordinary nonsingular point.

A cone point is distinct from a mirror boundary point, whose local group acts by a reflection. It also differs from a general metric cone singularity: an orbifold cone angle must be of the form $2\pi /m$ with positive integer $m$. A teardrop orbifold has exactly one such exceptional point on an underlying sphere.

The orbifold fundamental group is the group of deck transformations of the universal orbifold cover. It retains the local finite group data that the ordinary fundamental group of the underlying topological space forgets.^[ref]

For an oriented closed topological surface of genus $g$ with cone points of orders $m_1,\dots ,m_r$, the group presentation isEach local cone relation says $c_i^{m_i}=1$; it does not guarantee that $c_i$ has order exactly $m_i$ in the resulting global group. For the teardrop orbifold, a two-sphere with one cone point of order $m>1$, the remaining relation is $c_1=1$, so its orbifold fundamental group is trivial despite nontrivial local isotropy.^[ref]

For a complete hyperbolic surface orbifold, the universal cover is the hyperbolic plane. A torsion-free finite-index subgroup then acts freely and gives a smooth surface cover. The hyperbolic covering model is what identifies local cone stabilizers with actual finite subgroups in this case.

Pair the $2s$ edges of an oriented hyperbolic polygon by length-preserving identifications reversing their boundary directions. Label the pairs by generating elements $A_1,\dots ,A_s$ of a finite group $T$. Glue the corresponding edge of a copy labelled $g$ to its mate in the copy labelled $g{A}_k$. Left multiplication of labels defines the $T$ action. A vertex cycle of a paired polygon has angle sum ${\theta }_j$ and accumulated gluing word $C_j\in T$ of order $q_j$; above it lie $|T|/q_j$ vertices, each of cone angle $q_j{\theta }_j$. The action is free away from vertices, and the vertex stabilizers are conjugates of $⟨C_j⟩$. If the base cone has integer order $m_j=2\pi /{\theta }_j$ and $q_j\mid m_j$, this is an orbifold covering, with lifted cone order $m_j/q_j$. The lift is smooth exactly when $q_j=m_j$.

In the paired-polygon construction of finite surface covers, a subgroup $U$ acts freely precisely when the displayed intersection is trivial for every vertex word $C_j$ and every $g\in T$. The quotient map $M\to U\backslash M$ is then a normal covering map with deck group $U$ and $\chi (U\backslash M)=\chi (M)/|U|$. Normality of $U$ in $T$ is instead the criterion for the intermediate orbifold cover $U\backslash M\to T\backslash M$ to be normal. These are different covering maps.

For a paired $2s$-gon with $r$ base vertex cycles of a paired polygon, there are one face, $s$ edges and $r$ vertices on the base, so $\chi (M_0)=1-s+r$. The full finite-group construction has $|T|$ faces, $s|T|$ edges and ${\sum }_j|T|/q_j$ vertices, giving the displayed ordinary Euler characteristic. If all lifted angles are $2\pi$, the orbifold Euler characteristic formula gives equivalently $\chi (M)=|T|{\chi }_{\mathrm{orb}}(M_0)$.

A vertex cycle of a paired polygon is an equivalence class of vertices under the edge identifications. Its angle sum is the total angle at the resulting point of the quotient surface. Recording the edge labels encountered around that point gives a group word, whose order determines the number of turns required to close a lifted cycle. This explains the angle multiplier and lifted vertex count in the paired-polygon construction of finite surface covers.

Over a cone point of order $m$, a cycle of length $\ell$ in the monodromy on sheets has ramification degree $\ell$ and residual cone angle $2\pi \ell /m$. The lifted inherited hyperbolic metric is smooth precisely when every such cycle has length $m$. Equivalently the subgroup defining the cover meets no conjugate of a nonidentity power of a cone generator.

For a compact topological surface with cone-point orders ${\alpha }_1,\dots ,{\alpha }_k$, the orbifold Euler characteristic is ${\chi }_{\mathrm{orb}}=\chi (\text{underlying surface})-{\sum }_i(1-1/{\alpha }_i)$. An orbifold covering multiplies this quantity by its degree.

An $n$-sheeted surface cover of an orbifold of underlying genus $g_0$ with $r$ cone points has the displayed ordinary Euler characteristic, where $c_i$ is the number of cycles of the $i$th cone monodromy permutation. The ramification deficit over that point is $n-c_i$, so the formula follows by counting lifted vertices, edges and faces, or by the Riemann-Hurwitz formula. If every cone cycle has its full prescribed length, this equals $n{\chi }_{\mathrm{orb}}$.