Degree theory for orbifolds

In [3] Borzellino and Brunsden started to develop an elementary differential topology theory for orbifolds. In this paper we carry on their project by defining a mapping degree for proper maps between orbifolds, which counts preimages of regular values with appropriate weights. We show that the mapping degree satisfies the expected invariance properties, under the assumption that the domain does not have a codimension one singular stratum. We study properties of the mapping degree and compute the degree in some examples.


Introduction
An orbifold is a space that is locally homeomorphic to R n /G, where a finite group G acts linearly on R n . It is therefore natural to expect that many properties of manifolds are shared by orbifolds, such as the existence of an elementary differential topology. In this paper we contribute to the project of Borzellino and Brunsden [3] in this direction and we define a mapping degree for complete orbifold maps.
More precisely, for a proper, smooth orbifold map f : O → P and a regular value y ∈ P of f, we define the mod-2 degree of f at y, deg 2 (f; y) and, if O and P are oriented, the integer valued degree deg(f; y). These degrees are defined by a weighted count of the preimages of a given regular value: this takes into account the possibly non-trivial isotropy of the points involved.
Our first result in Section 3 is that the mod-2 degree deg 2 (f; y) is independent of the choice of regular value y ∈ P and of the proper homotopy class of f, provided P is connected and O does not contain a codimension-1 singular stratum. We show that the condition on the codimension-1 stratum is necessary by constructing in Section 5 an orbifold map whose degree depends on the regular value chosen. If the orbifolds P and O are oriented, this condition is always satisfied, since orientable orbifolds never have codimension-1 singular points. It follows that in the oriented case, if P is connected, then the integer degree deg(f; y) is well defined and always independent of both the choice of regular value y ∈ P and the proper homotopy class of f.
In Section 3 we also prove some properties of the degree, namely that non-zero degree implies that the underlying map of an orbifold map is surjective. We also show that the degree is multiplicative under compositions of maps.
Orbifolds frequently arise as quotients M//G, where a compact Lie group G acts effectively on a manifold M with finite stabilizers. These quotients are also a natural source of orbifold mappings. We show in Section 4 that an equivariant map between such manifolds with group actions induces a complete orbifold map. We use this to construct orbifold mappings between weighted projective spaces, and we calculate their degree in Section 5.

Orbifolds and orbifold maps
For us an (effective) orbifold O is a topological space O together with an atlas of orbifold charts around each point of O. An orbifold chart around a point x ∈ O consists of an open setŨ x ⊂ R n together with an effective action of the isotropy group, a finite group Γ x , fixing 0 ∈ R n , and a homeomorphism φ x fromŨ x /Γ x onto an open neighborhood of U x of x such that φ x (0) = x. Two chartsŨ x andŨ y with U x ∩ U y = ∅ need to satisfy the following compatibility condition in the atlas: there exist an orbifold chartŨ z , injective group homomorphisms ρ zx : Γ z → Γ x and ρ zy : Γ z → Γ y , and embeddings i zx :Ũ z →Ũ x and i zy :Ũ z →Ũ y , such that i zx (γw) = ρ zx (γ)(i zx (w)) and i zy (γw) = ρ zy (γ)(i zy (w)) for all γ ∈ Γ z andw ∈Ũ z .
Just as in the manifold case, every orbifold atlas lies in a unique maximal atlas and we will always assume our orbifolds to be equipped with a maximal atlas. The Bochner-Cartan linearization theorem shows that we may always choose charts in which the groups Γ x act linearly onŨ x = R n , i.e., the chart is a representation of Γ x .
The singular set Σ of O consists of the points with a non-trivial isotropy group. Let x ∈ Σ andx the lift of x to a chartŨ x centered at x. Then Γ x acts on TxŨ x and fixes a subspace TxŨ Γx x , cf. [6]. The singular dimension of x is defined to be sdim(x) = dim TxŨ Γx x and does not depend on the chosen chart. The singular set is the union Σ = n−1 i=0 Σ i of singular strata Σ i , where Σ i = {x ∈ Σ | sdim(x) = i}. Each stratum Σ i can be further decomposed: the connected components of Σ i all have a well defined isotropy group up to isomorphism. We will not use this decomposition.
It can be shown that each stratum Σ i is a boundary-less manifold of dimension i, cf. [5, Page 74 and onwards], whose tangent spaces are modeled on TxŨ Γx x . We will refer to points in Σ dim O as smooth points, and points in Σ i with i < dim O as non-smooth points.
We will also consider smooth orbifolds with boundary, by allowing the orbifold chartsŨ x in the definition of orbifold to be open subsets of the closed upper half-space [0, ∞) × R n−1 .
An orbifold is locally orientable if the groups Γ x act onŨ x by orientation preserving diffeomorphisms. An orbifold is orientable if in addition the embeddings i xy :Ũ x →Ũ y are orientation preserving. An orientation is then a consistent choice of orientation of the charts U x . We will need the following result on the structure of the singular set.
Proof. Clearly Σ dim O is a manifold and it is open in O. Moreover, since we demand that the isotropy groups Γ x act effectively, it is also dense. Now suppose that Σ dim O−1 = ∅. Let x ∈ Σ dim O−1 . LetŨ x be an orbifold chart centered at x on which the finite group Γ x acts. Without loss of generality, we can assumeŨ x ∼ = R n and that the action is linear. The action then fixes a hyperplane V ⊂ R n . Choose a Γ x invariant inner product. Then Γ x also leaves the line V ⊥ = {v ∈ R n | v, w = 0, for all w ∈ V } invariant and it must act effectively on V ⊥ . But this is only possible if Γ x = Z 2 and the action is by a reflection. The quotient of R n by this action is thus isomorphic to ( A full suborbifold S of the orbifold O consists of: Of course all maps in the above definition are required to be smooth. We refer to [3,4] for a general definition of suborbifold and a discussion of the properties of full suborbifolds. Given two orbifolds O and P, a smooth complete orbifold map between O and P is a triple f = (f, {f x }, {Θ x }) consisting of the following data: • a continuous map f : O → P between the underlying topological spaces; We identify two orbifold maps x and Θ f x = Θ g x . In particular, the maps f and g of the underlying topological spaces coincide for equivalent maps. We will often drop the adjective complete, and speak of a smooth (orbifold) map f.  The homotopy is said to be proper if the underlying map F is proper (this assumption is stronger than the assumption that Fi t is proper for all t, cf. [8]).
The statements below are the orbifold version of the well-known Sard's theorem and regular preimage theorem for manifolds. Given a smooth orbifold map f : O → P, a point x ∈ O is called regular if the differential is a surjective linear map. Herex denotes the lift of x toŨ x . A point y ∈ P is called a regular value if all x ∈ f −1 (y) are regular points. We refer the reader to [3] for proofs of the statements.  It was remarked in [3] that there are representation theoretic obstructions for a point of a smooth orbifold map f : O → P to be regular. We need the following version of this principle. Proposition 2.5. Let f : O → P be a smooth orbifold map. Let y ∈ P be a regular value and a smooth point of P. Then f −1 (y) is a full suborbifold and every Proof. LetŨ x be a chart centered at x andx be the lift of x. Then Γ x acts on TxU x by the differential. To avoid cumbersome notation we denote this action by left multiplication, i.e γ ·ṽ = T γxṽ. Following [3] we define K x = ker Txf x and N x = ker Θ x . Now define the linear Figure 1. In Section 5 we define complete orbifold maps between weighted projective spaces. This picture shows the map f (1,3) : 3) with isotropy Z 3 , depicted at the top on the right hand side. Even though the point is non-smooth, it is a regular value of f. The red curve on the right therefore consists of regular values. The preimage of every point on this curve consists of three points, except for the non-smooth point. Thus the cardinality of the preimage is not locally constant. The weighted cardinality however, is, as we show in Lemma 3.3.
operator A x by Here |N x | denotes the order of N x . The operator has the following properties. Ifṽ ∈ TxŨ x then Txf x (γ ·ṽ) = Txf x (ṽ) for all γ ∈ N x hence Txf x (A xṽ ) = 0, i.e. im A x ⊂ K x . Moreover, the operator A x commutes with the action and A 2 for all δ ∈ N x . So far we have not used the assumption that y is a smooth point. But if that is the case, then N x must be the full isotropy group, i.e.
Corollary 2.6. Let f : O → P be a smooth orbifold map between orbifolds of the same dimension. Let y ∈ P be a regular value and a smooth point of P. Then each x ∈ f −1 (y) is smooth.

Degree of an orbifold map
In the case of smooth maps between manifolds of the same dimension, if y is a regular value of a proper map, then the cardinality #f −1 (y) is a locally constant function. For an orbifold map, this is not going to be the case, unless we assign weights to the points in the preimage, cf. Figure 1.
The next Lemma tells us that the weighted cardinality is, in fact, an integer number. Proof. To avoid possible confusion, in this proof we label the isotropy groups by the orbifold to which they belong. Part of the data defining a smooth orbifold map is a group homomorphism . But S is a zero dimensional orbifold, hence a manifold. This implies that Γ S x is the trivial group and hence that Θ x is injective whenever x is a regular point. Then x | of a subgroup is always an integer, which was what was to be shown. Proof. Choose orbifolds charts U y and U x around around y and x ∈ f −1 (y), respectively. After shrinking U y and U x we may assume the following: U y only contains regular values, f −1 (U y ) ⊂ x∈f −1 (y) U x , and there exists a liftf x :Ũ x →Ũ y . If z is another regular value of f contained in U y , then z has |Γ y |/|Γ z | lifts toŨ y . Given that #f −1 x (y) is a local invariant, by possibly shrinking the orbifold chart around y we can assume that each one of these lifts has precisely one preimage inŨ x . When projecting down fromŨ x to U x , some of these preimages are identified by the action of the isotropy groups Γ x , so the number of points we are left with in U x is where w is a generic preimage of z in U x . By equivariance, all groups Γ w are isomorphic. So if we compute the weighted cardinality of z with the above formula in mind we see that Definition 3.4. Let f : O → P be a proper and smooth map between orbifolds of the same dimension, and let y ∈ P be a regular value of f. We define the mod-2 degree of f at y to be Proposition 3.5. The mod-2 degree is locally constant.
In fact we will show below that deg 2 (f; y) is constant in y if P is connected and O does not have a codimension-1 singular stratum. We start by showing homotopy invariance at regular values.
Lemma 3.6. Let f, g : O → P be smooth and proper orbifold maps between orbifolds of the same dimension. Let F : O × [0, 1] → P be a proper homotopy between f and g. Let y ∈ P be a smooth point. Suppose that y is regular for F, f, and g simultaneously, and that Then deg 2 (f; y) = deg 2 (g; y).
Proof. Since y is a regular point of f, g, and F simultaneously, Theorem 2.4 tells us that F −1 (y) is a full one-dimensional orbifold with boundary, while Proposition 2.5 implies that all points in f −1 (y)∪g −1 (y) are smooth, and thus the degree formulas just count the number of points in the preimage, possibly taking orientation into account.
The number of boundary components of a compact one dimensional manifold with boundary is even, which shows that deg 2 (f; y) = deg 2 (g; y).  In the next step we combine invariance at smooth, regular points with the local invariance of the mod-2 degree, in order to prove invariance at all regular points. Proof. In view of Lemma 3.7, the only thing that remains to prove is that the degree is invariant for regular but not necessarily smooth points. This follows from the local invariance: suppose y, z are regular but not necessarily smooth. Since the smooth stratum Σ P dim P is an open and dense subset of P , and regular values of f are also open and dense, there exist smooth and regular values y and z , sufficiently close to y and z, respectively, such that deg 2 (f; y) = deg 2 (f; y ) and deg 2 (f; z) = deg 2 (f; z ). If we combine this with the conclusion of Lemma 3.7, we see that deg 2 (f; y) = deg 2 (f; y ) = deg 2 (f; z ) = deg(f; z).
As in the manifold case, we can define an integer valued degree if the orbifolds are oriented.
For every x ∈ f −1 (y) choose charts U x and U y centered at x and y, respectively. Letf x be the lift of f in these charts and letx be the lift of x. The sign sgn(Txf x ), which tells us if the map is orientation preserving or reversing atx, does not depend on the choice of liftx. Definition 3.9. Let f : O → P be a proper and smooth map between oriented orbifolds of the same dimension and y ∈ P a regular value of f. Define the (integer valued or oriented) degree of f at y to be the oriented count of points in the preimage of y, namely: The integer valued degree shares the invariance properties of the mod-2 degree.
Theorem 3.10. Let f : O → P be a proper and smooth map between oriented orbifolds of the same dimension. Assume that P is connected and that y ∈ P a regular value of f. Then the degree deg(f; y) does not depend on the the regular value y, nor on the proper homotopy class of f.
Proof. The proof of Lemma 3.3 shows that also deg(f; y) is locally constant, since the sign sgn(Txf x ) is locally constant in each U x . Recall the proof of 3.6 needs assumption Σ O dim(O)−1 = ∅, because this implies that the preimage of a regular and smooth point of the homotopy is a compact, one-dimensional manifold with boundary. But this is automatically satisfied in this case, since O is assumed to be oriented, cf. Proposition 2.1. This implies that we can follow the proof of Lemma 3.6 to show homotopy invariance for the oriented degree. The only thing left to do is to keep track of the orientations. Since O and P are oriented, the compact one-dimensional manifold F −1 (y) is also oriented, and the oriented count of the boundary components is zero. The proofs of Lemma 3.7 and Theorem 3.8 work then ad verbatim for the oriented degree.
Since under the assumptions of Theorems 3.8 and 3.10 the degrees do not depend on the chosen regular value y, in what follows we will write deg 2 (f) = deg 2 (f; y) and deg(f) = deg(f; y), where y is any regular value of f. The following statement is an immediate corollary of the fact that the degree does not depend on the regular value chosen. Another consequence of the fact that the degree of an orbifold map is independent of the choice of regular value is the next proposition, which shows that the degree behaves multiplicatively with respect to composition. where we assume that the orbifolds are oriented for the integer degree to be well-defined.
Proof. We prove this for the integer degree, the case of the mod-2 degree is similar. Let z be a regular value of the composition gf. Then z must be a regular value of g, and all y ∈ g −1 (z) must be regular values for f and we get Notice that we have used the fact that the degree of f is independent of the regular value y.
Theorem 3.13. Let f, g : O → P be two proper orbifold maps whose underlying maps f, g : O → P are the same. Assume that P is connected and that Then and where we assume the orbifolds are oriented for the integer degree to be well-defined.
Proof. Since regular values of both f and g are open and dense in P, there is a common regular value. If we compute the mod-2 degree at this point, we see that the parameters involved are just the number of preimages and the order of the isotropy groups of these preimages, which are all independent of the chosen lifts. In the oriented degree case, the additional sign term appearing in the degree formula is also determined by the underlying map, which can be most easily seen at smooth points.

Quotient orbifolds
Orbifolds arise in a natural way if we take the quotient of a smooth manifold M by the effective and locally free action of a compact Lie group G. Weighted projective spaces fall into this category, as quotients of a circle action on an odd dimensional sphere. For a quotient orbifold O = M//G, the underlying space is just the topological quotient M/G and orbifold charts can be constructed as follows: the Slice Theorem provides for each x ∈ M a submanifold S x ∼ = R n of M containing x and which is invariant under the action of the stabilizer G x = Stab(x) (here n = dim M − dim G). Such a "slice" is the image, under the exponential map associated to an auxiliary Riemannian metric, of a fibre of an -tubular neighborhood of the orbit of x. An orbifold chart around [x] is given by (S x , G x , φ x : We now show that equivariant mappings induce orbifold maps between the quotients.
. Thus for every x ∈ M 1 , the group homomorphism Θ restricts to a group homomorphism Θ x : Sincef is neither immersive nor submersive at x, there is no apriori reason forf to map the slice S x to the slice Sf (x) . We now construct a mapf [x] that does map S x to Sf (x) . Recall that the subgroup Gf (x) ⊂ G 2 is finite. Therefore there exists a neighborhood U ⊂ G 2 of the identity such that U ∩ G f (x) = {e}, the trivial subgroup. By the definition of a slice, the set V = {γz | γ ∈ U z ∈ Sf (x) } is a neighborhood off (x). Sincef is continuous, we can shrink S x so thatf (S x ) ⊂ V . For each y ∈ S x , the orbit throughf (y) intersects the slice Sf (x) in a point and there exists a unique k(y) ∈ U such that k(y)f (y) ∈ Sf (x) (recall that Γ x ∩ U is the identity, cf. Figure 2). Now let γ ∈ G x and y ∈ S x . Then k(y)Θ x (γ) −1f (γy) = k(y)f (γ −1 γy) = k(y)f (y) ∈ Sf (x) .
Define the mapf [x] : S x → Sf (x) viaf [x] (y) = k(y)f (y). We need to check that this map is Θ x -equivariant: let γ ∈ G x and y ∈ S x , then by the transformation rule for k and the equivariance off we havef Hence we have constructed a Θ x -equivariant liftf [x] of f in the chart (S x , G x , φ x ) around each point [x] ∈ M 1 //G 1 , and thus defined a smooth orbifold map f.

Remark 4.2.
With a small additional effort, the result can also be proved if the groups G 1 , G 2 are not necessarily compact. It is crucial to demand that the actions are proper, i.e. the maps Φ i : Figure 2. This figure illustrates the construction in Proposition 4.1. A complete orbifold map f is constructed from an equivariant mapf . The red slice S x through x is not mapped to the blue slice S x . However, for every element y ∈ S x there exists a unique element k(y) ∈ G 2 , close to the identity, such that k(y) ∈ Sf (x) .

Examples
Example 5.1 (An orbifold map whose degree depends on the choice of regular value). The mod-2 degree of a proper map f : O → P is independent of the regular value y ∈ P if P is connected and O does not have a codimension 1 singularity. A hypothesis like this is necessary as the following example illustrates: Let S 1 ⊂ R 2 be the standard circle. Then Z 2 acts on it by reflection in the second coordinate. The underlying space of the orbifold S 1 //Z 2 is an orbifold is a closed interval, whose boundary points have Z 2 isotropy. The mapf : S 1 → S 1 given byf (x, y) = 1 √ x 2 +y 4 (x, y 2 ) satisfiesf (x, y) =f (x, −y), hence is equivariant if the domain is equipped with the action of Z 2 above and the codomain has a trivial group acting. By Proposition 4.1 the mapf descends to a complete orbifold map f : S 1 //Z 2 → S 1 (here Θ is the trivial homomorphism). Both (0, −1) and (0, 1) are regular values for this map. One easily checks that f −1 ((0, 1)) contains one preimage that is a smooth point. Hence deg 2 (f; (0, 1)) = 1. But deg 2 (f; (0, −1)) = 0 as this does not lie in the image of f. This map is depicted in Figure 3.
Example 5.2 (A contractible map with non-zero degree at all regular values). Consider once more the action of Z 2 on S 1 by reflection in the second coordinate, and the following two maps from S 1 to itself: Let Θ f : Z 2 → Z 2 be the trivial homomorphism and Θ g : Z 2 → Z 2 be the identity. Thenf and g are Θ f and Θ g equivariant, respectively, and by Proposition 4.1 they induce orbifold maps f, g : S 1 //Z 2 → S 1 //Z 2 . The underlying maps f and g are equal. All smooth points are regular and deg 2 (f; z) = deg 2 (g; z) for all smooth points. Notice that g is not contractible, while f is. In fact, g it is homotopic to the identity.

5.1.
Degrees of maps between weighted projective spaces. Let q = (q 0 , . . . , q n ) be an (n + 1)-tuple of positive integers. The group C * = C \ {0} acts on C n+1 \ {0} as follows: γ · (z 0 , . . . , z n ) = (γ q 0 z 0 , . . . , γ qn z n ). The weighted projective space CP n (q) is the quotient of Figure 4. This figure depicts the maps constructed in Example 5.2. The mapsf andĝ are Θ f and Θ g equivariant, respectively, and descend to orbifold maps f and g. The underlying maps f and g are equal and deg 2 (f; y) = deg 2 (g; y) for all regular values, that is, all the smooth points of S 1 //Z 2 . The orbifold map f is contractible while g is not.
C n+1 \ {0} by this action. We denote the equivalence class of (z 0 , . . . , z n ) ∈ C n+1 \ {0} by [z 0 : . . . : z n ] q . If q 0 = q 1 = . . . = q n = 1, then we recover the ordinary projective space. The action is proper and we will assume that the weights q i are coprime, so that the action is effective. The weighted projective space CP n (q) has then the structure of an effective orbifold. The mapsf q : We invoke Remark 4.2 to construct these maps, but notice that is also possible to normalise the coordinates and definef q andĝ r as maps from S 2n+1 onto S 2n+1 , with the compact Lie group S 1 acting on S 2n+1 via the same formula, and apply Proposition 4.1 directly. Here lcm(q) denotes the least common multiple of the q i 's, so in particular, when the q i 's are pairwise coprime, lcm(q) = q 0 · . . . · q n . The map f q has degree q 0 · . . . · q n : to see this, consider the orbifold chart (Ũ x , Γ x , φ x ) around x = [1 : 0 : . . . : 0] q , where U x = {[z 0 : · · · : z n ] q ∈ CP n (q) : z 0 = 0} and Γ x = Z q 0 acts onŨ x = C n by ξ · (w 1 , . . . , w n ) = (ξ q 1 w 1 , . . . , ξ qn w n ). The map φ x : we see that f q lifts, in these charts, to the map f q x (w 1 , . . . , w n ) = (w q 1 1 , . . . , w qn n ). Hence The claim about the degree of the second map, g q , can be verified as follows. The composition g q • f q maps [z 0 : . . . : z n ] to [z lcm(q) 0 : . . . : z lcm(q) n ] and it is a degree lcm(q) n self-map of CP n . In view of the multiplicativity of the degree (Proposition 3.12), we can conclude that deg(g q ) = lcm(q) n q 0 · . . . · q n We can also introduce a second weighted projective space CP n (r), with different weights, and consider the composition h rq = f r • g q : CP n (q) → CP n (r), whose underlying map satisfies Again by the multiplicativity of the degree, Proposition 3.12, we see that deg(h rq ) = lcm(q) n q 0 · . . . · q n · r 0 · . . . · r n .
We would like to use this example to underline the fact that regular values of an orbifold map are not necessarily smooth points, and conversely. For instance, let q = (1, . . . , 1, k) and consider the map f q defined above. It lifts to the identity in a neighbourhood of [0 : . . . : 0 : 1] q . The point [0 : · · · : 0 : 1] q ∈ CP n (q) is thus a non-smooth point (it has nontrivial isotropy, namely Z k ), but it is a regular value of the mapping f q . Its preimage consists of only one point, namely [0 : . . . : 0 : 1] ∈ CP n , and by the weighted count in our definition it follows that the degree of f q at this point is k. On the other hand, the point [1 : 0 : . . . : 0] q is a smooth point of CP n (q), but it is not a regular value of f q , since in standard orbifold charts it lifts to (w 1 , . . . , w n ) → (w 1 , . . . , w k n ) in a neighbourhood of the origin of C n .