Journal of Lie Theory Volume 13 (2003) 311–327 \begin{matrix}c \\ ◯ \end{matrix} 2003 Heldermann Verlag

Determination of the Topological Structure of an Orbifold by its Group of Orbifold Diffeomorphisms

Joseph E. Borzellino and Victor Brunsden^{∗}

Communicated by F. Knop

Abstract. We show that the topological structure of a compact, locally smooth orbifold is determined by its orbifold diffeomorphism group. Let Diff_{Orb}^{r}(O) denote the C^{r} orbifold diffeomorphisms of an orbifold O. Suppose that Φ: Diff_{Orb}^{r}(O_{1}) → Diff_{Orb}^{r}(O_{2}) is a group isomorphism between the the orbifold diffeomorphism groups of two orbifolds O_{1} and O_{2} . We show that Φ is induced by a homeomorphism h: X_{O_{1}} → X_{O_{2}} , where X_{O} denotes the underlying topological space of O. That is, Φ(f) = hfh^{−1} for all f ∈ Diff_{Orb}^{r}(O_{1}). Furthermore, if r > 0, then h is a C^{r} manifold diffeomorphism when restricted to the complement of the singular set of each stratum.

1. Introduction

Given a topological space X with some geometric structure (including topological structures, differentiable structures, symplectic structures and contact structures) and the group of transformations that preserve these structures (the group of homeomorphisms, diffeomorphisms, symplectic diffeomorphisms and contact diffeomorphisms), one can ask whether these groups of structure preserving transformations determine the corresponding structures. The topological case has been studied by Gerstenhaber [10],[11], Fine and Schweigert [9], Rubin [15], Wechsler [19], and Whittaker [20]. The differentiable case has been studied by Banyaga [1], [2], Filipkiewicz [8], Rubin [15], Rybicki [16] and Takens [17]. The symplectic and contact cases have been studied by Banyaga [3],[4], [5]. Rubin [15] has also studied many other variants of this question including the PL, Lipschitz and quasiconformal cases. A key ingredient in our proof in the orbifold case will be a theorem of Rubin:

Theorem 1.1. (Rubin [15]) Let X_{i}, (i = 1, 2) be locally compact Hausdorff spaces and G_{i} subgroups of the group of homeomorphisms of X_{i} such that for

ISSN 0949–5932 / $2.50 \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag every open set T ⊂ X_{i} and x ∈ T the set {g(x) | g ∈ G_{i} and g |_{(X_{i}−T)}= Id} is somewhere dense. Then if Φ : G_{1} → G_{2} is a group isomorphism, then there is a homeomorphism h between X_{1} and X_{2} such that for every g ∈ G_{1}, Φ(g) = hgh^{−1}.

∗ We would like to thank A. Banyaga for helpful conversations leading to improvements of the original manuscript. We would also like to thank the referee for their careful reading of the original manuscript and for making many helpful suggestions.

Recall that a subset S of a topological space X is called somewhere dense if the interior of its closure is nonempty. That is, int(cl(S)) ≠ Ø. Our theorem is the following:

Theorem 1.2. Let O_{1} and O_{2} be two compact, locally smooth orbifolds. Fix r ≥ 0. Suppose that Φ : Diff_{Orb}^{r}(O_{1}) → Diff_{Orb}^{r}(O_{2}) is a group isomorphism. Then Φ is induced by a homeomorphism h : X_{O_{1}} → X_{O_{2}} . That is, Φ(f) = hfh^{−1} for all f ∈ Diff_{Orb}^{r}(O_{1}). Furthermore, if r > 0, h is a C^{r} manifold diffeomorphism when restricted to the complement of the singular set of each stratum.

Here, Diff_{Orb}^{r}(O) denotes the C^{r} orbifold diffeomorphism group and X_{O} the underlying topological space of an orbifold O. We review the definitions of these notions in the next few sections. The restriction in Theorem 1.2 to compact orbifolds cannot be removed as the following example shows.

Example 1.3. Let O_{1} = (0, 1) and O_{2} = [0, 1], the open and closed unit intervals. These orbifolds have the same homeomorphism group, but are clearly not homeomorphic spaces.

In general, the homeomorphism h in Theorem 1.2 is not necessarily an orbifold homeomorphism. To see this, consider the following

Example 1.4. Let O_{i}, (i = 1, 2) be two so–called ℤ_{p_{i}} –teardrops (see Example 4.5) with p_{1} ≠ p_{2}. It is clear that the homeomorphism groups of O_{i} are each isomorphic to the subgroup of the homeomorphism group of the 2–sphere S^{2} which fix the north pole. To see this, just observe that any homeomorphism of S^{2} that fixes the north pole can be locally lifted to a p_{i}–fold covering of a neighborhood of the north pole. Note, however that the orbifolds themselves are not orbifold homeomorphic, even though their underlying spaces X_{O_{i}} = S^{2}, are topologically homeomorphic.

In light of Theorem 1.2 and Example 1.4, it is natural to ask what happens if we fix an orbifold O and consider a group automorphism Φ : Diff_{Orb}^{r}(O) → Diff_{Orb}^{r}(O). Theorem 1.2 guarantees that there exists a topological homeomorphism h : X_{O} → X_{O} such that Φ(f) = hfh^{−1} for all f ∈ Diff_{Orb}^{r}(O). The following theorem shows that, in general, h ∉ Diff_{Orb}^{r}(O), and thus it is possible that some automorphisms of Diff_{Orb}^{r}(O) may not be inner automorphisms.

Proposition 1.5. For each n > 1 there exists a compact connected orbifold O of dimension n, such that the group of automorphisms Aut(Diff_{Orb}^{r}(O)) ≠ Inn(Diff_{Orb}^{r}(O)), the group of inner automorphisms.

Proof. Parametrize S^{2} with spherical coordinates (θ, ϕ), 0 ≤ θ < 2π, −π/2 ≤ ϕ ≤ π/2. Let A = (θ, −π/2) be the north pole and B = (θ, π/2) be the south pole. Give S^{2} the structure of a (p, q)–football orbifold F (see Example 4.5) with p ≠ q so the singular set = {A} ∪ {B}. It is not hard to see that Diff_{Orb}^{r}(F) is isomorphic to the group of C^{r} diffeomorphisms of S^{2} which fix A and B pointwise. Consider the group automorphism Φ : Diff_{Orb}^{r}(F) → Diff_{Orb}^{r}(F) defined by (Φ(f)) = g ∘ f ∘ g^{−1} where g(θ, ϕ) = (θ, −ϕ). Φ ∉ Inn(Diff_{Orb}^{r}(F)). To see this, suppose f is an orbifold diffeomorphism with support in a neighborhood U of A and B ∈ int(O − U) and Ψ is an inner automorphism. Ψ(f) has support in a neighborhood of A and B ∈ int(O − supp(Ψ(f)). However Φ(f) has support in a neighborhood of B, hence Φ cannot be an inner automorphism. Higher dimensional examples can be constructed by considering products with spheres F × S^{n}.

Remark 1.6. The work of [5], [8], [15], [16] collectively show that such examples do not exist in the topological (with or without boundary), differentiable, PL, Lipschitz, symplectic and contact categories. In addition, Theorem 1.2 implies that one–dimensional orbifold examples do not exist. This follows since the only non-trivial orbifolds are closed rays and closed intervals, and a topological homeomorphism of such a 1–orbifold is also an orbifold homeomorphism.

If one considers two non–homeomorphic Riemannian manifolds with trivial isometry group one is easily convinced that if the automorphism group of a particular structure is not rich enough then the underlying topological structure cannot be determined at all. In fact, in our case, it is a very interesting question to determine the conditions necessary to guarantee that h is an orbifold homeomorphism. It turns out that, in order to guarantee that h is an orbifold homeomorphism, one must introduce a more general notion of orbifold diffeomorphism group, the unreduced orbifold diffeomorphism group Diff_{unred}^{r} O. There is a natural projection Diff_{unred}^{r} O → Diff_{Orb}^{r}(O). The details can be found in [6]. Before proceeding with the proof of our theorem we need to review some definitions involving the orbifold category.

2. Orbifold Preliminaries

Orbifolds. Our definition is modeled on the definition in Thurston [18]. A (topological) orbifold O, consists of a paracompact, Hausdorff topological space X_{O} called the underlying space, with the following local structure. For each x ∈ X_{O} and neighborhood U of x, there is a neighborhood U_{x} ⊂ U, an open set Ũ_{x} ≅ ℝ^{n}, a finite group Γ_{x} acting continuously and effectively on Ũ_{x} which fixes 0 ∈ Ũ_{x}, and a homeomorphism ϕ_{x} : Ũ_{x}/Γ_{x} → U_{x} with ϕ_{x}(0) = x. These actions are subject to the condition that for a neighborhood U_{z} ⊂ U_{x} with corresponding Ũ_{z} ≅ ℝ^{n}, group Γ_{z} and homeomorphism ϕ_{z} : Ũ_{z}/Γ_{z} → U_{z}, there is an embedding ψ̃ : Ũ_{z} → Ũ_{x} and an injective homomorphism f : Γ_{z} → Γ_{x} so that ψ̃ is equivariant with respect to the f (that is, for γ ∈ Γ_{z}, ψ̃(γy) = f(γ)ψ̃(y) for all y ∈ Ũ_{z} ), such that the following diagram commutes:

\xymatrix{Ũ_{z} \ar[r]^{ψ̃} \ar[d] & Ũ_{x} \ar[d] \\ Ũ_{z}/Γ_{z} \ar[r]^{ψ=ψ̃/Γ_{z}} \ar[dd]^{ϕ_{z}} & U_{x}/f(Γ_{z}) \ar[d] \\  & Ũ_{x}/Γ_{x} \ar[d]^{ϕ_{x}} \\ U_{z} \ar[r] & U_{z}}

The covering {U_{x}} of X_{O} is not an intrinsic part of the orbifold structure. We regard two coverings to give the same orbifold structure if they can be combined to give a larger covering still satisfying the definitions.

Let 0 ≤ r ≤ ∞. An orbifold O is a C^{r} orbifold if each Γ_{x} acts C^{r} – smoothly and the embedding ψ̃ is C^{r}. Locally Smooth Orbifolds. We say that an orbifold O is locally smooth if the action of Γ_{x} on Ũ_{x} ≅ ℝ^{n} is an orthogonal action for all x ∈ O. That is, for each x ∈ O, there exists a representation L : Γ_{x} → 𝕆(n) such that if γ · y denotes the Γ_{x} action on Ũ_{x}, then we have γ · y = L(γ)y for all y ∈ Ũ_{x}. Orbifold Strata and Isotropy Groups. Let O be a connected n-dimensional locally smooth orbifold. Given a point x ∈ O, there is a neighborhood U_{x} of x which is homeomorphic to a quotient Ũ_{x}/Γ_{x} where Ũ_{x} is homeomorphic to ℝ^{n} and Γ_{x} is a finite group acting orthogonally on ℝ^{n}. The definition of orbifold implies that the germ of this action in a neighborhood of the origin of ℝ^{n} is unique. We define the isotropy group of x to be the group Γ_{x}. The singular set of O is the set of points x ∈ O with Γ_{x} ≠ {1}. Denote the singular set of O by Σ_{1}. Then Σ_{1} is also a (possibly disjoint) union ⋃_{l_{1}} Σ^{(l_{1})} of connected locally smooth orbifolds of strictly lower dimension (though different components may have different dimensions). See the section of examples. Each of the orbifolds Σ_{1}^{(l_{1})} has a singular set ⋃_{l_{2}} Σ_{1}^{(l_{1})(l_{2})}. Define the singular set of Σ_{1} to be Σ_{2} = ⋃_{(l_{1})(l_{2})} Σ_{1}^{(l_{1})(l_{2})}. Proceeding inductively, we get a stratification of O:

O = Σ_{0} ⊃ Σ_{1} ⊃ Σ_{2} ⊃ ⋯ Σ_{k−1} ⊃ Σ_{k} = Ø for some k ≤ n + 1

By a result of M.H.A Newman [7], the singular set of a topological orbifold is a closed nowhere dense set. In the locally smooth case, the proof is much easier. See Proposition 3.1 and the remark that follows.

3. Elementary Properties of Locally Smooth Orbifolds

The two results that we shall need involving locally smooth orbifolds are the following:

Proposition 3.1. If O is locally smooth then in each local orbifold chart Ũ_{x} the fixed point set S_{x} = {y ∈ Ũ_{x} | Γ_{x} · y = y} is a topological submanifold of Ũ_{x}.

Proof. Let Ũ be an orbifold chart for x and Γ_{x} its isotropy group. Since O is locally smooth there is a Γ_{x}-equivariant embedding F : Ũ → ℝ_{L}^{n}, where ℝ_{L}^{n} denotes ℝ^{n} with the orthogonal Γ_{x}-action L. Since we can regard L as a representation L : Γ_{x} → 𝕆(n), we will use the notation L(γ) = L_{γ}. Thus, we have F (γ · x) = L_{γ}(F (x)). If y ∈ S_{x}, and z = F (y) then we have that z = L_{γ}(z), hence F (S_{x}) ⊂ ⋂_{γ∈Γ_{x}} ker(L_{γ} − I). Let W = ⋂_{γ∈Γ_{x}} ker(L_{γ} − I) and let w ∈ W, with F (v) = w for some v ∈ Ũ. Then

v = F^{−1}(w) = F^{−1}L_{γ}(w) = F^{−1}L_{γ}F (v) = F^{−1}F (γ · v) = γ · v

for all γ ∈ Γ_{x}, hence v ∈ S_{x}. Thus we have shown F (S_{x}) = W. Since W is a subspace, we have that S_{x} = F^{−1}(W) is a submanifold of Ũ. This completes the proof.

Remark 3.2. It follows easily from this result, that the singular set of a locally smooth orbifold is closed and nowhere dense. This is because the intersection of the singular set with a fundamental chart is closed and nowhere dense by Proposition 3.1, and O is a Baire space (O is locally compact Hausdorff).

Proposition 3.3. If O is a smooth C^{r} orbifold with r > 0, then it is locally smooth.

Proof. Let Γ_{x} be the isotropy group of x and B a neighborhood of x with a neighborhood B̃ of 0 in ℝ^{n} together with a homeomorphism ϕ_{x} : B̃/Γ_{x} → B where Γ_{x} acts C^{r} -smoothly on B̃. We denote the action of Γ_{x} by (γ, y) → γ ·y for all γ ∈ Γ_{x} and y ∈ B̃. Without loss of generality we may assume that ϕ_{x}(0) = x and thus Γ_{x} · 0 = 0. Let L_{γ} : T_{0}B̃ → T_{0}B̃ be the linearization at 0 of y → γ · y. Note that L_{γ}, being the linearization at 0, is a fixed linear map, and is therefore C^{∞}. Define F : B̃ → ℝ^{n} by

F (y) = \frac{1}{|Γ_{x}|} ∑_{η∈Γ_{x}} L_{η}(η^{−1} · y)

Then F is C^{r} since L_{η} is C^{∞} and the action of Γ_{x} is C^{r}. Also, dF (0) = Id and F (γ · y) = L_{γ}(F (y)). To see the last statement, note that

F (γ · y) = \frac{1}{|Γ_{x}|} ∑_{η∈Γ_{x}} L_{η}(η^{−1}γ · y) = \frac{1}{|Γ_{x}|} ∑_{η∈Γ_{x}} L_{η}((γ^{−1}η)^{−1} · y)

= \frac{1}{|Γ_{x}|} ∑_{μ∈Γ_{x}} L_{γμ}(μ^{−1} · y) where μ = γ^{−1}η

= \frac{1}{|Γ_{x}|} ∑_{μ∈Γ_{x}} L_{γ}(L_{μ}(μ^{−1} · y)) = L_{γ}(\frac{1}{|Γ_{x}|} ∑_{μ∈Γ_{x}} L_{μ}(μ^{−1} · y)) = L_{γ}(F (y))

So by the inverse function theorem, there is a neighborhood C̃ of 0 in B̃ on which F is an equivariant C^{r} diffeomorphism. F conjugates the action of Γ_{x} to the linear action L_{γ}. Since the linear action L_{γ} is linearly conjugate to an orthogonal action, the proof is complete.

4. Examples

Example 4.1. (Hemisphere) Let O = (S^{n}, can)/G, n > 1, be the n-dimensio-nal hemisphere of constant curvature 1 (topologically O is just the closed n–disk D^{n}). G = ℤ_{2} ⊂ 𝕆(n+1) is the group generated by reflection through an equatorial (n − 1)–sphere. In this case Σ_{1} is the equatorial (n − 1)–sphere. More generally, one might consider the quotient of S^{n} by the group generated by reflections in all coordinate n-planes of ℝ^{n+1}, getting an orbifold whose topological structure is that of a so-called manifold with corners.

Example 4.2. (Football) Let O be a ℤ_{p}–football. O = (S^{2}, can)/G, where G ⊂ 𝕆(3) is rotation around the z–axis in ℝ^{3}, through an angle of 2π/p. Here Σ_{1} = {north pole} ∪ {south pole}.

Example 4.3. (ℤ_{p}–hemisphere) Let O be a ℤ_{p}–football/G, where G is reflection in the equator of the football that does not contain the singular points. Topologically, O is D^{2}. Note that the singular set Σ_{1} = {equator} ∪ {point}, thus it is possible for different components of the singular set to have different dimensions.

Example 4.4. (Pillow) Let O = ℝ^{2}/G, where G is the crystallographic group generated by reflecting an equilateral triangle or square in each of its sides to produce a tiling of ℝ^{2}. Then O is just the closed triangle or square, with singular set the boundary of the tiling region. The stratification of O is as follows:

O = Σ_{0} ⊃ Σ_{1} = {the boundary of the triangle or square} ⊃

Σ_{2} = {the vertices} ⊃ Σ_{3} = Ø

Here, Σ_{1} is union of the closed line segments making up the boundary of the triangle or square and each of these line segments is a 1–dimensional orbifold with 2 singular points. One should observe that Σ_{1} is not a 1–dimensional orbifold but a union of 1–dimensional orbifolds. The lowest dimensional stratum has dimension 0. Note that the manifold Σ_{1} − Σ_{2} is a union of open line segments. If one only quotients out by the index 2 subgroup G_{0} of orientation preserving elements of G then O becomes topologically a 2–sphere. The complement of the singular set is topologically ℝ^{2} − {2 points or 3 points}.

Example 4.5. (Teardrop and (p, q)-footballs) Let O be a ℤ_{p}–teardrop. The underlying space of this orbifold is S^{2} with a single conical singularity of order p at the north pole. One may also construct a (p, q)-football whose underlying space is also S^{2} with two conical singularities, one of order p at the north pole and the other of order q ≠ p at the south pole.

Example 4.6. Consider the group G = ℤ_{2} × ℤ_{2} generated by rotations of π about the three coordinate axes of ℝ^{3}. If we consider the quotient of the 2–sphere S^{2}/G, we get a 2–dimensional orbifold O whose underlying space is topologically the 2–sphere with 3 singular points. The sin–suspension Σ_{sin}O = S^{3}/ΣG is an orientable 3–dimensional orbifold. ΣG denotes the suspension of the action on S^{2} to S^{3} . In this case, Σ_{1} is the union of the 3 line segments joining the suspension points and passing through one of the singular points of O. Σ_{2} is just the two suspension points.

Example 4.7. Let L_{p} = S^{3}/G be a 3–dimensional lens space. Suspend the action of G to an action ΣG on the 4–sphere S^{4} . Let O = S^{4}/ΣG. Then the underlying space of O is not a manifold (or manifold with boundary).

5. Maps Between Orbifolds

Orbifold Maps. A map f : O → O′ of C^{r} orbifolds is a C^{r} orbifold map if for each x ∈ X_{O} there are open neighborhoods U_{x} and V_{y} of x and y = f(x) in O and O′ respectively, open sets Ũ_{x} and Ṽ_{y} in ℝ^{n} with finite groups Γ_{x} and Γ_{y} acting C^{r} on Ũ_{x} and Ṽ_{y} respectively, a homomorphism Θ : Γ_{x} → Γ_{y} and a C^{r} map f̃: Ũ_{x} → Ṽ_{y} equivariant with respect to Θ, (that is, the following diagram commutes:

Ũ_{x} ⟶^{f̃}  Ṽ_{y}
↓  ↓
Ũ_{x}/Γ_{x} ⟶ Ṽ_{y}/Θ(Γ_{x})
|  ↓
|  Ṽ_{y}/Γ_{y}
↓  ↓
U_{x} ⟶^{f}  V_{y}

We will write C_{Orb}^{r}(O, O′) for the set of C^{r} orbifold maps.

It is immediate from the definition that composition of C^{r} orbifold maps gives another C^{r} orbifold map. We may therefore define the category C^{r} Orb with objects the C^{r} orbifolds and morphisms the C^{r} maps between them, although we will not use this terminology. We now define the objects in the automorphism group of the orbifold structure. Orbifold Homeomorphisms and Diffeomorphisms. For any topological space X, let H(X) denote its group of homeomorphisms. For a topological orbifold O, the group of orbifold homeomorphisms, H_{Orb}(O) will be the subgroup of H(X_{O}) so that f, f^{−1} ∈ C_{Orb}^{0}(X_{O}, X_{O}). If O is a C^{r} orbifold, Diff_{Orb}^{r}(O) is the subgroup of H_{Orb}(O) with f, f^{−1} ∈ C_{Orb}^{r}(O). We will also use Diff_{Orb}^{0}(O) for H_{Orb}(O). Let

Diff_{Orb}^{r}(O, Σ_{m}) = {f ∈ Diff_{Orb}^{r}(O) | f(x) = x for all x ∈ Σ_{m}}

be the subgroup of Diff_{Orb}^{r}(O) fixing the entire stratum Σ_{m} pointwise.

Lemma 5.1. Any element of Diff_{Orb}^{r}(O) leaves Σ_{i} invariant (as a set), where Σ_{i} is any substratum of O.

Proof. We show first that Σ_{1} is invariant. Since Σ_{1} = {x ∈ O | Γ_{x} ≠ {1}}, just note that if f is an orbifold diffeomorphism then by definition there is an isomorphism of the isotropy subgroups Γ_{x} and Γ_{f(x)}. Hence singular points get sent to singular points. To see the general case, it suffices to show the invariance of Σ_{2}. Let y ∈ Σ_{2} with isotropy subgroup Γ_{y}. Then there is an orbifold chart U about y which contains x ∈ Σ_{1} − Σ_{2} with Γ_{x} ⊊ Γ_{y}. Denote the action of Γ_{y} on Ũ by α, and the action of Θ(Γ_{y}) on f̃(Ũ) by α^{Θ}. Note that the equivariance of f̃ on Ũ implies that f̃∘ α_{γ} ∘ f̃^{−1} = α_{Θ(γ)}^{Θ}. Thus, the action of Θ(Γ_{x}) on f̃(Ũ) is the restriction of α^{Θ} to Θ(Γ_{x}). Since f preserves Σ_{1} and Θ(Γ_{x}) is a proper subgroup of Θ(Γ_{y}), we see that the singular set Σ_{2} of Σ_{1} is preserved. This completes the proof.

Using this lemma it is easily verified that

Diff_{Orb}^{r}(O) ▷ Diff_{Orb}^{r}(O, Σ_{k−1}) ▷ ⋯ ▷ Diff_{Orb}^{r}(O, Σ_{2}) ▷ Diff_{Orb}^{r}(O, Σ_{1})

where G ▷ H means that H is normal subgroup of G.

6. Extending Orbifold Diffeomorphisms

For any subgroup G of the homeomorphism group H(X) of a topological space X, let G_{c} ⊂ G denote those elements of G with compact support in X. Let G_{0} be the subgroup of G_{c} whose elements are isotopic to the identity through elements of G with compactly supported isotopy. For any self-map f : X → X of a topological space X, let the support supp(f) = cl{x ∈ X | f(x) ≠ x} where cl(S) denotes the closure of the set S. By compactly supported isotopy we mean an isotopy f : [0, 1] × X → [0, 1] × X, such that supp(f) ⊂ [0, 1] × X is compact.

Proposition 6.1. (Extension of Orbifold Diffeomorphisms) The group Diff^{r}(O − Σ)_{c} is a subgroup of Diff_{Orb}^{r}(O) for any C^{r} orbifold O, 0 ≤ r ≤ ∞. Moreover, if 1 ≤ r ≤ ∞, then for each component A = Σ_{m}^{(l_{1})(l_{2})···(l_{m})} of Σ_{m}, each f ∈ Diff^{r}(A−Σ_{A})_{0}, and open neighborhood of supp(f) in O, there is an extension g ∈ Diff_{Orb}^{r}(O)_{0} such that supp(g) ⊂ U. Here, Σ_{A} denotes the singular set of A

Proof. Note that Diff^{r}(O−Σ)_{c} is the group of compactly supported diffeomorphisms of the manifold X_{O} − Σ. If f ∈ Diff^{r}(O − Σ)_{c}, then supp(f) is a compact subset of X_{O} − Σ disjoint from Σ and so in any sufficiently small neighborhood U of Σ, f(x) = x for all x ∈ U − Σ. Therefore we can define an extension \overline{f} of f to X_{O} by

\overline{f}(x) = \begin{cases}f(x) & if x ∈ X_{O} − Σ \\ x & if x ∈ Σ \end{cases}

It is clear that \overline{f} ∈ Diff_{Orb}^{r}(O).

Since the rest of the argument is fairly technical, we will first give an outline of the proof that follows. Many of the ideas that we use here are the same ones used to show that for a smooth manifold M, the group, Diff(M)_{0}, of diffeomorphisms isotopic to the identity through compactly supported isotopies, has the so-called fragmentation property [5]. Let f ∈ Diff^{r}(A − Σ_{A})_{0}. In the final part of the proof we use the isotopy of f to the identity to show that it is sufficient to find an extension g for f close to the identity. To obtain g, we first construct a manifold M̃, and a map from M̃ to an appropriate neighborhood of A. By construction, the manifold M̃ contains a copy the stratum. Using the fact that a local chart at the identity in the group of compactly supported diffeomorphisms is given by the vector fields with compact support, we extend the vector field on A that corresponds to f to a compactly supported equivariant vector field on M̃. This enables us to extend f to an equivariant diffeomorphism on M̃, which projects to an extension of f to O. The details now follow.

Let N(Σ_{A}) be a (small) closed neighborhood of Σ_{A} in O. Let {V_{i}}_{i∈I} be a covering of A − N(Σ_{A}) by orbifold charts such that V_{i} ∩ Σ_{A} = Ø which are themselves covered by {Ṽ_{i}}_{i∈I} equipped with group actions Γ_{i} and projections π_{i} : Ṽ_{i} → V_{i}. For each i ∈ I choose a Dirichlet fundamental domain D̃_{i} and let ι_{i} : V_{i} → D̃_{i} be the function assigning the unique x̃ ∈ D̃_{i} with π_{i}(x̃) = x for each x ∈ V_{i}. Since Γ_{x} = Γ_{y} for all x, y in A − Σ_{A}, we let Γ = Γ_{x}. Define a C^{r} manifold M̃ with a C^{r} Γ action via

M̃ = ⋃_{i∈I} Ṽ_{i}/ ∼

where ∼ is the equivalence relation we now define. For x̃_{i} ∈ Ṽ_{i}, x̃_{j} ∈ Ṽ_{j},

1. If i = j, x̃_{i} ∼ x̃_{i}

2. If i ≠ j, x̃_{i} ∼ x̃_{j} if and only if π_{i}(x̃_{i}) = π_{j}(x̃_{j}), Ψ_{i}(γ) · x̃_{i} ∈ D̃_{i} and Ψ_{j}(γ) · x̃_{j} ∈ D̃_{j} for some fixed γ ∈ Γ, where Ψ_{i} : Γ → Γ_{i}, Ψ_{j} : Γ → Γ_{j} are the natural identification homomorphisms.

The action of Γ on M̃ is given by the action of Ψ_{i}(Γ) on Ṽ_{i}. Let M = ⋃_{i∈I} V_{i}. The functions ι_{i}, by definition, glue together to give a function

ι : M → M̃

which restricts to a C^{r} embedding ι of A − N(Σ_{A}) into M̃. To see this, just note that by construction, x̃ ∈ ι(A − N(Σ_{A})) ⇔ Γx̃ = x̃. Without loss of generality, we can assume that M̃ is an equivariant tubular neighborhood of ι(A − N(Σ_{A})) with projection ρ : M̃ → ι(A − N(Σ_{A})). Moreover, M̃/Γ = M is an orbifold that is homeomorphic to a neighborhood of A − N(Σ_{A}) in O. Identify M with this neighborhood and let π : M̃ → M be the quotient map.

Fix 1 ≤ r ≤ ∞. By results in [12], we may assume that M̃ carries a C^{∞} structure M̃^{∞} which is C^{r} diffeomorphic to M̃. That is, there exists a C^{r} diffeomorphism Δ : M̃ → M̃^{∞}. In addition, the paper [14] allows us to assume that Γ is equivalent to a C^{∞} Γ action on M̃^{∞}. Endow M̃^{∞} with a C^{∞} Riemannian metric which is equivariant with respect to the induced orthogonal Γ action on T M̃^{∞}. This makes ρ : M̃^{∞} → ι(A − N(Σ_{A})) a Riemannian submersion and ι(A − N(Σ_{A})) a totally geodesic submanifold of M̃^{∞} (being the fixed point set of Γ acting by isometries). In what follows we will identify M̃ with M̃^{∞} via Δ and A − N(Σ_{A}) with its image ι(A − N(Σ_{A})) in M̃.

It is well known, see for example [5], that for a manifold N the group Diff^{r}(N)_{0} carries the structure of an infinite dimensional manifold whose local model T_{f} Diff^{r}(N)_{0} is the vector space χ^{r}(N) of C^{r} vector fields on N with compact support in the uniform C^{r} topology. In addition, for each f ∈ Diff^{r}(N)_{0} there is a neighborhood U_{f} of 0 ∈ T_{f} Diff^{r}(N)_{0} ≅ χ^{r}(N) and a smooth open embedding

\overline{exp}_{f} : U_{f} ⊂ T_{f} Diff^{r}(N)_{0} → Diff^{r}(N)_{0}

defined by (\overline{exp}_{f}(ν))(x) = exp_{f(x)}(ν(x)) for ν ∈ U_{f}, where exp is the Riemannian exponential map.

Let f ∈ Diff^{r}(A − Σ_{A})_{0} and f_{t}(x) = f(t, x), t ∈ [0, 1] a C^{r} compactly supported isotopy with f_{0}(x) = Id and f_{1}(x) = f(x). Choose N(Σ_{A}) so that supp(f_{t}) ⊂ [0, 1]×(A−N(Σ_{A})). We can regard f_{t} as a C^{r} path in Diff^{r}(A−Σ_{A})_{0} joining Id to f. Let U_{0} be a neighborhood of 0 in χ^{r}(A−N(Σ_{A})) small enough so that \overline{exp}_{Id_{A−N(Σ_{A})}} |_{U_{0}} is an embedding. Denote by χ_{A−N(Σ_{A})}^{r}(M̃) the C^{r} sections of T_{A−N(Σ_{A})}(M̃) with compact support equipped with the uniform C^{r} topology on A − N(Σ_{A}), and let V_{0} be a neighborhood of 0 in χ_{A−N(Σ_{A})}^{r}(M̃) so that V_{0} ∩ χ^{r}(A − N(Σ_{A})) = U_{0}. Let Ω̃_{0} ⊂ Diff^{r}(M̃)_{0} be a neighborhood of IdM̃ so that

(\overline{exp}_{IdM̃} )^{−1}(Ω̃_{0})|_{A−N(Σ_{A})} = V_{0}

Let

Ω_{t} = \overline{exp}_{Id_{A−N(Σ_{A})}}(U_{0}) ∘ f_{t}

The collection Ω_{t}, t ∈ [0, 1] forms an open cover of the curve f_{[0,1]} ⊂ Diff^{r}(A − N(Σ_{A}))_{0}. Since f_{[0,1]} is compact and the isotopy f_{t} is continuous in t, there exists a partition 0 = t_{0} < t_{1} < ⋯ < t_{n} = 1 of [0, 1] so that ⋃_{i} Ω_{t_{i}} is a cover of f_{[0,1]} with f_{t_{i+1}} ∘ f_{t_{i}}^{−1} ∈ Ω_{0} for i = 0, …, n − 1. Thus,

(\overline{exp}_{Id_{A−N(Σ_{A})}})^{−1}(f_{t_{i+1}} ∘ f_{t_{i}}^{−1}) ∈ U_{0}, i = 0, …, n − 1

Let g_{i} = f_{t_{i+1}} ∘ f_{t_{i}}^{−1}, then f_{1} = g_{n−1} ∘ g_{n−2} ∘ ⋯ ∘ g_{0} and let

ν_{i} = (\overline{exp}_{Id_{A−N(Σ_{A})}})^{−1}(g_{i})

Define \overline{ν}_{i}(y) to be the unique horizontal lift of ν_{i}(ρ(y)) to T_{y}M̃ for y ∈ M̃ where the lift is taken with respect to the Γ equivariant Riemannian submersion ρ : M̃ → A − N(Σ_{A}). By construction, \overline{ν}_{i}(y) ∈ χ^{r}(M̃) is Γ equivariant and ρ_{∗}\overline{ν}_{i} = ν_{i}.

Now, let η_{i} : M̃ → [0, 1] be C^{r}, Γ equivariant functions with η_{i}|_{A−N(Σ_{A})} = 1 and η_{i} = 0 outside some compact neighborhood of supp(g_{i}) in M̃. By decreasing the mesh of the partition if necessary, we may ensure that ν̃_{i} = η_{i}\overline{ν}_{i} ∈ U_{0} for i = 0, …, n − 1. Then g̃_{i} = \overline{exp}_{IdM̃} (ν̃_{i}) is a Γ equivariant extension of g_{i} to a relatively compact neighborhood of supp(g_{i}) ⊂ A − N(Σ_{A}) in M̃ and f̃_{1} = g̃_{n−1} ∘ g̃_{n−2} ∘ ⋯ ∘ g̃_{0}, defines the required extension of f_{1}. This completes the proof.

7. Local Contractions

The proof of the main theorem will require that there are enough local orbifold diffeomorphisms whose behavior under the group isomorphism can be controlled. For a locally compact Hausdorff space X, a subgroup G ⊂ H(X) and x ∈ X, we say that g_{x} ∈ G is a local contraction about x if:

1. x ∈ supp(g_{x}) and supp(g_{x}) is compact 2. for all open neighborhoods V and W of x in supp(g_{x}) with \overline{W} ⊂ V ⊂ \overline{V} ⊂

int(supp(g_{x})) there is an N ∈ ℕ so that g_{x}^{n}(\overline{V}) ⊂ W for all n > N.

3. g_{x}(x) = x

While it is not hard to show that (3) is a consequence of (1) and (2), we wanted to make this explicit.

Proposition 7.1. If O is locally smooth, then for each x ∈ O and neighborhood U of x there is a local contraction about x with support in U.

Proof. The result is clearly local so there is no loss in generality if U is assumed to be in an orbifold chart. As O is locally smooth, there is an orthogonal action L of Γ_{x} on ℝ^{n}, and an orbifold homeomorphism f : ℝ^{n}/Γ_{x} → U sending 0 to x. Let χ : [0, ∞) → [0, 1] be a smooth, decreasing function with χ|_{[0,r/2)} = 1 and χ|_{[r,∞)} = 0 where r > 0 is such that B_{r}(0) ⊂ (f ∘ π)^{−1}(U), π : ℝ_{L}^{n} → ℝ_{L}^{n}/Γ_{x} is the projection and B_{r}(0) is the ball of radius r about 0 with respect to the Euclidean metric. Define the vector field

ν(x) = −χ(|x|)x

Since Γ_{x} is an orthogonal action, ν is a Γ_{x} invariant vector field on ℝ^{n} and so the flow g_{t} generated by ν will also be Γ_{x} invariant. g_{1} is clearly a local contraction about 0 supported in (f ∘ π)^{−1}(U) and may be extended outside of this set to all of ℝ^{n} by the identity. Since g_{1} is Γ_{x} equivariant, π ∘ g_{1} ∘ π^{−1} is well defined. Thus g_{x} = f ∘ π ∘ g_{1} ∘ (f ∘ π)^{−1} is an orbifold homeomorphism of U. Extending g_{x} outside of U by the identity gives the required local contraction. This completes the proof.

8. Proof Of Theorem 1.2

Before the main part of the proof, we need to record two observations that will be used throughout.

Remark 8.1. 1. For any open subset U of an orbifold O, x ∈ U if and only if there is a neighborhood V of x so that V − Σ ⊂ U − Σ.

2. For an open subset U as above, x ∈ cl(U) if and only if (V − Σ) ∩ (U − Σ)

≠ Ø for all neighborhoods V of x. These follow trivially from the fact that the singular set Σ of an orbifold O is nowhere dense.

Proof. Let O_{1} and O_{2} be two compact, locally smooth orbifolds and let Φ : Diff_{Orb}^{r}(O_{1}) → Diff_{Orb}^{r}(O_{2}) be a group isomorphism. Lemma 5.1 implies that Diff_{Orb}^{r}(O_{i})|_{O_{i}−Σ_{i}} is a subgroup of Diff^{r}(O_{i} − Σ_{i}). Let T be an open subset of O_{i} − Σ_{i} and assume x ∈ T. Since O is locally compact, we may choose a compact neighborhood U ⊂ T of x. Proposition 6.1 implies that Diff^{r}(O_{i}−Σ_{i})_{c} ⊂ Diff_{Orb}^{r}(O_{i}). Thus, given y ∈ U with y ≠ x, there is g ∈ Diff^{r}(O_{i} − Σ_{i})_{c} with support in U so that g(x) = y. This follows from the local homogeneity of the manifold O_{i} − Σ_{i}. Thus, we conclude that the groups Diff_{Orb}^{r}(O_{i}) satisfy the hypotheses of Rubin’s theorem and so we have a homeomorphism h : O_{1} − Σ_{1} → O_{2}−Σ_{2} such that for every f ∈ Diff_{Orb}^{r}(O_{1}) we have Φ(f) = hfh^{−1}. Note that this implies that the singular sets of O_{i} are either both empty or are both non-empty. To see this, suppose Σ_{1} ≠ Ø and that Σ_{2} = Ø. Then O_{2} = O_{2} − Σ_{2} is a closed manifold. O_{1} − Σ_{1}, however, is a non–compact manifold, and this contradicts the existence of a homeomorphism h : O_{1} − Σ_{1} → O_{2} − Σ_{2} guaranteed by Rubin’s theorem. Since Rubin’s theorem implies Theorem 1.2 when Σ_{1} = Σ_{2} = Ø (the manifold case), we need only concern ourselves with case when Σ_{1} and Σ_{2} are non–empty.

Since we are about to embark on a long technical argument, we first give an outline of the rest of the proof. We use the existence of local contractions to extend the homeomorphism h : O_{1} − Σ_{1} → O_{2} − Σ_{2} to a bijection \overline{h} : O_{1} → O_{2} which induces the group isomorphism Φ : Diff_{Orb}^{r}(O_{1}) → Diff_{Orb}^{r}(O_{2}). This is actually a delicate argument. If we knew a priori that a local contraction g_{x} at x was sent to a local contraction under Φ, then we could easily define our required extension by sending x ∈ Σ_{1} to the unique fixed point of Φ(g_{x}). The problem with doing this is that the behavior of Φ(g_{x}) on the singular set is not well determined by knowledge of h ∘ g_{x} ∘ h^{−1}, which is only defined on the complement of the singular set. This means that until we establish the existence of an appropriate extension of the homeomorphism h, we do not know that local contractions are sent to local contractions by Φ. To define the extension of the homomorphism h, we show that Φ(g_{x}) possesses a unique fixed point y ∈ int(supp(Φ(g_{x}))), and then define an extension \overline{h} via \overline{h}(x) = y. We then show that this extension is independent of the choice of local contraction g_{x}. Next, we verify that our extension is continuous and has continuous inverse, and thus is a homeomorphism which induces the group isomorphism Φ. Lastly, we show \overline{h} and (\overline{h})^{−1} are smooth on the non-singular part of each stratum.

To extend the homeomorphism h : O_{1} − Σ_{1} → O_{2} − Σ_{2} to a bijection \overline{h} : O_{1} → O_{2} which induces the group isomorphism Φ : Diff_{Orb}^{r}(O_{1}) → Diff_{Orb}^{r}(O_{2}), let x ∈ Σ_{1}, and let U_{x} be a relatively compact open neighborhood of x in O_{1}. By Proposition 7.1, there exists a g_{x} ∈ Diff_{Orb}^{r}(O) which is a local contraction about x with support in U_{x}. Let ĝ_{x} = Φ(g_{x}), and Û_{x} = int ( cl(h(U_{x} − Σ_{1}))). Note that by Rubin’s theorem, we have that ĝ_{x} = hg_{x}h^{−1} on O_{2} − Σ_{2}. It follows that supp(ĝ_{x}) ⊂ cl(Û_{x}). Because h is not defined on all of O_{2}, this statement needs justification. Consider the set S = {z ∈ O_{2} − Σ_{2} | hg_{x}h^{−1}z ≠ z}. It’s not hard to see that S ⊂ h( supp(g_{x}) − Σ_{1}). Thus,

cl(S) ⊂ cl (h(supp(g_{x}) − Σ_{1})) ⊂ cl (h(U_{x} − Σ_{1})) = cl(Û_{x})

By definition, we have that

supp(ĝ_{x}) − Σ_{2} = supp(hg_{x}h^{−1}) − Σ_{2} ⊂ cl(S)

Since Σ_{2} is nowhere dense we have that supp(ĝ_{x}) = cl(supp(ĝ_{x})−Σ_{2}), from which it follows that supp(ĝ_{x}) ⊂ cl(Û_{x}).

We now extend h by showing that ĝ_{x} possesses a unique fixed point y ∈ int(supp(ĝ_{x})) and then define an extension \overline{h} via \overline{h}(x) = y.

Since O_{2} is locally compact, we may choose a relatively compact open subset Ŵ ⊂ Û_{x} of O_{2} with

x ∈ int ( cl(h^{−1}(Ŵ − Σ_{2})))

For any neighborhood V of x with cl(V) − Σ_{1} ⊂ h^{−1}(Ŵ − Σ_{2}) there is an m > 0 so that

g_{x}^{m}(h^{−1}(Ŵ − Σ_{2})) ⊂ V ⊂ int ( cl(h^{−1}(Ŵ − Σ_{2})))

since g_{x} is a local contraction about x. Thus, for any N > 0,

x ∈ ⋂_{n<N} g_{x}^{mn}( cl (h^{−1}(Ŵ − Σ_{2}))) ≠ Ø

which implies

⋂_{n<N} cl (g_{x}^{mn}(h^{−1}(Ŵ − Σ_{2}))) ≠ Ø

and so by definition of ĝ_{x} and h,

⋂_{n<N} cl (h^{−1}(ĝ_{x}^{mn}(Ŵ − Σ_{2}))) ≠ Ø

which in turn implies,

⋂_{n<N} h^{−1}(ĝ_{x}^{mn}(Ŵ) − Σ_{2}) ≠ Ø

It now follows that

Ø ≠ ⋂_{n<N} h ∘ h^{−1}(ĝ_{x}^{mn}(Ŵ) − Σ_{2}) ⊂ ⋂_{n<N} ĝ_{x}^{mn}(Ŵ)

and so

⋂_{n<N} ĝ_{x}^{mn}(cl(Ŵ)) ≠ Ø

Then the collection of closed sets {ĝ_{x}^{mn}(cl(Ŵ))} has the finite intersection property, and so by compactness of O_{2} we have

Y_{x} = ⋂_{n>0} ĝ_{x}^{mn}(cl(Ŵ)) ≠ Ø.

By construction, Y_{x} = ⋂_{m>0} ĝ_{x}^{mn}(cl(Ŵ)) is a compact, ĝ_{x} invariant set. We claim that Y_{x} is independent of g_{x} and the subset Ŵ. To see this, suppose that g′_{x} is another local contraction with fixed point x, and Ŵ′ ⊂ O_{2} is a compact subset of int ( supp(Φ(g′_{x}))) satisfying the same requirement of Ŵ as above. As both g_{x} and g′_{x} are local contractions, for any n > 0 there is an m > 0 so that:

g_{x}^{m}( int(cl(h^{−1}(Ŵ − Σ_{2})))) ⊂ g_{x}^{′n}( int(cl(h^{−1}(Ŵ′ − Σ_{2}))))

and for any m > 0 there is an n > 0 so that:

g_{x}^{′n}( int(cl(h^{−1}(Ŵ′ − Σ_{2})))) ⊂ g_{x}^{m}( int(cl(h^{−1}(Ŵ − Σ_{2}))))

Therefore ⋂_{n>0} ĝ_{x}^{n}(Ŵ) ⊂ ⋂_{m>0} ĝ_{x}^{′m}(Ŵ′) ⊂ ⋂_{n>0} ĝ_{x}^{n}(Ŵ) which shows the independence of Y_{x} on the local contraction.

The next step in the proof is to show that if x ≠ x′ then Y_{x} ∩ Y_{x′} = Ø. Let g_{x} and g′_{x′} be local contractions about x and x′ respectively with disjoint supports such that supp(g_{x}) ⊂ U and supp(g′_{x′}) ⊂ U′ where U and U′ are open sets with U ∩ U′ = Ø, U = int(cl(U)) and U′ = int(cl(U′)). Therefore h(U − Σ_{1}) ∩ h(U′ − Σ_{1}) = Ø and by the remark above, if z ∈ int ( cl(h(U − Σ_{1}))), then there is a neighborhood V of z so that V − Σ_{2} ⊂ int ( cl(h(U − Σ_{1}))) − Σ_{2} = h(U − Σ_{1}). Therefore z ∉ int ( cl(h(U′ − Σ_{1}))). Thus,

int ( cl(h(U − Σ_{1}))) ⋂ int ( cl(h(U′ − Σ_{1}))) = Ø

Since Y_{x} ⊂ int ( cl(h(U − Σ_{1}))) and Y_{x′} ⊂ int ( cl(h(U′ − Σ_{1}))), Y_{x} ∩ Y_{x′} = Ø. Therefore for any two such subsets Y_{x} and Y_{x′} of O_{2}, if Y_{x} ∩ Y_{x′} ≠ Ø then Y_{x} = Y_{x′} and x = x′.

Given a k ∈ Diff_{Orb}^{r}(O_{1}), x ∈ Σ_{1} and a local contraction g_{x} about x, the orbifold diffeomorphism k ∘ g_{x} ∘ k^{−1} is a local contraction about k(x). Hence Φ(k ∘ g_{x} ∘ k^{−1}) will have invariant set Y_{k(x)}. Since Φ is a group isomorphism between Diff_{Orb}^{r}(O_{1}) and Diff_{Orb}^{r}(O_{2}), the invariant set of Φ(k ∘ g_{x} ∘ k^{−1}) will be Φ(k)(Y_{x}). Therefore Φ(k)(Y_{x}) = Y_{k(x)} for all x ∈ Σ_{1}. We will use this below to prove that the sets Y_{x} consist of a single point.

To show this last assertion, let y ∈ Y_{x}, and ĝ_{y} ∈ Diff_{Orb}^{r}(O_{2}) be a local contraction about y. Let g_{y} = Φ^{−1}(ĝ_{y}) and then by definition y ∈ ĝ_{y}^{n}(Y_{x}) = Y_{g_{y}^{n}(x)} for all n ≥ 0. Hence Y_{x} ∩ ĝ_{y}^{n}(Y_{x}) ≠ Ø for all n ≥ 0 and so Y_{x} = ĝ_{y}^{n}(Y_{x}) for all n ≥ 0. If z ∈ Y_{x} ∩ supp(ĝ_{y}) then for any neighborhood V of y in O_{2}, there is an n > 0 so that ĝ_{y}^{n}(z) ∈ V which implies that Y_{x} ∩ int(supp(ĝ_{y})) = {y}. Since ĝ_{y} was essentially arbitrary, this implies that Y_{x} = {y}, that is, the invariant set Y_{x} of g_{x} consists of a single point.

We now define the extension \overline{h} of h to all of O_{1} by the following:

\overline{h}(x) = \begin{cases}h(x), & if x ∈ O_{1} − Σ_{1} \\ Y_{x}, & if x ∈ Σ_{1} \end{cases}

By construction, \overline{h} is an bijection inducing the group isomorphism. Similarly we can construct an bijection \overline{h^{−1}}. Continuity of \overline{h} follows from the following. Given x ∈ O_{1} and a neighborhood U_{x} of x, then there is a local contraction g_{x} about x with support in U_{x} (by Proposition 7.1). By construction, x ∈ int(supp(g_{x})) and so the collection

B = ⋃_{x∈O_{1}} ⋃_{U_{x}∋x} {int(supp(g_{x})) | int(supp(g_{x})) ⊂ U_{x}}

forms a base for the topology of O_{1}. Let Fix(f) = {x ∈ O | f(x) = x}. Note that:

Fix(Φ(f)) = \overline{h}(Fix(f))

and that for any local contraction g_{x}, Fix(g_{x}) = (O_{1} − int(supp(g_{x}))) ∪ {x}. Thus

\overline{h}((O_{1} − int(supp(g_{x}))) ∪ \{x\}) = \overline{h}(Fix(g_{x})) = Fix(Φ(g_{x}))

= (O_{2} − int(supp(Φ(g_{x})))) ∪ \{\overline{h}(x)\}

so

\overline{h}(O_{1} − int(supp(g_{x}))) = O_{2} − int ( supp(Φ(g_{x})))

and therefore

\overline{h}( int(supp(g_{x}))) = int ( supp(Φ(g_{x})))

and so \overline{h} maps basic open sets to basic open sets and so \overline{h} is continuous. Similarly, \overline{h^{−1}} is continuous. Note that by construction

\overline{h} ∘ \overline{h^{−1}} = Id on O_{2} − Σ_{2}

and

\overline{h^{−1}} ∘ \overline{h} = Id on O_{1} − Σ_{1}.

Since O_{2}−Σ_{2} is dense in O_{2} and O_{1}−Σ_{1} is dense in O_{1}, we have that \overline{h}∘\overline{h^{−1}} = Id on O_{2} and \overline{h^{−1}} ∘ \overline{h} = Id on O_{1}. Hence \overline{h^{−1}} = (\overline{h})^{−1} and so \overline{h} is a homeomorphism that induces the group isomorphism Φ.

We note that it is only at this stage of the proof that we know that Φ(g_{x}) is a local contraction if g_{x} is a local contraction. We will now proceed to the smoothness assertions of Theorem 1.2.

To prove that \overline{h} and (\overline{h})^{−1} are smooth on the non-singular part of each stratum, we follow closely the argument given in [1]. Note that it is enough to show for any C^{r} function f on O_{2}, and connected component A = Σ_{k}^{(l_{1})···(l_{k})} of some stratum of O_{1} that

f ∘ \overline{h}|_{A−Σ_{A}} ∈ C^{r}(A − Σ_{A}).

We will use the notation as in the proof of Proposition 6.1. Note that A − Σ_{A} is a priori a connected manifold. Let ζ be any C^{r} vector field on A − Σ_{A} with compact support contained in A − N(Σ_{A}). Let \overline{ζ} be the Γ equivariant horizontal lift of i_{∗}ζ to M̃. By choosing an appropriate equivariant cutoff function whose support is contained in a relatively compact neighborhood of A − N(Σ_{A}) in M̃, we obtain a Γ equivariant vector field ζ̃ on M̃. Let z̃_{t} be the 1–parameter group of diffeomorphisms generated by ζ̃, and let z_{t} = π ∘ z̃_{t}. For each t, Φ(z_{t}) = \overline{h} ∘ z_{t} ∘ (\overline{h})^{−1} ∈ Diff_{Orb}^{r}(O_{2}) and the map:

(t, x) ↦ \overline{h} ∘ z_{t} ∘ (\overline{h})^{−1}

is continuous. Moreover, when restricted to \overline{h}(A − Σ_{A}), for fixed t, Φ(z_{t}) ∈ Diff^{r}(\overline{h}(A − Σ_{A}))_{0}. Hence, we have a continuous action of ℝ on \overline{h}(A − Σ_{A}) by C^{r} diffeomorphisms. By Montgomery-Zippin [13, p. 208 - 214], it follows that \overline{h} ∘ z_{t} ∘ (\overline{h})^{−1} is C^{r} in both t and x. Therefore the restriction of \overline{h} ∘ z_{t} ∘ (\overline{h})^{−1} to \overline{h}(A − Σ_{A}) is a 1-parameter group of diffeomorphisms, which has an infinitesimal generator ξ\overline{h} defined by:

\frac{d}{dt}(\overline{h} ∘ z_{t} ∘ (\overline{h})^{−1})|_{\overline{h}(A−Σ_{A})} = ξ\overline{h}(\overline{h} ∘ z_{t} ∘ (\overline{h})^{−1}|_{\overline{h}(A−Σ_{A})})

By construction of the vector field ξ\overline{h} it is easily seen that

ξ(f ∘ \overline{h})|_{A−Σ_{A}} = \frac{d}{dt}|_{t=0}(f ∘ \overline{h}) ∘ z_{t}|_{A−Σ_{A}}

= lim_{t→0} \frac{f ∘ \overline{h} ∘ z_{t} ∘ (\overline{h})^{−1} ∘ \overline{h} − f ∘ \overline{h}}{t}|_{A−Σ_{A}}

= lim_{t→0} \frac{f ∘ Φ(z_{t}) ∘ \overline{h} − f ∘ \overline{h}}{t}|_{A−Σ_{A}}

= (lim_{t→0} \frac{f ∘ Φ(z_{t}) − f}{t} ) ∘ \overline{h}|_{A−Σ_{A}}

= (\frac{d}{dt}|_{t=0}(f ∘ Φ(z_{t}))) ∘ \overline{h}|_{A−Σ_{A}}

= ξ\overline{h}(f) ∘ \overline{h}|_{A−Σ_{A}}

To compute higher derivatives, we can iterate this formula:

ξ_{2}(ξ_{1}(f ∘ \overline{h}))|_{A−Σ_{A}} = ξ_{2}(ξ_{1,\overline{h}}(f) ∘ \overline{h})|_{A−Σ_{A}}
= (ξ_{2,\overline{h}}(ξ_{1,\overline{h}}(f)) ∘ \overline{h}|_{A−Σ_{A}}

Let x ∈ Σ_{1} and Ũ_{x} → U_{x} an orbifold chart around x. Then there is a connected component A = Σ_{k}^{(l_{1})···(l_{k})} for which x ∈ A − Σ_{A}. Since A − Σ_{A} is a manifold, we can choose ξ_{i} to be vector fields which agree with the coordinate vector fields in the neighborhood of x (for local coordinates in A − Σ_{A} around x). Thus, we can obtain continuous partial derivatives up to order r of f ∘ \overline{h}. Therefore, \overline{h} is C^{r} when restricted to the component A − Σ_{A} of the singular set. This completes the proof of the main theorem.

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Joseph E. Borzellino Department of Mathematics California Polytechnic State University San Luis Obispo, CA 93407 jborzell@calpoly.edu Victor Brunsden Department of Mathematics and Statistics Penn State Altoona 3000 Ivyside Park Altoona, PA 16601 vwb2@psu.edu

Received August 3, 2001 and in final form October 10, 2002