Orbifold finiteness under geometric and spectral constraints
The class of Riemannian orbifolds of dimension defined by a lower bound on the sectional curvature and the volume and an upper bound on the diameter has only finitely many members up to orbifold homeomorphism. Furthermore, any class of isospectral Riemannian orbifolds with a lower bound on the sectional curvature is finite up to orbifold homeomorphism.
2010 Mathematics Subject Classification:Primary: 53C23; Secondary: 53C20,57R18, 58J53
The question of how the geometry of a Riemannian manifold controls its topology is of long-standing interest. One particular problem is finding geometric constraints to define a class of manifolds which is finite up to homotopy, homeomorphism or diffeomorphism.
A convenient notation for these classes is to write for the class of all Riemannian manifolds with , and . Where a value is replaced with “” the condition is understood to be deleted.
The first such result is that of Weinstein, who showed that, for , , the class of unformly pinched positively curved manifolds of even dimension, has only finitely many members up to homotopy . Shortly after this, Cheeger showed that has finitely many members up to diffeomorphism .
Grove and Petersen removed the upper bound on sectional curvature, and obtained finiteness of up to homotopy . Shortly afterwards, in collaboration with Wu, this result was improved to show finiteness up to homeomorphism . As long as the dimension is not four, the work of Kirby and Siebenmann  implies finiteness up to diffeomorphism.
The homeomorphism finiteness result was generalized to Alexandrov geometry by Perelman with his Stability Theorem , which showed that , the corresponding class of Alexandrov spaces, is finite up to homeomorphism.
The present work generalizes the homeomorphism finiteness result of Grove, Petersen and Wu to the area of Riemannian orbifolds. An orbifold is a mild generalization of a manifold, and, to give just a few examples, the concept has found applications in Thurston’s work on the Geometrization Conjecture , the construction of a new positively curved manifold by Dearricott  and Grove–Verdiani–Ziller , and string theory, such as Dixon, Harvey, Vafa and Witten’s conformal field theory built on a quotient of a torus . The same convenient notation can be used for orbifolds, here replacing with .
The first finiteness result for orbifolds is that of Fukaya , who generalized the result of Cheeger, showing that a subclass of is finite up to orbifold diffeomorphism. Fukaya used a much more restrictive definition of orbifold, considering only the orbit spaces of global actions by finite groups on Riemannian manifolds. This corresponds to what Thurston called a “good” orbifold .
Working in dimension two, Proctor and Stanhope showed that is finite up to orbifold diffeomorphism , providing a first generalization of the result of Grove, Petersen and Wu. The homeomorphism finiteness result was then shown in all dimensions by Proctor, provided the orbifold has only isolated singularities . Here the assumption that the only singularities are isolated is removed.
For any , the class has only finitely many members up to orbifold homeomorphism.
This result completes the generalization of Grove–Petersen–Wu’s homeomorphism finiteness.
By Weyl’s asymptotic formula, which Farsi has shown is valid for orbifolds , a Laplace isospectral class of orbifolds has fixed volume and dimension. Stanhope has shown that, in the presence of a lower bound on Ricci curvature, such a class has a uniform upper bound on its diameter , and so, just as in , the following corollary is clear.
Any class of Laplace isospectral orbifolds with a uniform lower bound on its sectional curvature has only finitely many members up to orbifold homeomorphism.
This generalizes the similar result of Brooks, Perry and Petersen for Laplace isospectral manifolds . While one cannot hear the shape of an orbifold, one can, at least in the presence of a lower sectional curvature bound, know that there are only finitely many possibilities.
2. Gromov–Hausdorff topologies
A general approach for proving finiteness results such as the Main Theorem [13, 19, 21] is to proceed via a compactness or precompactness result for the class. A particularly useful topology (in fact, a metric) on the set of isometry classes compact metric spaces was proposed by Gromov . Gromov’s metric generalizes the Hausdorff metric on the closed subsets of a compact metric space.
Let and be metric spaces. A function (not necessarily continuous) is called an Gromov–Hausdorff –approximation if, for all , and an –neighborhood of the image of covers all of .
The Gromov–Hausdorff distance between two compact metric spaces and is the infimum of the set of all such that there are Gromov–Hausdorff –approximations and .
The equivariant Gromov–Hausdorff topology was first defined by Fukaya , and achieved its final form some years later in his work with Yamaguchi . Consider the set of ordered pairs where is a compact metric space and is a closed group of isometries of . Say that two pairs are equivalent if they are equivariantly isometric up to an automorphism of the group. Let be the set of equivalence classes of such pairs.
Let . An equivariant Gromov–Hausdorff –approximation is a triple of functions , and such that
is an Gromov–Hausdorff –approximation;
if , then ; and
if , then .
Note that these functions need not be morphisms from the relevant category. The equivariant Gromov–Hausdorff distance is defined from these approximations just as with the standard Gromov–Hausdorff distance.
Convergence of non-compact spaces can also be defined by adding a basepoint. Such sequences are said to converge if the closed metric balls around the basepoint converge. Where equivariant convergence of non-compact spaces is considered in the present work, the basepoint will always be fixed by the group. In this case, convergence also reduces to the convergence of closed balls.
By [9, Proposition 3.6], given a sequence in , if the sequence of underlying metric spaces converges in the Gromov–Hausdorff topology to a compact metric space then there is a subsequence which converges in the equivariant Gromov–Hausdorff topology.
3. Alexandrov geometry
Certain curvature conditions define precompact subsets of the set of all compact metric spaces. For example, Gromov showed that the class of all Riemannian manifolds of dimension , with diameter less than , and with Ricci curvature greater than is precompact . Strengthening the curvature condition to require a lower bound on the sectional curvature provides much more structure on the limit spaces, and it is in this context that Alexandrov geometry was first studied.
It is possible to show that, for a Riemannian manifold, the condition that sectional curvature be can be expressed as a triangle-comparison condition. Grove and Petersen showed  that the closure of is contained within the class of all complete length metric spaces satisfying this triangle-comparison condition. It is natural, then, to study this class in its own right.
An Alexandrov space of finite dimension is a locally complete, locally compact, connected length space, with a lower curvature bound in the triangle-comparison sense. By convention, a –dimensional Alexandrov space is either a one-point or a two-point space.
Many fundamental results in this area were proved by Burago, Gromov and Perelman . They showed that the class of all Alexandrov spaces is closed under passing to Gromov–Hausdorff limits, and under quotients by isometric group actions.
Let be an Alexandrov space, and let . Then, also by , there is a uniquely defined tangent cone at , , which can be obtained as a limit object by rescaling around . is itself an Alexandrov space, with curvature .
The most important singularities of an Alexandrov space are its extremal subsets, introducted by Perelman and Petrunin . The distance functions in an Alexandrov space have well-defined gradients, and it is possible to flow along these gradients. The gradient flow gives a natural way to understand an extremal subset.
Let be an Alexandrov space. A subset is extremal if, for every , the flow along the gradient of preserves .
Trivial examples of extremal sets are the empty set, and the entire space . Any point having a space of directions with diameter is extremal, as is the boundary of an Alexandrov space. Where a compact Lie group acts on an Alexandrov space by isometries, the closure of the orbit-type strata in the orbit space are also extremal sets , an example of particular interest for the topic under discussion.
Extremal sets survive the passage to Gromov–Hausdorff limits, and so for any extremal set , and any point , there is a well defined tangent subcone which is also extremal. Conversely, if is a closed subset of such that is extremal for each , then is an extremal subset.
A crucial advance in the understanding of Alexandrov spaces was made by Perelman with his proof of the stability theorem . The author recommends the treatment by Kapovitch  for those who wish to learn more about this deep result. The statement of the theorem given here is a relative version of Perelman’s original theorem. It was proved by Kapovitch for the case where only one extremal subset is under consideration, but as shown by Searle and the author , it is in fact true in greater generality.
Let be a sequence of Alexandrov spaces of dimension with curvature uniformly bounded from below, converging to an Alexandrov space of the same dimension. Let be a family of extremal sets in indexed by a set , converging to a family of extremal sets in .
Let be a function with . Let be a sequence of Gromov–Hausdorff –approximations.
Then for all large there exist homeomorphisms , –close to .
This result implies all the previously known finiteness results for manifolds, other than Cheeger diffeofiniteness in dimension four. It also has a vital application in Alexandrov geometry. Consider the construction of the tangent cone to an Alexandrov space by the convergence of the sequence obtained by rescaling the metric around a certain point. By Theorem 3.3, the local structure of the space is controlled by the tangent cone.
Let be an Alexandrov space, and let . Then for some , for all . Furthermore, and the homeomorphism can be chosen so that, for every extremal set , is mapped to .
These small conical neighborhoods are extremely useful in the study of Alexandrov spaces, and so it will be convenient to make the following definition.
An open subset of an Alexandrov space is called cone-like around if , and there is a homeomorphism with being the vertex of the cone and for each extremal set .
Finally, the following result on equivariant convergence of Alexandrov spaces, which is due to the author, will be central in constructing orbifold category homeomorphisms. The original result is for equicontinuous sequences of actions. This hypothesis is always satisfied for a finite group.
 Let be a finite Lie group and let be a sequence of pointed Alexandrov spaces of dimension and curvature bounded below by . Let act isometrically on each of , fixing . Suppose the sequence converges to an action of on another -dimensional pointed Alexandrov space in the pointed equivariant Gromov–Hausdorff topology.
Then for large the spaces are equivariantly homeomorphic to .
In the proof of that theorem, after passing to a subsequence, the convergent sequence is reformulated as a Hausdorff convergent sequence of invariant subspaces of an enveloping metric space with an isometric action of . Then the orbit spaces converge in the Hausdorff sense inside . If are Hausdorff approximations which are also homeomorphisms, then they can be lifted to equivariant homeomorphisms . This provides some flexibility in the construction of the , since there is usually some freedom in the precise choice of .
Orbifolds were first introduced by Satake under the name V-manifolds , as topological spaces locally modelled on a quotient of Euclidean space by a finite group. Some basic facts about orbifolds are reviewed here. The reader may refer to, among others, the book by Adem, Leida and Ruan  or Thurston’s notes  for further information.
A smooth –dimensional orbifold chart over a topological space is a triple such that is a connected open subset of , is a finite group of smooth automorphisms of and is a –invariant map inducing a homeomorphism .
For convenience, a chart will sometimes be referred to as being over a point . This will mean that the chart is over some neighborhood of .
Let and be open subsets of a topological space , and let and be orbifold charts of dimension over and respectively. The charts are called compatible if, for every , there is a neighborhood of and an orbifold chart over such that there are smooth embeddings and with and .
As usual, an orbifold atlas on a space will mean a collection of compatible charts covering . Now the definition of an orbifold can be made.
A smooth orbifold of dimension is a paracompact Hausdorff space equipped with an atlas of orbifold charts of dimension .
An orbifold diffeomorphism (respectively homeomorphism) is a homeomorphism of the underlying topological space which can locally be lifted to an equivariant diffeomorphism (respectively homeomorphism) of charts.
Let be an orbifold, let , and let be a chart over with . The isotropy group of will be called the local group at , and will be written as . It is uniquely defined up to conjugacy in , and choosing a different chart does not change the isomorphism type of the group.
In fact, one can always choose a linear chart over such that the group of automorphisms is isomorphic to . By this is meant a chart of the form where the action of is via a faithful orthogonal representation . Such a chart will be referred to as a linear chart around . The representation is also uniquely determined up to isomorphism, and will be called the local action at . The differential of the action of at the origin of the chart is also isomorphic to .
A Riemannian metric on an orbifold can be given by fixing a finite atlas and a partition of unity with respect to the corresponding cover, and choosing Riemannian metrics on the charts which are invariant with respect to the finite group action. An orbifold equipped with a Riemannian metric is called a Riemannian orbifold. Once the metric on the orbifold is given it can be lifted to the maximal atlas in a canonical manner. The various notions of curvature at points of an orbifold can then be defined by reference to the curvature of the charts.
It is straightforward to see that an orbifold with sectional curvature is also an Alexandrov space with curvature . The tangent cone at any point of an orbifold is then well-defined, and coincides with the usual notion of tangent space for orbifolds. The notion of an extremal set now finds a very natural application in orbifolds.
Let be an orbifold of dimension , a finite group, and a linear representation of . Let be the closure of all points with local action . Then is an extremal set of .
The result is clear where . Let and consider the local action at by . The tangent cone at is the cone on the quotient of the unit sphere by . Consider the image of those points in the unit sphere having isotropy isomorphic to . The closure of the cone on this set is and by induction it is extremal in . Since is closed, it is extremal.∎
The following proposition now shows that a linear chart around can be extended over any cone-like set around .
Let be an orbifold, and let . Let be a cone-like set around . Then there is a linear chart over around .
Consider the differential of the local action of on . The quotient of this action is the tangent cone at , .
Let be a homeomorphism carrying each extremal set in to . Note that because is cone-like, preserves the local action at every point.
Using a maximal atlas, cover by the ranges of all possible linear charts, . Discard any such that is not the range of a linear chart in .
Observe that this reduced family still covers . Suppose some is not in any element of the reduced family. Then for every such that , is not a linear chart. But is covered by some linear chart, and the intersection of the range of this chart with is also covered by a linear chart. Then because it too is covered by a linear chart. It follows that for some , and is in the reduced family.
Select a countable subcover, , and write for . Let be the local group acting on the charts and . The charts can be glued together to construct a chart over all of . The gluing requires copies of . The manner of this gluing gives a set of instructions which allows one to glue the charts together to obtain the desired chart .
Since this chart is built by gluing together charts from the orbifold atlas, it is compatible with the atlas. ∎
5. Proof of Main Theorem
By the Stability Theorem 3.3, contains only finitely many topological types. To prove the main theorem, it is therefore sufficient to prove the following.
Let be a compact topological space. Then, up to orbifold homeomorphism, there are only finitely many orbifold structures on which belong to .
Aiming for a contradiction, let be a sequence of orbifolds in , all of which have underlying topological space , and no two of which are orbifold homeomorphic. By compactness of , a subsequence of converges in the Gromov–Haudorff sense to some which also has underlying space . Abusing notation, the subsequence will still be written as . Since there will be many more instances of passing to subsequences, this abuse of notation will be repeated throughout the proof.
By Stanhope  there is a uniform upper bound on the order of the local group of a point in . Recall a finite group has only finitely many linear representations in a given dimension. It follows that all the possible local actions up to isomorphism can be listed by , for where is some finite number. Let be , the closure of the subset of with local group action isomorphic to . By Proposition 4.3 the are extremal sets.
Passing to a subsequence times if necessary, one may assume that each sequence converges to an extremal subset . Now, by the relative stability theorem , there are homeomorphisms which are Gromov–Hausdorff approximations and carry each of the onto the .
To prove the result, it is now sufficient to show that given by is an orbifold homeomorphism.
Let be a set of points in such that is covered by cone-like metric balls centered at . Then the sets are also cone-like around , and cover . Denote these sets by .
By Proposition 4.4 each is covered by a chart . By passing to a subsequence, we may assume that the form a convergent sequence in the pointed equivariant Gromov–Hausdorff topology, converging to some object .
Now, by Theorem 3.6, and are equivariantly homeomorphic by some . The induce homeomorphisms which are Hausdorff approximations witnessing the Hausdorff convergence of the orbit spaces inside the enveloping orbit space.
Write for the isometry induced by .
^α_i\arrow[two heads]d & ~U^α\arrowlF_i[swap]≅ \arrow[two heads]d~U
^α_i/ Γ_p^α_i \arrowdμ^α_i[anchor=center,rotate=-90,yshift=-1ex,xshift=-0.2ex]≅ & ~U^α/ Γ_p^α \arrowlf_i[swap]≅
Now the gap may be filled in by a homeomorphism given by . The make up a sequence of Gromov–Hausdorff approximations, and the sequence converges to some isometry . Then the may be adjusted slightly, setting , and the adjusted to induce the new .
This gives a non-smooth orbifold chart over , such that the are orbifold homeomorphisms. The maps are then also orbifold homeomorphisms. ∎
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