Orbifold homeomorphism finiteness based on geometric constraints Authors Emily Proctor Department of Mathematics Middlebury College Original Paper

First Online: 24 May 2011 Received: 01 March 2011 Accepted: 19 April 2011 DOI :
10.1007/s10455-011-9270-4

Cite this article as: Proctor, E. Ann Glob Anal Geom (2012) 41: 47. doi:10.1007/s10455-011-9270-4
Abstract We show that any collection of n -dimensional orbifolds with sectional curvature and volume uniformly bounded below, diameter bounded above, and with only isolated singular points contains orbifolds of only finitely many orbifold homeomorphism types. This is a generalization to the orbifold category of a similar result for manifolds proven by Grove, Petersen, and Wu. It follows that any Laplace isospectral collection of orbifolds with sectional curvature uniformly bounded below and having only isolated singular points also contains only finitely many orbifold homeomorphism types. The main steps of the argument are to show that any sequence from the collection has subsequence that converges to an orbifold, and then to show that the homeomorphism between the underlying spaces of the limit orbifold and an orbifold from the subsequence that is guaranteed by Perelman’s stability theorem must preserve orbifold structure.

Keywords Orbifold Global Riemannian geometry Alexandrov space Spectral geometry

References 1.

Adem A., Leida J., Ruan Y.: Orbifolds and Stringy Topology. Cambridge Tracts in Mathematics, 171. Cambridge University Press, Cambridge (2007)

CrossRef 2.

Borzellino J.: Orbifolds of maximal diameter. Indiana Univ. Math. J.

42 (1), 37–53 (1993)

MathSciNet MATH CrossRef 3.

Brooks B., Perry P., Petersen P.: Compactness and finiteness for isospectral manifolds. J. Reine Angew. Math.

426 , 67–89 (1992)

MathSciNet MATH 4.

Burago D., Burago Y., Ivanov S.: A Course in Metric Geometry. Graduate Studies in Mathematics, 33. American Mathematical Society, Providence, RI (2001)

5.

Cheeger J.: Finiteness theorems for Riemannian manifolds. Am. J. Math.

92 , 61–74 (1970)

MathSciNet MATH CrossRef 6.

Chiang, Y.-J.: Spectral geometry of V-manifolds and its application to harmonic maps. Differential geometry: Partial differential equations on manifolds, Los Angeles, CA, 1990. Proceedings of Pure Symposia in Pure Mathametics, vol. 54, pp. 93–99. American Mathematical Society, Providence, RI (1993)

7.

Dryden E., Gordon C., Greenwald S., Webb D.L.: Asymptotic expansion of the heat kernel for orbifolds. Mich. Math. J.

56 , 205–238 (2008)

MathSciNet MATH CrossRef 8.

Farsi C.: Orbifold spectral theory. Rocky Mt. J. Math.

31 , 215–235 (2001)

MathSciNet MATH CrossRef 9.

Fukaya K.: Theory of convergence for Riemannian orbifolds. Jpn. J. Math.

12 (1), 121–160 (1986)

MathSciNet MATH 10.

Greene R.E., Wu H.: Approximation theorems,

C
^{∞} convex exhaustions and manifolds of positive curvature. Bull. Am. Math. Soc.

81 , 101–104 (1975)

MathSciNet MATH CrossRef 11.

Gromov M.: Groups of polynomial growth and expanding maps. Inst. Hautes Etud. Sci. Publ. Math.

53 , 53–73 (1981)

MathSciNet MATH CrossRef 12.

Grove K., Petersen P.: Bounding homotopy types by geometry. Ann. Math.

128 , 195–206 (1988)

MathSciNet MATH CrossRef 13.

Grove K., Petersen P., Wu J-Y.: Geometric finiteness theorems via contolled topology. Invent Math.

99 , 205–213 (1990)

MathSciNet MATH CrossRef 14.

Kapovitch V.: Regularity of noncollapsing limits of manifolds. Geom. Funct. Anal.

12 (1), 121–137 (2002)

MathSciNet MATH CrossRef 15.

Kapovitch, V.: Perelman’s stability theorem. Surveys in differential geometry. Surv. Differ. Geom, vol. XI, pp. 103–136, 11, International Press, Somerville, MA (2007)

16.

Kleiner, B., Lott, J.: Geometrization of three-dimensional orbifolds via Ricci flow, arXiv:1101.3733v2 [math:DG]

17.

Perelman, G.: Alexandrov’s spaces with curvatures bounded from below, II. preprint (1991)

18.

Perelman G.: Elements of Morse theory on Aleksandrov spaces. St. Petersburg Math. J.

5 (1), 205–213 (1994)

MathSciNet 19.

Perelman G., Petrunin A.: Extremal subsets in Aleksandrov spaces and the generalized Liberman theorem. St. Petersburg Math. J.

5 (1), 215–227 (1994)

MathSciNet 20.

Petersen P.: Riemannian Geometry. Graduate Texts in Mathematics, 171. Springer, New York (1998)

21.

Petrunin A.: Applications of quasigeodesics and gradient curves. Comparison geometry (Berkeley, CA, 1993–1994). Math. Sci. Res. Inst. Publ., vol. 30, pp. 203–219. Cambridge University Press, Cambridge (1997)

22.

Proctor E., Stanhope E.: Spectral and geometric bounds on 2-orbifold diffeomorphism type. Differ. Geom. Appl.

28 (1), 12–18 (2010)

MathSciNet MATH CrossRef 23.

Satake I.: On a generalization of the notion of a manifold. Proc. Natl. Acad. Sci. USA

42 , 359–363 (1956)

MathSciNet MATH CrossRef 24.

Stanhope E.: Spectral bounds on orbifold isotropy. Ann. Global Anal. Geom.

27 (4), 355–375 (2005)

MathSciNet MATH CrossRef 25.

Thurston, W.: The geometry and topology of 3-manifolds, Lecture Notes.

http://www.msri.org/publications/books/gt3m/ (1980)

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