Good coverings of Alexandrov spaces

Mitsuishi, Ayato; Yamaguchi, Takao

doi:10.1090/tran/7849

Good coverings of Alexandrov spaces
HTML articles powered by AMS MathViewer

by Ayato Mitsuishi and Takao Yamaguchi PDF

Trans. Amer. Math. Soc. 372 (2019), 8107-8130 Request permission

Abstract:

In the present paper, we define a notion of good coverings of Alexandrov spaces with curvature bounded below, and we prove that every Alexandrov space admits such a good covering and that it has the same homotopy type as the nerve of the good covering. We also prove a kind of stability of the isomorphism classes of the nerves of good coverings in the noncollapsing case. In the proof, we need a version of Perelman’s fibration theorem, which is also proved in this paper.

References

Raoul Bott and Loring W. Tu, Differential forms in algebraic topology, Graduate Texts in Mathematics, vol. 82, Springer-Verlag, New York-Berlin, 1982. MR 658304, DOI 10.1007/978-1-4757-3951-0
Dmitri Burago, Yuri Burago, and Sergei Ivanov, A course in metric geometry, Graduate Studies in Mathematics, vol. 33, American Mathematical Society, Providence, RI, 2001. MR 1835418, DOI 10.1090/gsm/033
Yu. Burago, M. Gromov, and G. Perel′man, A. D. Aleksandrov spaces with curvatures bounded below, Uspekhi Mat. Nauk 47 (1992), no. 2(284), 3–51, 222 (Russian, with Russian summary); English transl., Russian Math. Surveys 47 (1992), no. 2, 1–58. MR 1185284, DOI 10.1070/RM1992v047n02ABEH000877
Jeff Cheeger, Finiteness theorems for Riemannian manifolds, Amer. J. Math. 92 (1970), 61–74. MR 263092, DOI 10.2307/2373498
Michael Gromov, Curvature, diameter and Betti numbers, Comment. Math. Helv. 56 (1981), no. 2, 179–195. MR 630949, DOI 10.1007/BF02566208
Karsten Grove and Peter Petersen V, Bounding homotopy types by geometry, Ann. of Math. (2) 128 (1988), no. 1, 195–206. MR 951512, DOI 10.2307/1971439
Karsten Grove and Peter Petersen, A radius sphere theorem, Invent. Math. 112 (1993), no. 3, 577–583. MR 1218324, DOI 10.1007/BF01232447
Karsten Grove, Peter Petersen V, and Jyh Yang Wu, Geometric finiteness theorems via controlled topology, Invent. Math. 99 (1990), no. 1, 205–213. MR 1029396, DOI 10.1007/BF01234418
John Harvey and Catherine Searle, Orientation and symmetries of Alexandrov spaces with applications in positive curvature, J. Geom. Anal. 27 (2017), no. 2, 1636–1666. MR 3625167, DOI 10.1007/s12220-016-9734-7
V. Kapovitch, Regularity of limits of noncollapsing sequences of manifolds, Geom. Funct. Anal. 12 (2002), no. 1, 121–137. MR 1904560, DOI 10.1007/s00039-002-8240-1
Vitali Kapovitch, Perelman’s stability theorem, Surveys in differential geometry. Vol. XI, Surv. Differ. Geom., vol. 11, Int. Press, Somerville, MA, 2007, pp. 103–136. MR 2408265, DOI 10.4310/SDG.2006.v11.n1.a5
Kyung Whan Kwun, Uniqueness of the open cone neighborhood, Proc. Amer. Math. Soc. 15 (1964), 476–479. MR 161319, DOI 10.1090/S0002-9939-1964-0161319-3
Ayato Mitsuishi and Takao Yamaguchi, Stability of strongly Lipschitz contractible balls in Alexandrov spaces, Math. Z. 277 (2014), no. 3-4, 995–1009. MR 3229976, DOI 10.1007/s00209-014-1289-3
Ayato Mitsuishi and Takao Yamaguchi, Locally Lipschitz contractibility of Alexandrov spaces and its applications, Pacific J. Math. 270 (2014), no. 2, 393–421. MR 3253688, DOI 10.2140/pjm.2014.270.393
A. Mitsuishi and T. Yamaguchi, Lipschitz homotopy convergence of Alexandrov spaces, J. Geom. Anal. (to appear).
G. de Rham, Complexes à automorphismes et homéomorphie différentiable, Ann. Inst. Fourier (Grenoble) 2 (1950), 51–67 (1951) (French). MR 43468, DOI 10.5802/aif.19
G. Perelman, Spaces with curvature bounded below, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994) Birkhäuser, Basel, 1995, pp. 517–525. MR 1403952
G. Ya. Perel′man, Elements of Morse theory on Aleksandrov spaces, Algebra i Analiz 5 (1993), no. 1, 232–241 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 5 (1994), no. 1, 205–213. MR 1220498
G. Perelman, DC structure on Alexandrov space, preprint.
A. D. Aleksandrov, Quasigeodesics, Doklady Akad. Nauk SSSR (N.S.) 69 (1949), 717–720 (Russian). MR 0035042
Peter Petersen V, A finiteness theorem for metric spaces, J. Differential Geom. 31 (1990), no. 2, 387–395. MR 1037407
Anton Petrunin, Quasigeodesics in multidimensional Alexandrov spaces, ProQuest LLC, Ann Arbor, MI, 1995. Thesis (Ph.D.)–University of Illinois at Urbana-Champaign. MR 2693118
Anton Petrunin, Semiconcave functions in Alexandrov’s geometry, Surveys in differential geometry. Vol. XI, Surv. Differ. Geom., vol. 11, Int. Press, Somerville, MA, 2007, pp. 137–201. MR 2408266, DOI 10.4310/SDG.2006.v11.n1.a6
G. de Rham, Complexes à automorphismes et homéomorphie différentiable, Ann. Inst. Fourier (Grenoble) 2 (1950), 51–67 (1951) (French). MR 43468, DOI 10.5802/aif.19
L. C. Siebenmann, Deformation of homeomorphisms on stratified sets. I, II, Comment. Math. Helv. 47 (1972), 123–136; ibid. 47 (1972), 137–163. MR 319207, DOI 10.1007/BF02566793
Alan Weinstein, On the homotopy type of positively-pinched manifolds, Arch. Math. (Basel) 18 (1967), 523–524. MR 220311, DOI 10.1007/BF01899493
Takao Yamaguchi, Homotopy type finiteness theorems for certain precompact families of Riemannian manifolds, Proc. Amer. Math. Soc. 102 (1988), no. 3, 660–666. MR 928999, DOI 10.1090/S0002-9939-1988-0928999-X
T. Yamaguchi, Collapsing and essential coverings, arXiv:1205.0441 (2012).