Abstract
We provide an alternative proof that crosscaps are diffeomorphically stable.
Acknowledgements
We are grateful to Stefano Vidussi for several conversations about exotic differential structures on ℝP4. We are grateful to a referee of this paper for making us aware of the results in [21].
References
[1] D. Barden, The structure of manifolds. PhD thesis, Cambridge University, Cambridge, England.Search in Google Scholar
[2] Y. Burago, M. Gromov, G. Perelman, A. D. Aleksandrov spaces with curvatures bounded below. UspekhiMat. Nauk47 (1992), no. 2(284), 3–51, 222. MR1185284 Zbl 0802.53018Search in Google Scholar
[3] S. E. Cappell, J. L. Shaneson, Some new four-manifolds. Ann. of Math. (2) 104 (1976), 61–72. MR0418125 Zbl 0345.5700310.2307/1971056Search in Google Scholar
[4] J. Cerf, La stratification naturelle des espaces de fonctions différentiables réelles et le théorème de la pseudo-isotopie. Inst. Hautes Études Sci. Publ. Math. no. 39 (1970), 5–173. MR0292089 Zbl 0213.2520210.1007/BF02684687Search in Google Scholar
[5] J. Cheeger, Comparison and finiteness theorems for Riemannian manifolds. Thesis, Princeton University, 1967.Search in Google Scholar
[6] J. Cheeger, Finiteness theorems for Riemannian manifolds. Amer. J. Math. 92 (1970), 61–74. MR0263092 Zbl 0194.5290210.2307/2373498Search in Google Scholar
[7] R. Fintushel, R. J. Stern, An exotic free involution on S4. Ann. of Math. (2) 113 (1981), 357–365. MR607896 Zbl 0474.5701410.2307/2006987Search in Google Scholar
[8] K. Fukaya, Theory of convergence for Riemannian orbifolds. Japan. J. Math. (N.S.)12 (1986), 121–160. MR914311 Zbl 0654.5304410.4099/math1924.12.121Search in Google Scholar
[9] K. Fukaya, T. Yamaguchi, Isometry groups of singular spaces. Math. Z. 216 (1994), 31–44. MR1273464 Zbl 0797.5303310.1007/BF02572307Search in Google Scholar
[10] R. E. Greene, H. Wu, Integrals of subharmonic functions on manifolds of nonnegative curvature. Invent. Math. 27 (1974), 265–298. MR0382723 Zbl 0342.3100310.1007/BF01425500Search in Google Scholar
[11] K. Grove, P. Petersen, Manifolds near the boundary of existence. J. Differential Geom. 33 (1991), 379–394. MR1094462 Zbl 0729.5304510.4310/jdg/1214446323Search in Google Scholar
[12] K. Grove, P. Petersen, V, Bounding homotopy types by geometry. Ann. of Math. (2) 128 (1988), 195–206. MR951512 Zbl 0655.5303210.2307/1971439Search in Google Scholar
[13] K. Grove, P. Petersen, V, Volume comparison à la Aleksandrov. Acta Math. 169 (1992), 131–151. MR1179015 Zbl 0758.5303510.1007/BF02392759Search in Google Scholar
[14] K. Grove, K. Shiohama, A generalized sphere theorem. Ann. of Math. (2) 106 (1977), 201–211. MR0500705 Zbl 0341.5302910.2307/1971164Search in Google Scholar
[15] K. Grove, F. Wilhelm, Hard and soft packing radius theorems. Ann. of Math. (2)142 (1995), 213–237. MR1343322 Zbl 0846.5304210.2307/2118635Search in Google Scholar
[16] K. Grove, F. Wilhelm, Metric constraints on exotic spheres via Alexandrov geometry. J. Reine Angew. Math. 487 (1997), 201–217. MR1454266 Zbl 0882.53030Search in Google Scholar
[17] I. Hambleton, M. Kreck, P. Teichner, Nonorientable 4-manifolds with fundamental group of order 2. Trans. Amer. Math. Soc. 344 (1994), 649–665. MR1234481 Zbl 0830.57012Search in Google Scholar
[18] G. Higman, The units of group-rings. Proc. London Math. Soc. (2) 46 (1940), 231–248. MR0002137 Zbl 0025.24302 JFM 66.0104.0410.1112/plms/s2-46.1.231Search in Google Scholar
[19] V. Kapovitch, Perelman’s stability theorem. In: Surveys in differential geometry. Vol. XI, volume 11 of Surv. Differ. Geom., 103–136, Int. Press, Somerville, MA 2007. MR2408265 Zbl 1151.5303810.4310/SDG.2006.v11.n1.a5Search in Google Scholar
[20] M. A. Kervaire, J. W. Milnor, Groups of homotopy spheres. I.Ann. of Math. (2) 77 (1963), 504–537. MR0148075 Zbl 0115.4050510.1142/9789812836878_0002Search in Google Scholar
[21] K. Kuwae, Y. Machigashira, T. Shioya, Sobolev spaces, Laplacian, and heat kernel on Alexandrov spaces. Math. Z. 238 (2001), 269–316. MR1865418 Zbl 1001.5301710.1007/s002090100252Search in Google Scholar
[22] W. A. LaBach, On diffeomorphisms of the n-disk. Proc. Japan Acad. 43 (1967), 448–450. MR0224104 Zbl 0163.4530410.3792/pja/1195521566Search in Google Scholar
[23] N. Li, X. Rong, Relatively maximum volume rigidity in Alexandrov geometry. Pacific J. Math. 259 (2012), 387–420. MR2988498 Zbl 1272.5302910.2140/pjm.2012.259.387Search in Google Scholar
[24] B. Mazur, Relative neighborhoods and the theorems of Smale. Ann. of Math. (2) 77 (1963), 232–249. MR0150786 Zbl 0112.3830110.2307/1970215Search in Google Scholar
[25] J. Milnor, Lectures on the h-cobordism theorem. Princeton Univ. Press 1965. MR0190942 Zbl 0161.2030210.1515/9781400878055Search in Google Scholar
[26] J. Milnor, Whitehead torsion. Bull. Amer. Math. Soc. 72 (1966), 358–426. MR0196736 Zbl 0147.2310410.1090/S0002-9904-1966-11484-2Search in Google Scholar
[27] Y. Otsu, K. Shiohama, T. Yamaguchi, A new version of differentiable sphere theorem. Invent. Math. 98 (1989), 219–228. MR1016261 Zbl 0688.5301610.1007/BF01388850Search in Google Scholar
[28] Y. Otsu, T. Shioya, The Riemannian structure of Alexandrov spaces. J. Differential Geom. 39 (1994), 629–658. MR1274133 Zbl 0808.5306110.4310/jdg/1214455075Search in Google Scholar
[29] G. Perelmann, Alexandrov spaces with curvature bounded from below II. Preprint 1991.Search in Google Scholar
[30] A. Petrunin, Semiconcave functions in Alexandrov’s geometry. In: Surveys in differential geometry. Vol. XI, volume 11 of Surv. Differ. Geom., 137–201, Int. Press, Somerville, MA 2007. MR2408266 Zbl 1166.5300110.4310/SDG.2006.v11.n1.a6Search in Google Scholar
[31] C. Pro, M. Sill, F. Wilhelm, The diffeomorphism type of manifolds with almost maximal volume. Preprint.Search in Google Scholar
[32] W. Rudin, Principles of mathematical analysis. McGraw-Hill Book Co., New York-Auckland-Düsseldorf 1976. MR0385023 Zbl 0346.26002Search in Google Scholar
[33] K. Shiohama, T. Yamaguchi, Positively curved manifolds with restricted diameters. In: Geometry of manifolds (Matsumoto, 1988), volume 8 of Perspect. Math., 345–350, Academic Press 1989. MR1040534 Zbl 0697.53041Search in Google Scholar
[34] C. Sormani, G. Wei, Universal covers for Hausdorff limits of noncompact spaces. Trans. Amer. Math. Soc. 356 (2004), 1233–1270. MR2021619 Zbl 1046.5302710.1090/S0002-9947-03-03412-3Search in Google Scholar
[35] J. Stallings, Projective class groups and Whitehead groups. Mimeographed, Rice University, Houston, Texas.Search in Google Scholar
[36] N. Steenrod, The Topology of Fibre Bundles. Princeton Univ. Press 1951. MR0039258 Zbl 0054.0710310.1515/9781400883875Search in Google Scholar
[37] F. H. Wilhelm, Jr., Collapsing to almost Riemannian spaces. Indiana Univ. Math. J. 41 (1992), 1119–1142. MR1206342 Zbl 0771.5302810.1512/iumj.1992.41.41056Search in Google Scholar
[38] T. Yamaguchi, Collapsing and pinching under a lower curvature bound. Ann. of Math. (2) 133 (1991), 317–357. MR1097241 Zbl 0737.5304110.2307/2944340Search in Google Scholar
[39] T. Yamaguchi, A convergence theorem in the geometry of Alexandrov spaces. In: Actes de la Table Ronde de Géométrie Différentielle (Luminy, 1992), volume 1 of Sémin. Congr., 601–642, Soc. Math. France, Paris 1996. MR1427772 Zbl 0885.53041Search in Google Scholar
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