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Cohomogeneity one topological manifolds revisited

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Abstract

We prove a structure theorem for closed topological manifolds of cohomogeneity one; this result corrects an oversight in the literature. We complete the equivariant classification of closed, simply-connected cohomogeneity one topological manifolds in dimensions 5, 6, and 7 and obtain topological characterizations of these spaces. In these dimensions, these manifolds are homeomorphic to smooth manifolds.

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References

  1. Atiyah, M.F., Bott, R.: A Lefschetz fixed point formula for elliptic complexes: II. Applications. Ann. Math. 2(88), 451–491 (1968)

    MathSciNet  MATH  Google Scholar 

  2. Atiyah, M.F., Singer, I.M.: The index of elliptic operators. III. Ann. Math. 2(87), 546–604 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  3. Böhm, C.: Non-compact cohomogeneity one Einstein manifolds. Bull. Soc. Math. France 127(1), 135–177 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bredon, G.E.: On homogeneous cohomology spheres. Ann. Math. 2(73), 556–565 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bredon, G.E.: Introduction to Compact Transformation Groups. Acad. Press, New York (1972)

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  7. Burago, Y., Gromov, M., Perel’man, G.: A.D. Alexandrov spaces with curvatures bounded below, (Russian) Uspekhi Mat. Nauk 47(2) (284), 3–51, 222 (1992); translation. Russ. Math. Surv. 47(2), 1–58 (1992)

    Article  Google Scholar 

  8. Cannon, J.W.: Shrinking cell-like decompositions of manifolds. Codimension three. Ann. Math. (2) 110(1), 83–112 (1979)

  9. Dancer, A., Swann, A.: Quaternionic Kahler manifolds of cohomogeneity one. Int. J. Math. 10(5), 541–570 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  10. van Dantzig, D., van der Waerden, B.L.: Über metrisch homogene räume. Abh. Math. Sem. Univ. Hamburg 6(1), 367–376 (1928)

    Article  MathSciNet  MATH  Google Scholar 

  11. Dearricott, O.: A 7-manifold with positive curvature. Duke Math. J. 158(2), 307–346 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  12. Dynkin, E.B.: Maximal subgroups of the classical groups (Russian) Trudy Moskov. Mat. Obšč. 1, 39–166 (1952)

    Google Scholar 

  13. Edwards, R.D., Suspensions of Homology Spheres, preprint (2006). arXiv:math/0610573

  14. Edwards, R.D.: The topology of manifolds and cell-like maps. In: Proceedings of the International Congress of Mathematicians (Helsinki, 1978), pp. 111–127, Acad. Sci. Fennica, Helsinki (1980)

  15. Escher, C.M., Ultman, S.K.: Topological structure of candidates for positive curvature. Topol. Appl. 158(1), 38–51 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  16. Frank, P.: Cohomogeneity one manifolds with positive Euler characteristic. Transform. Groups 18(3), 639–684 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  17. Fukaya, K., Yamaguchi, T.: Isometry groups of singular spaces. Math. Z. 216(1), 31–44 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  18. Galaz-Garcia, F., Searle, C.: Cohomogeneity one Alexandrov spaces. Transform. Groups 16(1), 91–107 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  19. Goertsches, O., Mare, A.: Equivariant cohomology of cohomogeneity one actions. Topol. Appl. 167, 36–52 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  20. Grove, K., Verdiani, L., Ziller, W.: An exotic \(T_1{\mathbb{S}}^4\) with positive curvature. Geom. Funct. Anal. 21(3), 499–524 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  21. Grove, K., Wilking, B., Ziller, W.: Positively curved cohomogeneity one manifolds and 3-Sasakian geometry. J. Differ. Geom. 78(1), 33–111 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  22. Grove, K., Ziller, W.: Curvature and symmetry of Milnor spheres. Ann. Math. (2) 152(1), 331–367 (2000)

  23. Grove, K., Ziller, W.: Cohomogeneity one manifolds with positive Ricci curvature. Invent. Math. 149(3), 619–646 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  24. Hirsch, M.W., Mazur, B.: Smoothings of piecewise linear manifolds. Annals of Mathematics Studies, vol. 80, pp. ix+134. Princeton University Press, Princeton, N.J, University of Tokyo Press, Tokyo. (1974)

  25. Hirzebruch, F.: Pontrjagin classes of rational homology manifolds and the signature of some affine hypersurfaces. Proc. Liverpool Singul. Symp. II, 207–212 (1970)

  26. Hirzebruch, F.: The signature theorem: reminiscences and recreation, Prospects in mathematics (Proc. Sympos., Princeton Univ., Princeton, N.J., 1970), pp. 3–31. Ann. Math. Stud., No. 70, Princeton Univ. Press, Princeton, N.J. (1971)

  27. Hoelscher, C.A.: Classification of cohomogeneity one manifolds in low dimensions, Ph.D. Thesis, University of Pennsylvania (2007)

  28. Hoelscher, C.A.: Diffeomorphism type of six-dimensional cohomogeneity one manifolds. Ann. Glob. Anal. Geom. 38(1), 1–9 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  29. Hoelscher, C.A.: Classification of cohomogeneity one manifolds in low dimensions. Pacif. J. Math. 246(1), 129–185 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  30. Hoelscher, C.A.: On the homology of low-dimensional cohomogeneity one manifolds. Transform. Groups 15(1), 115–133 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  31. Jupp, P.: Classification of certain 6-manifolds. Proc. Camb. Philos. Soc. 73, 293–300 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  32. Kahn, P.J.: A note on topological Pontrjagin classes and the Hirzebruch index formula. Illinois J. Math. 16, 243–256 (1972)

    MathSciNet  MATH  Google Scholar 

  33. Kervaire, M.A., Milnor, J.W.: Groups of homotopy spheres. I. Ann. Math. 2(77), 504–537 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  34. Kirby, R.C., Scharlemann, M.G.: Eight faces of the Poincaré homology sphere, Geometric topology (Proc. Georgia Topology Conf., Athens, Ga., 1977), pp. 113–146, Academic, New York (1979)

  35. Kirby, R.C., Siebenmann, L.C.: Foundational essays on topological manifolds, smoothings, and triangulations. Princeton Univ. Press, Princeton (1977)

    Book  MATH  Google Scholar 

  36. Kobayashi, S.: Transformation groups in differential geometry, Reprint of the, 1972nd edn. Classics in Mathematics. Springer-Verlag, Berlin (1995)

    Google Scholar 

  37. Kobayashi, S., Nomizu, K.: Foundations of differential geometry, Vol. I. Reprint of the 1963 original. Wiley Classics Library. A Wiley-Interscience Publication. Wiley, New York (1996)

  38. Maunder, C.R.F.: On the Pontrjagin classes of homology manifolds. Topology 10, 111–118 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  39. Milnor, J.W., Stasheff, J.D.: Characteristic classes, Annals of Mathematics Studies, No. 76. Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo (1974)

  40. Montgomery, D., Yang, C.T.: Orbits of highest dimension. Trans. Am. Math. Soc. 87, 284–293 (1958)

    Article  MathSciNet  MATH  Google Scholar 

  41. Montgomery, D., Zippin, L.: A theorem on Lie groups. Bull. Am. Math. Soc. 48, 448–452 (1942)

    Article  MathSciNet  MATH  Google Scholar 

  42. Mostert, P.S.: On a compact Lie group acting on a manifold, Ann. of Math. (2) 65, 447–455 (1957). Errata, “On a compact Lie group acting on a manifold”. Ann. Math. (2) 66, 589 (1957)

  43. Munkres, J.R.: Obstructions to the smoothing of piecewise-differentiable homeomorphisms. Ann. Math. 2(72), 521–554 (1960)

    Article  MathSciNet  MATH  Google Scholar 

  44. Myers, S.B., Steenrod, N.E.: The group of isometries of a Riemannian manifold. Ann. Math. (2) 40(2), 400–416 (1939)

  45. Neumann, W.D.: \(3\)-Dimensional \(G\)-manifolds with \(2\)-dimensional orbits. In: Mostert, P.S. (Ed.) Proceedings of Conference on Transformation Groups, pp. 220–222. Springer Verlag, Berlin (1968)

  46. Okonek, C., Van de Ven, A.: Cubic forms and complex \(3\)-folds. Enseign. Math. 41(3–4), 297–333 (1995)

    MathSciNet  MATH  Google Scholar 

  47. Onishchik, A.L.: Topology of transitive transformation groups. Johann Ambrosius Barth Verlag GmbH, Leipzig (1994)

    MATH  Google Scholar 

  48. Parker, J.: \(4\)-Dimensional \(G\)-manifolds with \(3\)-dimensional orbits. Pacif. J. Math. 129(1), 187–204 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  49. Perelman, G.: Elements of Morse theory on Aleksandrov spaces. St. Petersburg Math. J. 5, 205–213 (1994)

    MathSciNet  Google Scholar 

  50. Rourke, C.P., Sanderson, B.J.: Introduction to piecewise-linear topology, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 69. Springer-Verlag, New York (1972)

    Book  MATH  Google Scholar 

  51. Schwachhöfer, L.J., Tuschmann, W.: Metrics of positive Ricci curvature on quotient spaces. Math. Ann. 330(1), 59–91 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  52. Searle, C.: Cohomogeneity and positive curvature in low dimensions, Math. Z. 214(3), 491–498; Corrigendum. Math. Z. 226(1), 165–167 (1993)

  53. Thom, R.: Les classes caractéristiques de Pontrjagin des variétés triangulées, 1958 Symposium internacional de topología algebraica International symposium on algebraic topology 54–67 Universidad Nacional Autónoma de México and UNESCO, Mexico City

  54. Wall, C.T.C.: Classification problems in differential topology. V. On certain 6-manifolds. Invent. Math. 1, 355–374 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  55. Weinberger, S.: The Topological Classification of Stratified Spaces. Chicago Lectures in Mathematics. University of Chicago Press, Chicago (1994)

    MATH  Google Scholar 

  56. Wolf, J.A.: Spaces of Constant Curvature. McGraw-Hill Book Co., New York (1967)

    MATH  Google Scholar 

  57. Wu, J.Y.: Topological regularity theorems for Alexandrov spaces. J. Math. Soc. Japan 49(4), 741–757 (1997)

    Article  MathSciNet  Google Scholar 

  58. Zagier, D.: Equivariant Pontrjagin Classes and Applications to Orbit Spaces. Applications of the \(G\)-Signature Theorem to Transformation Groups, Symmetric Products and Number Theory, Lecture Notes in Mathematics, vol. 290. Springer-Verlag, Berlin (1972)

  59. Zagier, D.: The Pontrjagin class of an orbit space. Topology 11, 253–264 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  60. Zhubr, A.V.: Closed simply connected six-dimensional manifolds: proofs of classification theorems. Algebra i Analiz 12(4), 126–230 (2000)

    MathSciNet  Google Scholar 

  61. Ziller, W.: On the geometry of cohomogeneity one manifolds with positive curvature, Riemannian topology and geometric structures on manifolds, vols. 233–262, Progr. Math., 271. Birkhäuser, Boston (2009)

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Acknowledgements

M. Zarei thanks the Institut für Algebra und Geometrie at the Karlsruher Institut für Technologie (KIT) for its hospitality while the work presented herein was carried out. Both authors would like to thank Anand Dessai, Marco Radeschi and Alexey V. Zhubr for helpful conversations on the proof of Theorem G, and Martin Herrmann, Wilderich Tuschmann and Burkhard Wilking for conversations on the smoothability of manifolds. The authors also thank Martin Kerin, Wolfgang Ziller and the referee for suggesting improvements to the exposition. M. Zarei was partially supported by the Ministry of Science, Research and Technology of Iran.

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Correspondence to Fernando Galaz-García.

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Galaz-García, F., Zarei, M. Cohomogeneity one topological manifolds revisited. Math. Z. 288, 829–853 (2018). https://doi.org/10.1007/s00209-017-1915-y

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