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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Atiyah, M.F., Bott, R.: A Lefschetz fixed point formula for elliptic complexes: II. Applications. Ann. Math. 2(88), 451–491 (1968)
Atiyah, M.F., Singer, I.M.: The index of elliptic operators. III. Ann. Math. 2(87), 546–604 (1968)
Böhm, C.: Non-compact cohomogeneity one Einstein manifolds. Bull. Soc. Math. France 127(1), 135–177 (1999)
Bredon, G.E.: On homogeneous cohomology spheres. Ann. Math. 2(73), 556–565 (1961)
Bredon, G.E.: Introduction to Compact Transformation Groups. Acad. Press, New York (1972)
Burago, D., Burago, Y., Ivanov, S.: A Course in Metric Geometry, Graduate Studies in Mathematics, 33. American Mathematical Society, Providence, RI (2001)
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)
Cannon, J.W.: Shrinking cell-like decompositions of manifolds. Codimension three. Ann. Math. (2) 110(1), 83–112 (1979)
Dancer, A., Swann, A.: Quaternionic Kahler manifolds of cohomogeneity one. Int. J. Math. 10(5), 541–570 (1999)
van Dantzig, D., van der Waerden, B.L.: Über metrisch homogene räume. Abh. Math. Sem. Univ. Hamburg 6(1), 367–376 (1928)
Dearricott, O.: A 7-manifold with positive curvature. Duke Math. J. 158(2), 307–346 (2011)
Dynkin, E.B.: Maximal subgroups of the classical groups (Russian) Trudy Moskov. Mat. Obšč. 1, 39–166 (1952)
Edwards, R.D., Suspensions of Homology Spheres, preprint (2006). arXiv:math/0610573
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)
Escher, C.M., Ultman, S.K.: Topological structure of candidates for positive curvature. Topol. Appl. 158(1), 38–51 (2011)
Frank, P.: Cohomogeneity one manifolds with positive Euler characteristic. Transform. Groups 18(3), 639–684 (2013)
Fukaya, K., Yamaguchi, T.: Isometry groups of singular spaces. Math. Z. 216(1), 31–44 (1994)
Galaz-Garcia, F., Searle, C.: Cohomogeneity one Alexandrov spaces. Transform. Groups 16(1), 91–107 (2011)
Goertsches, O., Mare, A.: Equivariant cohomology of cohomogeneity one actions. Topol. Appl. 167, 36–52 (2014)
Grove, K., Verdiani, L., Ziller, W.: An exotic \(T_1{\mathbb{S}}^4\) with positive curvature. Geom. Funct. Anal. 21(3), 499–524 (2011)
Grove, K., Wilking, B., Ziller, W.: Positively curved cohomogeneity one manifolds and 3-Sasakian geometry. J. Differ. Geom. 78(1), 33–111 (2008)
Grove, K., Ziller, W.: Curvature and symmetry of Milnor spheres. Ann. Math. (2) 152(1), 331–367 (2000)
Grove, K., Ziller, W.: Cohomogeneity one manifolds with positive Ricci curvature. Invent. Math. 149(3), 619–646 (2002)
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)
Hirzebruch, F.: Pontrjagin classes of rational homology manifolds and the signature of some affine hypersurfaces. Proc. Liverpool Singul. Symp. II, 207–212 (1970)
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)
Hoelscher, C.A.: Classification of cohomogeneity one manifolds in low dimensions, Ph.D. Thesis, University of Pennsylvania (2007)
Hoelscher, C.A.: Diffeomorphism type of six-dimensional cohomogeneity one manifolds. Ann. Glob. Anal. Geom. 38(1), 1–9 (2010)
Hoelscher, C.A.: Classification of cohomogeneity one manifolds in low dimensions. Pacif. J. Math. 246(1), 129–185 (2010)
Hoelscher, C.A.: On the homology of low-dimensional cohomogeneity one manifolds. Transform. Groups 15(1), 115–133 (2010)
Jupp, P.: Classification of certain 6-manifolds. Proc. Camb. Philos. Soc. 73, 293–300 (1973)
Kahn, P.J.: A note on topological Pontrjagin classes and the Hirzebruch index formula. Illinois J. Math. 16, 243–256 (1972)
Kervaire, M.A., Milnor, J.W.: Groups of homotopy spheres. I. Ann. Math. 2(77), 504–537 (1963)
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)
Kirby, R.C., Siebenmann, L.C.: Foundational essays on topological manifolds, smoothings, and triangulations. Princeton Univ. Press, Princeton (1977)
Kobayashi, S.: Transformation groups in differential geometry, Reprint of the, 1972nd edn. Classics in Mathematics. Springer-Verlag, Berlin (1995)
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)
Maunder, C.R.F.: On the Pontrjagin classes of homology manifolds. Topology 10, 111–118 (1971)
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)
Montgomery, D., Yang, C.T.: Orbits of highest dimension. Trans. Am. Math. Soc. 87, 284–293 (1958)
Montgomery, D., Zippin, L.: A theorem on Lie groups. Bull. Am. Math. Soc. 48, 448–452 (1942)
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)
Munkres, J.R.: Obstructions to the smoothing of piecewise-differentiable homeomorphisms. Ann. Math. 2(72), 521–554 (1960)
Myers, S.B., Steenrod, N.E.: The group of isometries of a Riemannian manifold. Ann. Math. (2) 40(2), 400–416 (1939)
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)
Okonek, C., Van de Ven, A.: Cubic forms and complex \(3\)-folds. Enseign. Math. 41(3–4), 297–333 (1995)
Onishchik, A.L.: Topology of transitive transformation groups. Johann Ambrosius Barth Verlag GmbH, Leipzig (1994)
Parker, J.: \(4\)-Dimensional \(G\)-manifolds with \(3\)-dimensional orbits. Pacif. J. Math. 129(1), 187–204 (1986)
Perelman, G.: Elements of Morse theory on Aleksandrov spaces. St. Petersburg Math. J. 5, 205–213 (1994)
Rourke, C.P., Sanderson, B.J.: Introduction to piecewise-linear topology, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 69. Springer-Verlag, New York (1972)
Schwachhöfer, L.J., Tuschmann, W.: Metrics of positive Ricci curvature on quotient spaces. Math. Ann. 330(1), 59–91 (2004)
Searle, C.: Cohomogeneity and positive curvature in low dimensions, Math. Z. 214(3), 491–498; Corrigendum. Math. Z. 226(1), 165–167 (1993)
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
Wall, C.T.C.: Classification problems in differential topology. V. On certain 6-manifolds. Invent. Math. 1, 355–374 (1966)
Weinberger, S.: The Topological Classification of Stratified Spaces. Chicago Lectures in Mathematics. University of Chicago Press, Chicago (1994)
Wolf, J.A.: Spaces of Constant Curvature. McGraw-Hill Book Co., New York (1967)
Wu, J.Y.: Topological regularity theorems for Alexandrov spaces. J. Math. Soc. Japan 49(4), 741–757 (1997)
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)
Zagier, D.: The Pontrjagin class of an orbit space. Topology 11, 253–264 (1972)
Zhubr, A.V.: Closed simply connected six-dimensional manifolds: proofs of classification theorems. Algebra i Analiz 12(4), 126–230 (2000)
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)
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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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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DOI: https://doi.org/10.1007/s00209-017-1915-y