Algebraic & Geometric Topology

Positive curvature and rational ellipticity

Manuel Amann and Lee Kennard

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Abstract

Simply connected manifolds of positive sectional curvature are speculated to have a rigid topological structure. In particular, they are conjectured to be rationally elliptic, ie to have only finitely many non-zero rational homotopy groups. In this article, we combine positive curvature with rational ellipticity to obtain several topological properties of the underlying manifold. These results include an upper bound on the Euler characteristic and new evidence for a couple of well-known conjectures due to Hopf and Halperin. We also prove a conjecture of Wilhelm for even-dimensional manifolds whose rational type is one of the known examples of positive curvature.

Article information

Source
Algebr. Geom. Topol., Volume 15, Number 4 (2015), 2269-2301.

Dates
Received: 9 June 2014
Revised: 22 October 2014
Accepted: 24 November 2014
First available in Project Euclid: 16 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1510841003

Digital Object Identifier
doi:10.2140/agt.2015.15.2269

Mathematical Reviews number (MathSciNet)
MR3402341

Zentralblatt MATH identifier
1325.53044

Subjects
Primary: 53C20: Global Riemannian geometry, including pinching [See also 31C12, 58B20]
Secondary: 57N65: Algebraic topology of manifolds 55P62: Rational homotopy theory

Keywords
positive curvature rational ellipticity torus symmetry Euler characteristic Wilhelm conjecture Halperin conjecture Hopf conjecture

Citation

Amann, Manuel; Kennard, Lee. Positive curvature and rational ellipticity. Algebr. Geom. Topol. 15 (2015), no. 4, 2269--2301. doi:10.2140/agt.2015.15.2269. https://projecteuclid.org/euclid.agt/1510841003


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References

  • C Allday, V Puppe, Cohomological methods in transformation groups, Cambridge Studies in Advanced Mathematics 32, Cambridge Univ. Press (1993)
  • M Amann, Positive quaternion Kähler manifolds, PhD thesis, Westfälische Wilhelms–Universität Münster (2009) Available at \setbox0\makeatletter\@url http://d-nb.info/996176438/34 {\unhbox0
  • M Amann, L Kennard, Topological properties of positively curved manifolds with symmetry, Geom. Funct. Anal. 24 (2014) 1377–1405
  • I Belegradek, V Kapovitch, Obstructions to nonnegative curvature and rational homotopy theory, J. Amer. Math. Soc. 16 (2003) 259–284
  • M Berger, Sur quelques variétés riemanniennes suffisamment pincées, Bull. Soc. Math. France 88 (1960) 57–71
  • A Blanchard, Sur les variétés analytiques complexes, Ann. Sci. Ecole Norm. Sup. 73 (1956) 157–202
  • C B öhm, B Wilking, Manifolds with positive curvature operators are space forms, Ann. of Math. 167 (2008) 1079–1097
  • W Browder, Fiberings of spheres and $H\!$–spaces which are rational homology spheres, Bull. Amer. Math. Soc. 68 (1962) 202–203
  • X Chen, Curvature and Riemannian submersions, PhD thesis, University of Notre Dame (2014) Available at \setbox0\makeatletter\@url http://search.proquest.com/docview/1547351162 {\unhbox0
  • O Dearricott, A $7$–manifold with positive curvature, Duke Math. J. 158 (2011) 307–346
  • J-H Eschenburg, Freie isometrische Aktionen auf kompakten Lie-Gruppen mit positiv gekrümmten Orbiträumen, Schriftenr. Math. Inst. Univ. Münster (II) 32, Mathematisches Institut, Universität Münster (1984)
  • Y Félix, S Halperin, J-C Thomas, Rational homotopy theory, Graduate Texts in Mathematics 205, Springer, New York (2001)
  • Y Félix, J Oprea, D Tanré, Algebraic models in geometry, Oxford Graduate Texts in Mathematics 17, Oxford Univ. Press (2008)
  • L,A Florit, W Ziller, Topological obstructions to fatness, Geom. Topol. 15 (2011) 891–925
  • D Gromoll, K Grove, A generalization of Berger's rigidity theorem for positively curved manifolds, Ann. Sci. École Norm. Sup. 20 (1987) 227–239
  • K Grove, Geometry of, and via, symmetries, from: “Conformal, Riemannian and Lagrangian geometry”, (A Freire, editor), Univ. Lecture Ser. 27, Amer. Math. Soc. (2002) 31–53
  • K Grove, Developments around positive sectional curvature, from: “Geometry, analysis, and algebraic geometry: forty years of the Journal of Differential Geometry”, (H-D Cao, S-T Yau, editors), Surv. Differ. Geom. 13, Int. Press, Somerville, MA (2009) 117–133
  • K Grove, S Halperin, Dupin hypersurfaces, group actions and the double mapping cylinder, J. Differential Geom. 26 (1987) 429–459
  • K Grove, C Searle, Positively curved manifolds with maximal symmetry-rank, J. Pure Appl. Algebra 91 (1994) 137–142
  • K Grove, L Verdiani, W Ziller, An exotic $T\sb 1\Bbb S\sp 4$ with positive curvature, Geom. Funct. Anal. 21 (2011) 499–524
  • S Halperin, Rational fibrations, minimal models, and fibrings of homogeneous spaces, Trans. Amer. Math. Soc. 244 (1978) 199–224
  • S Halperin, Polynomial growth and elliptic spaces, unpublished (2012)
  • I,M James (editor), Handbook of algebraic topology, North-Holland, Amsterdam (1995)
  • L Kennard, On the Hopf conjecture with symmetry, Geom. Topol. 17 (2013) 563–593
  • L Kennard, Positively curved Riemannian metrics with logarithmic symmetry rank bounds, Comment. Math. Helv. 89 (2014) 937–962
  • M Kerin, Some new examples with almost positive curvature, Geom. Topol. 15 (2011) 217–260
  • G Lupton, Note on a conjecture of Stephen Halperin's, from: “Topology and combinatorial group theory”, (P Latiolais, editor), Lecture Notes in Math. 1440, Springer, Berlin (1990) 148–163
  • G Lupton, Variations on a conjecture of Halperin, from: “Homotopy and geometry”, (J Oprea, A Tralle, editors), Banach Center Publ. 45, Polish Acad. Sci., Warsaw (1998) 115–135
  • A-L Mare, M Willems, Topology of the octonionic flag manifold, Münster J. Math. 6 (2013) 483–523
  • M Markl, Towards one conjecture on collapsing of the Serre spectral sequence, from: “Proceedings of the Winter School on Geometry and Physics”, (J Bureš, V Souček, editors), Rend. Circ. Mat. Palermo Suppl. 22 (1990) 151–159
  • W Meier, Rational universal fibrations and flag manifolds, Math. Ann. 258 (1982) 329–340
  • W Meier, Some topological properties of Kähler manifolds and homogeneous spaces, Math. Z. 183 (1983) 473–481
  • S Papadima, L Paunescu, Reduced weighted complete intersection and derivations, J. Algebra 183 (1996) 595–604
  • P Petersen, F Wilhem, An exotic sphere with positive curvature, preprint (2008)
  • T Püttmann, C Searle, The Hopf conjecture for manifolds with low cohomogeneity or high symmetry rank, Proc. Amer. Math. Soc. 130 (2002) 163–166
  • X Rong, X Su, The Hopf conjecture for manifolds with abelian group actions, Commun. Contemp. Math. 7 (2005) 121–136
  • H Shiga, M Tezuka, Rational fibrations, homogeneous spaces with positive Euler characteristics and Jacobians, Ann. Inst. Fourier $($Grenoble$)$ 37 (1987) 81–106
  • Z Su, Rational analogs of projective planes, Algebr. Geom. Topol. 14 (2014) 421–438
  • J-C Thomas, Rational homotopy of Serre fibrations, Ann. Inst. Fourier $($Grenoble$)$ 31 (1981) v, 71–90
  • J,M Wahl, Derivations, automorphisms and deformations of quasihomogeneous singularities, from: “Singularities, Part 2”, (P Orlik, editor), Proc. Sympos. Pure Math. 40, Amer. Math. Soc. (1983) 613–624
  • N,R Wallach, Compact homogeneous Riemannian manifolds with strictly positive curvature, Ann. of Math. 96 (1972) 277–295
  • G Walschap, Soul-preserving submersions, Michigan Math. J. 41 (1994) 609–617
  • B Wilking, Index parity of closed geodesics and rigidity of Hopf fibrations, Invent. Math. 144 (2001) 281–295
  • B Wilking, Torus actions on manifolds of positive sectional curvature, Acta Math. 191 (2003) 259–297
  • B Wilking, Positively curved manifolds with symmetry, Ann. of Math. 163 (2006) 607–668
  • B Wilking, Nonnegatively and positively curved manifolds, from: “Metric and comparison geometry”, (J Cheeger, K Grove, editors), Surv. Differ. Geom. 11, Int. Press, Somerville, MA (2007) 25–62
  • J,E Yeager, Geometric and topological ellipticity in cohomogeneity two, PhD thesis, University of Maryland (2012) Available at \setbox0\makeatletter\@url http://hdl.handle.net/1903/12662 {\unhbox0
  • O Zariski, P Samuel, Commutative algebra, I, Graduate Texts in Mathematics 28, Springer, New York (1975)
  • O Zariski, P Samuel, Commutative algebra, II, Graduate Texts in Mathematics 29, Springer, New York (1976)
  • W Ziller, Examples of Riemannian manifolds with non-negative sectional curvature, from: “Metric and comparison geometry”, (J Cheeger, K Grove, editors), Surv. Differ. Geom. 11, Int. Press, Somerville, MA (2007) 63–102
  • W Ziller, Riemannian manifolds with positive sectional curvature, from: “Geometry of manifolds with non-negative sectional curvature”, (R Herrera, L Hernández-Lamoneda, editors), Springer, Berlin (2014) 1–19