Almost non-negative curvature and rational ellipticity in cohomogeneity two
Annales de l'Institut Fourier, to appear, 19 p.

An extension of a fundamental conjecture by R. Bott suggests that all simply connected closed almost non-negatively curved manifolds M are rationally elliptic, i.e., all but finitely many homotopy groups of such M are finite. We confirm this conjecture when in addition M supports an isometric action with orbits of codimension at most two. Our proof uses the geometry of the orbit space to control the topology of the homotopy fiber of the inclusion map of an orbit in M, and is applicable to more general contexts.

D’après une extension d’une conjecture fondamentale de R. Bott, toute variété compacte (sans bord) simplement connexe M à courbure positive est rationellement elliptique, i.e., seul un nombre fini de groupes d’homotopie de M sont infinis. On montre cette conjecture dans le cas où M admet une action par isométries dont l’orbite principale a codimension au plus est de deux. Notre preuve utilise la géométrie de l’espace quotient pour contrôler la topologie de la fibre homotopique de l’inclusion d’une orbite dans M, et s’applique à des contextes plus généraux.

Classification: 53C20,  55P62,  57S15,  58E10
Keywords: Almost Non-negative Curvature, Rational ellipticity, Morse Theory, Cohomogeneity
@unpublished{AIF_0__0_0_A3_0,
     author = {Grove, Karsten and Wilking, Burkhard and Yeager, Joseph},
     title = {Almost non-negative curvature and rational ellipticity in cohomogeneity two},
     note = {to appear in \emph{Annales de l'Institut Fourier}},
}
Grove, Karsten; Wilking, Burkhard; Yeager, Joseph. Almost non-negative curvature and rational ellipticity in cohomogeneity two. Annales de l'Institut Fourier, to appear, 19 p.

[1] Ballmann, Werner Lectures on the Blaschke conjecture (2014) (http://people.mpim-bonn.mpg.de/hwbllmnn/archiv/blaschke1603.pdf)

[2] Berger, Marcel; Bott, Raoul Sur les variétés à courbure strictement positive, Topology, Volume 1 (1962), pp. 301-311 | Zbl 0112.13604

[3] Cheeger, Jeff; Fukaya, Kenji; Gromov, Mikhael Nilpotent structures and invariant metrics on collapsed manifolds, J. Am. Math. Soc., Volume 5 (1992) no. 2, pp. 327-372

[4] Félix, Yves; Halperin, Stephen Rational LS category and its applications, Trans. Am. Math. Soc., Volume 273 (1982) no. 1, pp. 1-38

[5] Félix, Yves; Halperin, Stephen; Thomas, Jean-Claude Rational homotopy theory, Graduate Texts in Mathematics, Volume 205, Springer, 2001, xxxiv+535 pages

[6] Félix, Yves; Halperin, Steve The depth and LS category of a topological space, Math. Scand., Volume 123 (2018) no. 2, pp. 220-238

[7] Félix, Yves; Halperin, Steve; Thomas, Jean-Claude Rational homotopy theory. II, World Scientific, 2015, xxxvi+412 pages

[8] Gromov, Mikhael Almost flat manifolds, J. Differ. Geom., Volume 13 (1978) no. 2, pp. 231-241

[9] Grove, Karsten; Halperin, Stephen Contributions of rational homotopy theory to global problems in geometry, Publ. Math., Inst. Hautes Étud. Sci., Volume 56 (1982), pp. 171-178 | Zbl 0508.55013

[10] Grove, Karsten; Halperin, Stephen Dupin hypersurfaces, group actions and the double mapping cylinder, J. Differ. Geom., Volume 26 (1987) no. 3, pp. 429-459

[11] Grove, Karsten; Verdiani, Luigi; Wilking, Burkhard; Ziller, Wolfgang Non-negative curvature obstructions in cohomogeneity one and the Kervaire spheres, Ann. Sc. Norm. Super. Pisa, Cl. Sci., Volume 5 (2006) no. 2, pp. 159-170

[12] Grove, Karsten; Ziller, Wolfgang Polar manifolds and actions, J. Fixed Point Theory Appl., Volume 11 (2012) no. 2, pp. 279-313

[13] Haefliger, André Groupoïdes d’holonomie et classifiants, Structure transverse des feuilletages (Toulouse, 1982) (Astérisque) Volume 116, Société Mathématique de France, 1984, pp. 70-97 | Zbl 0562.57012

[14] Hatcher, Allen Algebraic topology, Cambridge University Press, 2002, xii+544 pages

[15] Kapovitch, Vitali; Petrunin, Anton; Tuschmann, Wilderich Nilpotency, almost nonnegative curvature, and the gradient flow on Alexandrov spaces, Ann. Math., Volume 171 (2010) no. 1, pp. 343-373

[16] Lange, Christian Orbifolds from a metric viewpoint (2018) (https://arxiv.org/abs/1801.03472)

[17] Lytchak, Alexander Geometric resolution of singular Riemannian foliations, Geom. Dedicata, Volume 149 (2010), pp. 379-395 | Zbl 1207.53035

[18] Lytchak, Alexander On contractible orbifolds, Proc. Am. Math. Soc., Volume 141 (2013) no. 9, p. 3303-3304

[19] Lytchak, Alexander; Thorbergsson, Gudlaugur Curvature explosion in quotients and applications, J. Differ. Geom., Volume 85 (2010) no. 1, pp. 117-139

[20] Mendes, Ricardo A. E. Extending tensors on polar manifolds, Math. Ann., Volume 365 (2016) no. 3-4, pp. 1409-1424

[21] Mendes, Ricardo A. E.; Radeschi, Marco A slice theorem for singular Riemannian foliations, with applications, Trans. Am. Math. Soc., Volume 371 (2019) no. 7, pp. 4931-4949

[22] Molino, Pierre Riemannian foliations, Progress in Mathematics, Volume 73, Birkhäuser, 1988, xii+339 pages

[23] Münzner, Hans Friedrich Isoparametrische Hyperflächen in Sphären, Math. Ann., Volume 251 (1980) no. 1, pp. 57-71

[24] Radeschi, Marco Lecture notes on singular Riemannian foliations (https://www.marcoradeschi.com)

[25] Schwachhöfer, Lorenz J.; Tuschmann, Wilderich Metrics of positive Ricci curvature on quotient spaces, Math. Ann., Volume 330 (2004) no. 1, pp. 59-91

[26] Searle, Catherine; Wilhelm, Frederick How to lift positive Ricci curvature, Geom. Topol., Volume 19 (2015) no. 3, pp. 1409-1475

[27] Thurston, William P. Three-dimensional geometry and topology. Vol. 1, Princeton Mathematical Series, Volume 35, Princeton University Press, 1997, x+311 pages

[28] Yeager, Joseph Geometric and Topological ellipticity in cohomogeneity two (2012) (http://hdl.handle.net/1903/12662) (Ph. D. Thesis)