Almost non-negative curvature and rational ellipticity in cohomogeneity two
[Courbure presque positive et ellipticité rationelle en cohomogénéité deux]
Annales de l'Institut Fourier, Tome 69 (2019) no. 7, pp. 2921-2939.

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.

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.

Publié le :
DOI : 10.5802/aif.3340
Classification : 53C20, 55P62, 57S15, 58E10
Keywords: Almost Non-negative Curvature, Rational ellipticity, Morse Theory, Cohomogeneity
Mot clés : Courbure presque positive, Ellipticité rationelle, Théorie de Morse, Cohomogénéité
Grove, Karsten 1 ; Wilking, Burkhard 2 ; Yeager, Joseph 3

1 Department of Mathematics University of Notre Dame Notre Dame, IN 46556 (USA)
2 Department of Mathematics University of Münster Einsteinstrasse 62 48149 Münster (Germany)
3 Department of Mathematics Howard University 204 Academic Support Building B Washington, DC 20059 (USA)
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits
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Grove, Karsten; Wilking, Burkhard; Yeager, Joseph. Almost non-negative curvature and rational ellipticity in cohomogeneity two. Annales de l'Institut Fourier, Tome 69 (2019) no. 7, pp. 2921-2939. doi : 10.5802/aif.3340. https://aif.centre-mersenne.org/articles/10.5802/aif.3340/

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