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 
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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, to appear, 19 p.

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