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Topological and equivariant rigidity in t e presence of lower curvature bounds

The Grove program proposes to classify Riemannian manifolds of positive or non-negative sectional curvature with a "large" isometry group.

The simplest groups that one may consider acting on a manifold are compact abelian Lie groups, i.e. tori. Thanks to the work of several authors fairly complete classification results are available, provided that that either the manifold, or the torus acting upon it, has sufficiently large dimension. For the action of a circle and a 2-torus in dimensions 5 and 6 the usual methods have so far failed to yield topological and equivariant classification results.

Taking as departure point the Grove program and the well-developed theory of cohomological methods for smooth torus actions on smooth manifolds, the project aims, on the one hand, at applying and developing equivariant topological methods in the context of Riemannian manifolds with a lower sectional curvature bound and, on the other hand, at studying closed Alexandrov spaces of cohomogeneity one. The primary goals are, respectively, to obtain a topological and equivariant classification of closed, simply connected 6-manifolds with an effective, isometric 2-torus action, and to classify closed, positively curved Alexandrov spaces of cohomogeneity one.

## Publications

In this work, it is shown that a simply-connected, rationally-elliptic torus orbifold is equivariantly rationally homotopy equivalent to the quotient of a product of spheres by an almost-free, linear torus action, where this torus has rank equal to the number of odd-dimensional spherical factors in the product. As an application, simply-connected, rationally-elliptic manifolds admitting slice-maximal torus actions are classified up to equivariant rational homotopy. The case where the rational-ellipticity hypothesis is replaced by non-negative curvature is also discussed, and the Bott Conjecture in the presence of a slice-maximal torus action is proved.

 Journal Int. Math. Res. Not. IMRN Volume 18 Pages 5786--5822 Link to preprint version Link to published version

In this paper, we study smooth, semi-free actions on closed, smooth, simply connected manifolds, such that the orbit space is a smoothable manifold. We show that the only simply connected 5-manifolds admitting a smooth, semi-free circle action with fixed-point components of codimension 4 are connected sums of $$S^3$$-bundles over $$S^2$$. Furthermore, the Betti numbers of the 5-manifolds and of the quotient 4-manifolds are related by a simple formula involving the number of fixed-point components. We also investigate semi-free $$S^3$$ actions on simply connected 8-manifolds with quotient a 5-manifold and show, in particular, that the Pontrjagin classes, the $$\hat A$$ -genus and the signature of the 8-manifold must all necessarily vanish.

In this article, a six-parameter family of highly connected 7-manifolds which admit an SO(3)-invariant metric of non-negative sectional curvature is constructed and the Eells-Kuiper invariant of each is computed. In particular, it follows that all exotic spheres in dimension 7 admit an SO(3)-invariant metric of non-negative curvature.

An upper bound is obtained on the rank of a torus which can act smoothly and effectively on a smooth, closed, simply connected, rationally elliptic manifold. In the maximal-rank case, the manifolds admitting such actions are classified up to equivariant rational homotopy type.

Let $$M^n, n \in \{4,5,6\}$$, be a compact, simply connected n-manifold which admits some Riemannian metric with non-negative curvature and an isometry group of maximal possible rank. Then any smooth, effective action on $$M^n$$ by a torus $$T^{n-2}$$ is equivariantly diffeomorphic to an isometric action on a normal biquotient. Furthermore, it follows that any effective, isometric circle action on a compact, simply connected, non-negatively curved four-dimensional manifold is equivariantly diffeomorphic to an effective, isometric action on a normal biquotient.

 Journal Math. Z. Volume 276 Pages 133--152 Link to preprint version Link to published version
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## Team Members

Dr. Fernando Galaz García