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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 a previous work, a six-parameter family of highly connected 7-manifolds which admit an $\mathrm{SO}(3)$-invariant metric of non-negative sectional curvature was constructed. Each member of this family is the total space of a Seifert fibration with generic fibre $\mathbb S^3$ and, in particular, has the cohomology of an $\mathbb S^3$-bundle over $\mathbb S^4$. In the present article, the linking form of these manifolds is computed and used to demonstrate that the family contains infinitely many manifolds which are not even homotopy equivalent to an $\mathbb S^3$-bundle over $\mathbb S^4$, the first time that any such spaces have been shown to admit non-negative sectional curvature.

In this short note we observe the existence of free, isometric actions of finite cyclic groups on a family of 2-connected 7-manifolds with non-negative sectional curvature. This yields many new examples including fake, and possible exotic, lens spaces.

We show that, for each $n\geqslant 1$, there exist infinitely many spin and non-spin diffeomorphism types of closed, smooth, simply-connected $(n+4)$-manifolds with a smooth, effective action of a torus $T^{n+2}$ and a metric of positive Ricci curvature invariant under a $T^{n}$-subgroup of $T^{n+2}$. As an application, we show that every closed, smooth, simply-connected $5$- and $6$-manifold admitting a smooth, effective torus action of cohomogeneity two supports metrics with positive Ricci curvature invariant under a circle or $T^2$-action, respectively.

 Journal Proc. Amer. Math. Soc. Volume In press. Link to preprint version

We obtain a Central Limit Theorem for closed Riemannian manifolds, clarifying along the way the geometric meaning of some of the hypotheses in Bhattacharya and Lin's Omnibus Central Limit Theorem for Fréchet means. We obtain our CLT assuming certain stability hypothesis for the cut locus, which always holds when the manifold is compact but may not be satisfied in the non-compact case.

Let $(M,g)$ be a smooth Riemannian manifold and $G$ a compact Lie group acting on $M$ effectively and by isometries. It is well known that a lower bound of the sectional curvature of $(M,g)$ is again a bound for the curvature of the quotient space, which is an Alexandrov space of curvature bounded below. Moreover, the analogous stability property holds for metric foliations and submersions.

The goal of the paper is to prove the corresponding stability properties for synthetic Ricci curvature lower bounds. Specifically, we show that such stability holds for quotients of $RCD^{*}(K,N)$-spaces, under isomorphic compact group actions and more generally under metric-measure foliations and submetries. An $RCD^{*}(K,N)$-space is a metric measure space with an upper dimension bound $N$ and weighted Ricci curvature bounded below by $K$ in a generalized sense. In particular, this shows that if $(M,g)$ has Ricci curvature bounded below by $K\in \mathbb{R}$ and dimension $N$, then the quotient space is an $RCD^{*}(K,N)$-space. Additionally, we tackle the same problem for the $CD/CD^*$ and $MCP$  curvature-dimension conditions.

We provide as well geometric applications which include: A generalization of Kobayashi's Classification Theorem of homogenous manifolds to $RCD^{*}(K,N)$-spaces with \emph{essential minimal dimension} $n\leq N$; a structure theorem for $RCD^{*}(K,N)$-spaces admitting actions by \emph{large (compact) groups}; and geometric rigidity results for orbifolds such as Cheng's Maximal Diameter and Maximal Volume Rigidity Theorems.

Finally, in two appendices  we apply the methods of the paper to study quotients by isometric group actions  of discrete spaces  and of (super-)Ricci flows.

 Journal J. Funct. Anal. Publisher Elsevier Volume 275 Pages 1368-1446 Link to preprint version Link to published version

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.

Alexandrov spaces are complete length spaces with a lower curvature bound in the triangle comparison sense. When they are equipped with an effective isometric action of a compact Lie group with one-dimensional orbit space they are said to be of cohomogeneity one. Well-known examples include cohomogeneity-one Riemannian manifolds with a uniform lower sectional curvature bound; such spaces are of interest in the context of non-negative and positive  sectional curvature.  In the present article we classify closed, simply-connected  cohomogeneity-one Alexandrov spaces in dimensions $5$, $6$ and  $7$. This yields, in combination with previous results for manifolds and Alexandrov spaces, a complete classification of  closed, simply-connected  cohomogeneity-one Alexandrov spaces in dimensions at most $7$.

We prove that sufficiently collapsed, closed and irreducible three-dimensional Alexandrov spaces are modeled on one of the eight three-dimensional Thurston geometries. This extends a result of Shioya and Yamaguchi, originally formulated for Riemannian manifolds, to the Alexandrov setting.

 Journal Indiana Univ. Math. J. Volume In press. Link to preprint version

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