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We classify simply connected, closed cohomogeneity one manifolds with singly generated or4-periodic rational cohomology and positive Euler characteristic. 

Extending existing work in small dimensions, Dessai computed the Euler characteristic, signature, and elliptic genus for [Formula: see text]-manifolds of positive sectional curvature in the presence of torus symmetry. He also computes the diffeomorphism type by restricting his results to classes of manifolds known to admit non-negative curvature, such as biquotients. The first part of this paper extends Dessai’s calculations to even dimensions up to [Formula: see text]. In particular, we obtain a first characterization of the Cayley plane in such a setting. The second part studies a closely related family of manifolds called positively elliptic manifolds, and we prove a conjecture of Halperin in this context for dimensions up to [Formula: see text] or Euler characteristics up to [Formula: see text]. 

A rational projective plane ([Formula: see text]) is a simply connected, smooth, closed manifold [Formula: see text] such that [Formula: see text]. An open problem is to classify the dimensions at which such a manifold exists. The Barge–Sullivan rational surgery realization theorem provides necessary and sufficient conditions that include the Hattori–Stong integrality conditions on the Pontryagin numbers. In this paper, we simplify these conditions and combine them with the signature equation to give a single quadratic residue equation that determines whether a given dimension supports a [Formula: see text]. We then confirm the existence of a [Formula: see text] in two new dimensions and prove several non-existence results using factorization of the numerators of the divided Bernoulli numbers. We also resolve the existence question in the Spin case, and we discuss existence results for the more general class of rational projective spaces. 

In this paper we study sectional curvature bounds for Riemannian manifolds with density from the perspective of a weighted torsion-free connection introduced recently by the last two authors. We develop two new tools for studying weighted sectional curvature bounds: a new weighted Rauch comparison theorem and a modified notion of convexity for distance functions. As applications we prove generalizations of theorems of Preissman and Byers for negative curvature, the (homeomorphic) quarter-pinched sphere theorem, and Cheeger’s finiteness theorem. We also improve results of the first two authors for spaces of positive weighted sectional curvature and symmetry. 

A famous conjecture of Hopf states that $$\mathbb{S}^{2}\times \mathbb{S}^{2}$$ does not admit a Riemannian metric with positive sectional curvature. In this article, we prove that no manifold product $$N\times N$$ can carry a metric of positive sectional curvature admitting a certain degree of torus symmetry. 

We prove an obstruction at the level of rational cohomology to the existence of positively curved metrics with large symmetry rank. The symmetry rank bound is logarithmic in the dimension of the manifold. As one application, we provide evidence for a generalized conjecture of H. Hopf, which states that no symmetric space of rank at least two admits a metric with positive curvature. Other applications concern product manifolds, connected sums, and manifolds with nontrivial fundamental group.