We classify all compact simply connected biquotients of dimension 6 and 7. For each 6-dimensional biquotient, all pairs of groups (G, H) and homomorphisms H → G × G giving rise to it are classified.

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An n-dimensional manifold M is said to be rationally 4-periodic if there is an element e ∈ H 4 (M ; Q) with the property that cupping with e, · ∪ e : H * (M ; Q) → H * +4 (M ; Q) is injective for 0 < * ≤ dim M − 4 and surjective when 0 ≤ * < dim M −4. We classify all compact simply connected biquotients which are rationally 4-periodic. In addition, we show that if a simply connected rationally elliptic CW-complex X of dimension at least 6 is rationally 4-periodic, then the cohomology ring is either singly generated, or X is rationally homotopy equivalent to S 2 × HP n , S 3 × HP n , or S 3 × S 3 . arXiv:1605.07694v3 [math.DG]

A Riemannian manifold is said to be almost positively curved if the sets of points for which all 2-planes have positive sectional curvature is open and dense. We show that the Grassmannian of oriented 2planes in R 7 admits a metric of almost positive curvature, giving the first example of an almost positively curved metric on an irreducible compact symmetric space of rank greater than 1. The construction and verification rely on the Lie group G 2 and the octonions, so do not obviously generalize to any other Grassmannians. arXiv:1707.07590v1 [math.DG]

In previous work, the second author and others have found conditions on a homogeneous space G/H which imply that, up to stabilization, all vector bundles over G/H admit Riemannian metrics of non-negative sectional curvature. One important ingredient of their approach is Segal's result that the set of vector bundles of the form G×H V for a representation V of H contains inverses within the class. We show that this approach cannot work for biquotients G/ /H , where we consider vector bundles of the form G ×H V . We call such vector bundles biquotient bundles. Specifically, we show that in each dimension n ≥ 4 except n = 5 , there is a simply connected biquotient of dimension n with a biquotient bundle which does not contain an inverse within the class of biquotient bundles. In addition, we show that for n ≥ 6 except n = 7 , there are infinitely many homotopy types of biquotients with the property that no non-trivial biquotient bundle has an inverse. Lastly, we show that every biquotient bundle over every simply connected biquotient M n = G/ /H with G simply connected and with n ∈ {2, 3, 5} has an inverse in the class of biquotient bundles.2010 Mathematics Subject classification. Primary 53C20.

Suppose φ 3 : Sp(1) → Sp(2) denotes the unique irreducible 4dimensional representation of Sp(1) = SU (2) and consider the twoWe show that the biquotient H 1 \Sp(3)/H 2 admits a quasi-positively curved Riemannian metric. arXiv:1609.07216v1 [math.DG]

We classify simply connected, closed cohomogeneity one manifolds with singly generated or 4-periodic rational cohomology and positive Euler characteristic.We denote by QP n k any smooth, simply connected, closed manifold whose rational cohomology is isomorphic to the truncated polynomial algebra Q[x]/(x n+1 ) where the generator x has degree k. Note that such a manifold is a rational sphere or point if k is odd by the graded commutativity of the cup product. If k is even, then a QP n k has even dimension kn and positive Euler characteristic n + 1.Prototypical examples are simply connected, closed manifolds with the rational cohomology (equivalently, rational homotopy type) of a compact rank one symmetric space: a rational sphere is a QP 1 k , a rational CP n is a QP n 2 , a rational HP n is a QP n 4 , and a rational Cayley plane is a QP 2 8 . We call the parameters n and k standard if they correspond to a rank one symmetric space. The classification of parameters (n, k) for which a QP n k exists is reduced by way of the Barge-Sullivan rational realization theorem to a number theoretic problem (see Su [Su14]), but this problem is hard and there is no classification (cf. [FS16,KS]).This paper is motivated in part by the attempt to find highly symmetric models for QP n k with non-standard parameters. It is known that no such manifold admits a homogeneous or biquotient structure (see and Totaro [Tot02]). Our first theorem is a similar, negative result:

A Riemannian manifold is called almost positively curved if the set of points for which all 2-planes have positive sectional curvature is open and dense. We find three new examples of almost positively curved manifolds: Sp(3)/Sp(1) 2 , and two circle quotients of Sp(3)/Sp(1) 2 . We also show the quasi-positively curved metric of Tapp [26] on Sp(n + 1)/Sp(n − 1)Sp( 1) is not almost positively curved if n ≥ 3.