In this note we show that every (real or complex) vector bundle over a compact rank one symmetric space carries, after taking the Whitney sum with a trivial bundle of sufficiently large rank, a metric with nonnegative sectional curvature. We also examine the case of complex vector bundles over other manifolds, and give upper bounds for the rank of the trivial bundle that is necessary to add when the base is a sphere.

Abstract. In this note we consider submersions from compact manifolds, homotopy equivalent to the Eschenburg or Bazaikin spaces of positive curvature. We show that if the submersion is nontrivial, the dimension of the base is greater than the dimension of the fiber. Together with previous results, this proves the Petersen-Wilhelm Conjecture for all the known compact manifolds with positive curvature.

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Abstract. We bound the dimension of the fiber of a Riemannian submersion from a positively curved manifold in terms of the dimension of the base of the submersion and either its conjugate radius or the length of its shortest closed geodesic.

We show that the moduli space of metrics of nonnegative sectional curvature on every homotopy RP 5 has infinitely many path components. We also show that in each dimension 4k + 1 there are at least 2 2k homotopy RP 4k+1 s of pairwise distinct oriented diffeomorphism type for which the moduli space of metrics of positive Ricci curvature has infinitely many path components. Examples of closed manifolds with finite fundamental group with these properties were known before only in dimensions 4k + 3 ≥ 7. Outline of proofsThe goal of this section is to give an outline of the proofs of Theorem A and Theorem B, all the references and details can be found in the following sections.

Positive $$k\mathrm{th}$$ k th -intermediate Ricci curvature on a Riemannian n-manifold, to be denoted by $${{\,\mathrm{Ric}\,}}_k>0$$ Ric k > 0 , is a condition that interpolates between positive sectional and positive Ricci curvature (when $$k =1$$ k = 1 and $$k=n-1$$ k = n - 1 respectively). In this work, we produce many examples of manifolds of $${{\,\mathrm{Ric}\,}}_k>0$$ Ric k > 0 with k small by examining symmetric and normal homogeneous spaces, along with certain metric deformations of fat homogeneous bundles. As a consequence, we show that every dimension $$n\ge 7$$ n ≥ 7 congruent to $$3\,{{\,\mathrm{mod}\,}}4$$ 3 mod 4 supports infinitely many closed simply connected manifolds of pairwise distinct homotopy type, all of which admit homogeneous metrics of $${{\,\mathrm{Ric}\,}}_k>0$$ Ric k > 0 for some $$k<n/2$$ k < n / 2 . We also prove that each dimension $$n\ge 4$$ n ≥ 4 congruent to 0 or $$1\,{{\,\mathrm{mod}\,}}4$$ 1 mod 4 supports closed manifolds which carry metrics of $${{\,\mathrm{Ric}\,}}_k>0$$ Ric k > 0 with $$k\le n/2$$ k ≤ n / 2 , but do not admit metrics of positive sectional curvature.

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.

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