Research

Given two closed manifolds B+ and B-, and disk bundles DB+ and DB- over B+ and B- respectively, assume that the boundaries of the disk bundles are diffeomorphic, say by a diffeomorphism f.  Then one can form a closed manifold M by gluing these two disk-bundles together along their boundaries using f.  The resulting manifold is called a double disk bundle.

Suppose M is a closed simply connected manifold.  Does it admit the structure of a double disk bundle?  We find obstructions at the level of both the rational and Z/2Z cohomology ring.  For example, if M is 6-dimensional, we show that H^*(M) must be torsion free with third rational Betti number at most 2.

Given two closed manifolds B+ and B-, and disk bundles DB+ and DB- over B+ and B- respectively, assume that the boundaries of the disk bundles are diffeomorphic, say by a diffeomorphism f.  Then one can form a closed manifold M by gluing these two disk-bundles together along their boundaries using f.  The resulting manifold is called a double disk bundle.

Suppose M is a closed simply connected manifold with the rational cohomology of a sphere.  If M is, in addition, highly connected, then we prove that the torsion in the cohomology ring must be cyclic.

If we replace the assumption that M is highly connected with the assumption that M is even-dimensional, we prove that M must, in fact, be homeomorphic to a sphere.

Along the way, we prove the following classification theorem for sphere bundles.   Suppose L -> B is a bundle whose fiber is an even dimensional sphere S^{2k}.  If L has the integral cohomology ring of a product of S^{2k} and another sphere, then either B is a homotopy sphere, or k = 1 and B is diffeomorphic to SU(3)/SO(3).  Conversely, in both of these situations, B has such a sphere bundle over it.

A version of the article is actually already available on the arxiv, but we are currently working on greatly expanding it before submitting.

A Riemannian manifold M is said to have positive kth-intermediate Ricci curvature, denoted Ric_k > 0,  if for any point p in M and any set {x, v_1,..., v_k} of k+1 orthonormal tangent vectors in TpM, that the sum sec(x, v_1) + sec(x, v_2) + ... + sec(x, v_k) is positive (where sec denotes the sectional curvature).  So, positive 1st-intermediate Ricci curvature just means positive sectional curvature and positive (dim M -1)th intermediate Ricci curvature just means positive Ricci curvature.

Apart from the known examples of positively curved Riemannian manifolds, there are just 3 known examples with Ric_2 > 0.  They are diffeomorphic to S^2 x S^2, S^2 x S^3, and S^3 x S^3.

We find infinitely more examples, including S^6 x S^7 and S^7 x S^7 and some inhomogeneous spaces.  Our examples are in dimension 10, 11, 12, 13, and 14.

In fact, we fully classify homogeneous spaces satisfying an certain condition which implies Ric_2 > 0.

This article contains an appendix by Michael Albanese with a proof of an unpublished result of Hirzebruch giving an obstruction to a 4k-dimensional manifold to admitting an almost complex structure.

Given a biquotient G // H of compact Lie groups H and G, one can construct a vector bundle over G // H by taking a representation V of H and considering the quotient space G x_H V of the diagonal action of H on G x V.  Is the tangent bundle T(G // H) always isomorphic to such a vector bundle?  For homogeneous spaces, the answer is well known to be "yes", but we find infinite families of examples which show that this is no longer true for biquotients, regardless of their presentation as a biquotient.

This requires structure results on how the class of biquotient vector bundles changes as the presentation of the biquotient changes.  We prove two such results:  pulling back to covers can only increase this class and the "reduced" form already contains the whole class.

We also find infinitely many examples where the answer depends on the presentation as a biquotient.

We show that in dimension up to 5, apart from CP^2 # CP^2, every simply connected biquotient has at least one presentation for which the tangent bundle is a biquotient bundle.

Eschenburg spaces are free circle quotients of the compact Lie group SU(3).  An infinite subfamily of them are known to admit a Riemannian metric with positive sectional curvature.   But what about the rest?

Kerin showed that Eschenburg's metric is almost positively curved (meaning that the set of points having a least one 2-plane of zero curvature has measure 0) for one example.  Wilking, using a different metric on SU(3), found one other example with almost positive curvature.  Prior to our work, these two examples were the only known Eschenburg spaces with almost positive curvature (except for those which admit metrics of strictly positive curvature.)

We show that, among the Eschenburg spaces which are not known to admit a metric of positive curvature, there are infinitely many examples which do admit a metric of almost positive curvature.  Our examples are precisely those who "natural" isometry groups acts with cohomogeneity <= 2, and include both Kerin's and Wilking's example.

We also completely characterize the curvature of all Eschenburg spaces equipped with Eschenburg's metric, confirming that Kerin's example is the unique one with almost positive curvature (which doesn't have strictly positive curvature.)

A manifold M is called a double disk bundle if it is diffeomorphic to the union of two disk bundles glued together along their common boundary.  The Double Soul Conjecture asserts that a simply connected closed Riemannian manifold of non-negative sectional curvature has the structure of a double disk bundle.

If one generalizes the conjecture by drops the hypothesis that M be simply connected, the previously, only a single counterexample was known, three dimensional Poincaré dodecahedral space.

We fully classify the 3-manifolds which are counterexamples to this generalized conjecture, finding infinitely many examples.

We also find an infinite family of flat examples which exist in arbitrarily large dimensions.

Suppose M is a closed simply connected manifold with the rational cohomology of a sphere of odd dimension.  Suppose a compact Lie group G acts on M with a codimension one orbit.  What can we say about G, M, and the G action on M?  The main result of this article is that we can completely recover the diffeomorphism type of M, and that we can recover G and the action up to so-called normal extensions and up to cover.

M must be diffeomorphic to a sphere, the Wu manifold SU(3)/SO(3), a Brieskorn variety with certain parameters, an infinite family in dimension 7, or one of two homogeneous rational 11-spheres.  Conversely, each of these manifolds does in fact admit such a cohomogeneity one action.

The case where M is even dimensional is Paper (9) below.

Given two closed manifolds B+ and B-, and disk bundles DB+ and DB- over B+ and B- respectively, assume that the boundaries of the disk bundles are diffeomorphic, say by a diffeomorphism f.  Then one can form a closed manifold M by gluing these two disk-bundles together along their boundaries using f.  The resulting manifold is called a double disk bundle.  In this paper, we address the following question.  If M is a closed simply connected manifold of dimension at most 7, and it admits a double disk bundle decomposition, what can be said about M?  We fully answer this question in dimension at most 5, proving a diffeomorphism classification of such manifolds.

In dimension 6, we show that if the third rational Betti number of M is non-zero, then M must be diffeomorphic to S^3 x S^3.  In addition, we show that if the second and third rational Betti numbers of M are zero, then M must be diffeomorphic to S^6.

In dimension 6 and 7, we also show that rational ellipticity can be read off from the Betti numbers in the following sense:  If M admits a double disk bundle decomposition, then M is rationally elliptic if and only if it has the rational Betti numbers of a rationally elliptic space.

Given a biquotient G // H of compact Lie groups H and G, one can construct a vector bundle over G // H by taking a representation V of H and considering the quotient space G x_H V of the diagonal action of H on G x V.   Given such a biquotient vector bundle, can one find another biquotient vector bundle G x_H W for which the Whitney sum of the two vector bundles is trivial?  For homogeneous spaces, the answer was previously known to be "yes".  We find infinite families of biquotients for which the answer is "no".

We identify a property, which we call Property (*), involving the cup product H^2 x H^2 -> H^4 which guarantees a very strong version of the answer "no".  Specifically, if a biquotient has the form G // T for a torus T and it has Property (*), we show the only biquotient vector bundles with biquotient vector bundle inverses are trivial.

As part of an undergraduate research project, we analyzed a certainly family of biquotients of the form SU(5)/Sp(2)S^1 called Bazaikin spaces.  The embedding of S^1 is parameterized by five integers, and for some special choices of parameters, the corresponding spaces were previously known to admit Riemannian metrics of positive sectional curvature at every point.  More precisely, beginning with a single metric on SU(5), the induced metric on each of these special Bazaikin spaces is positively curved.  For a general Bazaikin space, the induced metric is always non-negatively curved, but the full classification of "how much" positive curvature each example had was incomplete.  

We showed:  there is a unique (up to equivalence) choice of parameters for which the corresponding Bazaikin space has a zero-curvature plane at every point.  Every other Bazaikin space has at least one point at which all two-planes are positively curved.  There is a unique (up to equivalence) choice of parameters for which the corresponding Bazaikin space is almost positively curved (in the sense of measure theory) but not positively curved.  Moreover, the example with zero-curvature planes everywhere and the almost positively curved example are diffeomorphic.

A Riemannian manifold is said to be almost positively curved if the collection of points for which all two-planes have positive sectional curvature is open and dense.  Which manifolds admit such metrics?  We show the homogeneous space Sp(3)/Sp(1)Sp(1) admits a cohomogeneity two almost positively curved metric.  Moreover, with respect to this metric, there are two distinct free isometric S^1 actions, and the resuling quotient spaces also admit almost positively curved metrics. 

The two circle actions extend to free isometric actions by SU(2).  One of the SU(2) quotients was already known to admit a metric of strictly positive sectional curvature, while the other was previously known to admit an almost positively curved metric.

If a compact Lie group G acts on a smooth manifold M with a co-dimension one orbit, then the action is said to be cohomogeneity one.  Suppose M is a closed simply connected manifold, even dimensional, and has rational cohomology ring generated by a single element. If, in addition, M supports a cohomogeneity one action, then what can be said about M, G, and the action?  We completely classify the possibilities, showing that M must be diffeomorphic to a compact rank one symmetric space, a real Grassmannian of two-planes in R^{2k+1}, or to the rank two symmetric space G_2 / SO(4).  Moreover, in the first two cases, the G action is equivalent to a linear action (and the action has been previously classified), and when M is G_2 / SO(4), G = SU(3) and the action is equivalent to left multiplication by G.

We also classify the diffeomorphism type of M if H^*(M;Q) is rationally 4-periodic.  In this instance, the condition of being rationally 4-periodic and not singly generated is equivalent to H^*(M;Q) being isomorphic to H^*(S^2 x HP^k; Q).  In particular, we show that M is diffeomorphic to the product S^2 x HP^k, to the unique non-trivial HP^k bundle over S^2, or to S^2 x (G_2 / SO(4) ).  Conversely, we show that each of these spaces admits a cohomogeneity one action.

This paper is a mixture of general topology and analysis.  Given a possibly divergent series \sum a_n, with each a_n a positive real number, one can associate to it a topology on the natural numbers N = {1,2,...} as follows.  Given a proper subset A of N, we declare A to be closed if \sum_{n \in A} a_n converges.  Then one can ask how the topological properties on N relate to the analytic properties of a_n.  We characterize the notion of compactness, connectedness, separation axioms, etc. in terms of analytic properties of the series.  For example, we show the series converges if and only if the resulting topology on N is discrete.

We also study continuous maps between these kinds of topological spaces.  In particular, if N_p corresponds to the p-series \sum 1/n^p, we show that for 0 < p < q < 1, that the only continuous functions from N_p to N_q are constants.

A Riemannian manifold is said to be almost positively curved if the collection of points for which all two-planes have positive sectional curvature is open and dense.  Which manifolds admit such metrics?  A conjecture of Hopf predicts that a compact symmetric space of rank larger than one should not admit a metric of strictly positive curvature.  We find the first example of an irreducible compact symmetric space of rank larger than one which admits an almost positively curved metric: the Grassmannian of two-planes in R^7.  If one drops "irreducible", there are just two more examples, both due to Wilking.

We also determine the topology of the set of points which have at least one zero-curvature plane: it is a union of a copy of the Grassmannian of two-planes in R^6 together with a copy of CP^2 x S^5, and these two submanifolds intersect in a copy of W_(1,-1,0), the unique Aloff-Wallach space which does not admit a homogeneous metric of positive sectional curvature.

A Riemannian manifold is said to be quasi-positively curved if it has non-negative sectional curvature everywhere and if there is exists a point for which all two-planes at that point are positively curved. Which manifolds admit such a metric?  In Paper (2) below, we had shown that several biquotients of the form Sp(3)/Sp(1)Sp(1) admit such metrics.  It was clear that the techniques could, in principle, be applied to one more biquotient of that form, but that the calculations would likely be more complicated.  In this paper, we completed those calculations.

A biquotient is any manifold which is diffeomorphic to the quotient of a Riemannian homogeneous space by a free isometric action.  A topological space X is called rationally 4-periodic if H^4( X ; Q ) contains an element e for which cupping with e induces isomorphism H^k( X ; Q ) -> H^{k+4}( X ; Q ) for "most" values of k.   Which closed simply connected biquotients are rationally 4-periodic?  This paper gives a complete characterization of these biquotients:  either H^*( X ; Q) is singly generated (and hence, previously classified), or X is diffeomorphic to S^3 x S^3, or H^*( X ; Q ) is isomorphic to H^*( S^n x HP^m ; Q) for n = 2 or n = 3.

Moreover, when n = 3, we show that in each fixed dimension, there are only finitely many possible actions, and we obtain a complete diffeomorphism classification:  all such examples are S^3 bundles over one of HP^m, G_2 / SO(4), or the biquotient Delta SU(2) \ Spin(24+1) / Spin(4k-1).

When n = 2, in each dimension there are infinitely many actions under consideration.  Nonetheless, we show the set of diffeomorphism types in each dimension is finite.  We are, unfortunately, not able to find a complete diffeomorphism classification in this case.

A biquotient is any manifold which is diffeomorphic to the quotient of a Riemannian homogeneous space by a free isometric action.  Which manifolds have the structure of a biquotient?  We answer this question for simply connected closed manifolds of dimension 6 and 7.   We obtain a full diffeomorphism classification in dimension 6, including all possible ways of writing a biquotient in so-called reduced form.  In dimension 7, we are able to classify all possible actions, but only offer partial results in the diffeomorphism classification.

A biquotient is any manifold which is diffeomorphic to the quotient of a Riemannian homogeneous space by a free isometric action.   Biquotients have an alternate description as the quotient of a Lie group G by a subgroup H of G x G, acting via left and right multiplication.  Supposing that G is a compact Lie group of rank 3 and H has a cover isomorphic to SU(2)^2,  what are the possible biquotients?  This paper gives a full classification.

By far the most difficult case is when G = Spin(7).  Here, we use Clifford algebras to embed Spin(7) into SO(8), and then carry about the analysis there.

A biquotient is any manifold which is diffeomorphic to the quotient of a Riemannian homogeneous space by a free isometric action.   A Riemannian manifold is said to be quasi-positively curved if it has non-negative sectional curvature everywhere and if there is exists a point for which all two-planes at that point are positively curved.   Which biquotients have such a metric?  This paper shows that if one considers biquotients of the form Sp(3) // Sp(1)^2, then there are precisely 19 possible quotients, and that at least 11 of them admit such a metric.  Prior to this work, only 3 of the 11 were known to admit such a metric.

A biquotient is any manifold which is diffeomorphic to the quotient of a Riemannian homogeneous space by a free isometric action.  Supposing M is a closed simply connected manifold of dimension at most 5, can M be given the structure of a biquotient?  In how many different ways?  We completely solve this problem, showing that M must either be diffeomorphic to a symmetric space of rank at most 2, or that M must be diffeomorphic to CP^2 # \pm CP^2, or to the unique non-trivial linear S^3-bundle over S^2.  Morever, we find all possible biquotient structures on each of these manifolds.