arXiv research

Highly connected orbifolds are rare but exist.

New metrics found for certain highly connected manifolds.

The paper finds manifold structures on complex spaces.

Study shows non-trivial moduli spaces for certain highly connected manifolds.

We prove the existence of Sasakian metrics with positive Ricci curvature on certain highly connected odd dimensional manifolds. In particular, we show that manifolds homeomorphic to the 2k-fold connected sum of S^{2n-1} x S^{2n} admit Sasakian metrics with positive Ricci curvature for all k. Furthermore, a formula for …

We introduce a new and rich class of graph coloring manifolds via the Hom complex construction of Lovasz. The class comprises examples of Stiefel manifolds, series of spheres and products of spheres, cubical surfaces, as well as examples of Seifert manifolds. Asymptotically, graph coloring manifolds provide examples of…

Constructs 7-manifolds with SO(3) invariant non-negative curvature.

Study mapping class groups of complex manifolds, identifying key subgroups and abelianizations.

The paper classifies Poincaré complexes as topological manifolds.

We show that the space of nonpositively curved metrics of a negatively curved manifold is highly non connected.

The paper constructs homotopically rigid spaces for applications in algebraic topology.

For $k \ge 2,$ let $M^{4k-1}$ be a $(2k{-}2)$-connected closed manifold. If $k \equiv 1$ mod $4$ assume further that $M$ is $(2k{-}1)$-parallelisable. Then there is a homotopy sphere $Σ^{4k-1}$ such that $M \sharp Σ$ admits a Ricci positive metric. This follows from a new description of these manifolds as the boundarie…

The study finds infinitely many 7-manifolds with non-negative curvature but not homotopy equivalent to bundles.

An immersion of a compact manifold is tight if it admits the minimal total absolute curvature over all immersions of the manifold. A prominent result in the study of minimal total absolute curvature immersions is the theorem of Chern and Lashof, which characterizes minimal total absolute curvature immersions, and tight…

We show that the space of negatively curved metrics of a closed negatively curved Riemannian $n$-manifold, $n\geq 10$, is highly non-connected.

We study and in some cases classify highly connected manifolds which admit a Riemannian metric with positive $p$-curvature. The $p$-curvature was defined and studied by the second author. It turns out that positivity of $p$-curvature could be preserved under surgeries of codimension at least $p+3$. This gives a key to …

We prove a homological stability theorem for moduli spaces of high-dimensional, highly connected manifolds, with respect to forming the connected sum with the product of spheres $S^{p}\times S^{q}$, for $p < q < 2p - 2$. This result is analogous to recent results of S. Galatius and O. Randal-Williams regarding the homo…

Study bundles over surfaces with specific fibers, determining characteristic numbers and obstructions.

Let P be a closed smooth (4j-2)-connected 8j-manifold. We complete Wilkens' classification of the manifolds P for j = 1,2 and give an alternative proof to Wall's classification of the manifolds for j > 2. The Hopf-invariant-one dimensions (j=1,2) are characteristed by the fact that the quadratic linking functions which…

Proves boundaries of certain manifolds, solving long-standing questions.

Even-dimensional simply connected manifolds that are rational homology spheres and double disk bundles are homeomorphic to spheres.

We prove that the space of complete, finite volume, pinched negatively curved Riemannian metrics on a smooth high-dimensional manifold is either empty or it is highly non-connected, provided their behavior at infinity is similar.

Study cyclic group actions on specific high-dimensional manifolds.

New spaces help connect manifold structures on equivariant Poincaré spaces.

We continue our study of contact structures on manifolds of dimension at least five using complex surgery theory. We show that in each dimension 2q+1 > 3 there are 'maximal' almost contact manifolds to which there is a Stein cobordism from any other (2q+1)-dimensional contact manifold. We show that the product M x S^2 …

Let X be a finite CW complex or compact Lipschitz neighborhood retract with universal cover Z; let M be a compact orientable manifold of dimension at least 2 and nonempty boundary. We establish the existence of an isoperimetric profile for functions from M to Z, in the metric and cellular senses, and show that they are…

New technique identifies submanifolds in symmetric spaces based on Ricci curvature.

We prove that the braided Thompson's groups $V_{\rm br}$ and $F_{\rm br}$ are of type $F_\infty$, confirming a conjecture by John Meier. The proof involves showing that matching complexes of arcs on surfaces are highly connected. In an appendix, Zaremsky uses these connectivity results to exhibit families of subgroups …

In this paper, we propose new conditions guaranteeing that the trajectories of a mechanical control system can track any curve on the configuration manifold. We focus on systems that can be represented as forced affine connection control systems and we generalize the sufficient conditions for tracking known in the lite…

All parabolic geometries, i.e. Cartan geometries with homogeneous model a real generalized flag manifold, admit highly interesting classes of distinguished curves. The geodesics of a projective class of connections on a manifold, conformal circles on conformal Riemannian manifolds, and Chern--Moser chains on CR--manifo…

A clustering algorithm partitions a set of data points into smaller sets (clusters) such that each subset is more tightly packed than the whole. Many approaches to clustering translate the vector data into a graph with edges reflecting a distance or similarity metric on the points, then look for highly connected subgra…

We classify compact 2-connected homogeneous spaces with the same rational cohomology as a product of spheres. This classification relies on spectral sequences, homotopy theory, and representation theory. We then apply this classification to two geometric problems. The first problem is the classification of all isoparam…

Proves metrics with positive intermediate Ricci curvature on complex manifolds.

We show that a certain geometric property, the QSF introduced by S. Brick and M. Mihalik, is universally true for {\ibf all} finitely presented groups $Γ$. One way of defining this property is the existence of a smooth compact manifold $M$ with $π_1 M = Γ$, such that $\tilde M$ is geometrically simply-connected ({\it i…

We establish the existence, finiteness, and uniqueness up to scaling of various isoperimetric profiles of a group, in all dimensions. We also show that these profiles all coincide in dimensions 4 and higher; in particular, the nth Dehn function is equal to FV^{n+1} for n at least 3. Even for dimension 3, there is signi…

Highly twisted knots can be geometrically triangulated.

New proof of index theorem for topological manifold bundles.

New classification and realization results for 3D and higher PD complexes with highly connected covers.

The main result is that an s-cobordism (topological or smooth) of 4-manifolds has a product structure outside a ``core'' sub s-cobordism. These cores are arranged to have quite a bit of structure, for example they are smooth and abstractly (forgetting boundary structure) diffeomorphic to a standard neighborhood of a 1-…

Parallel spinors help characterize G2* structures and isotropic forms.

New methods train neural networks without changing weights, achieving similar or higher performance.

The paper studies connectivity properties of Morse complexes as simplicial complexes grow.

Homological stability for sequences of groups is often proved by studying the spectral sequence associated to the action of a typical group in the sequence on a highly-connected simplicial complex whose stabilizers are related to previous groups in the sequence. In the case of mapping class groups of manifolds, suitabl…

We study the complex of partial bases of a free group, which is an analogue for $\Aut(F_n)$ of the curve complex for the mapping class group. We prove that it is connected and simply connected, and we also prove that its quotient by the Torelli subgroup of $\Aut(F_n)$ is highly connected. Using these results, we give a…

The economical world consists of a highly interconnected and interdependent network of firms. Here we develop temporal and structural network tools to analyze the state of the economy. Our analysis indicates that a strong clustering can be a warning sign. Reduction in diversity, which was an essential aspect of the dyn…

Extends Killing superalgebras to higher dimensions and signatures.

Study solutions to metrizability equation on manifolds.

Let p be an odd regular prime, and assume that the Lichtenbaum-Quillen conjecture holds for K(Z[1/p]) at p. Then the p-primary homotopy type of the smooth Whitehead spectrum Wh(*) is described. A suspended copy of the cokernel-of-J spectrum splits off, and the torsion homotopy of the remainder equals the torsion homoto…