arXiv research

Compact 4-manifolds can be described with a small set of data.

Minimal topology on surface homeomorphisms proven.

The compact-open topology is minimal on diffeomorphism and homeomorphism groups of most smooth manifolds.

We show that any homomorphism from the homeomorphism group of a compact 2-manifold, with the compact-open topology, or equivalently, with the topology of uniform convergence, into a separable topological group is automatically continuous.

Compactness proven for CMC surfaces with bounded topology and boundary length.

We conjecture that the automorphism group of a topological parallelism on real projective 3-space is compact. We prove that at least the identity component of this group is, indeed, compact.

We survey results on compact Clifford-Klein forms of homogeneous spaces, with a focus on recent contributions and organized around approaches via topology, geometry and dynamics. In addition, we survey results on moduli spaces of compact forms.

Strong rigidity proven for non-compact surfaces.

Let $G$ be a compact Lie group. (Compact) topological $G$-manifolds have the $G$-homotopy type of (finite-dimensional) countable $G$-CW complexes (2.5). This partly generalizes Elfving's theorem for locally linear $G$-manifolds [Elf96], wherein the Lie group $G$ is linear (such as compact).

Study characterizes compact Einstein-type manifolds with boundary.

Study confirms Chern's conjecture on compact Hessian manifolds and classifies their topologies.

Study string topology on symmetric spaces, showing non-triviality and nilpotency results.

Study topological properties of foliations induced by closed 1-forms on orbifolds.

We prove that there does not exist any connected topological proper loop homeomorphic to a quasi-simple Lie group and having a compact Lie group as the group topologically generated by its left translations. Moreover, any connected topological loop homeomorphic to the 7-sphere and having a compact Lie group as the grou…

Study classifies 3D Hessian manifolds, proving their topology.

Study finds limits on curvature and shape of certain spacetimes.

We develop the basic topological properties of compact polygons, i.e. of compact topological Tits buildings of rank two. It is proved that the Coxeter diagram of such a building is always crystallographic, that is, compact connected n-gons exist only for n=3,4,6. We classify compact polygons which admit a transitive gr…

Universal spaces for finite topological spaces simplify shape descriptions.

Suppose M is a non-compact connected smooth n-manifold. Let D(M) denote the group of diffeomorphisms of M endowed with the compact-open C^\infty-topology and D^c(M) denote the subgroup consisting of diffeomorphisms of M with compact support. Let D(M)_0 and D^c(M)_0 be the connected components of id_M in D(M) and D^c(M)…

Study shows conditions for continuity of foliated homeomorphisms action on space of leaves.

Undergraduate thesis explores topological barriers to compact Ricci solitons in 4D.

For a non-compact n-manifold M let H(M) denote the group of homeomorphisms of M endowed with the Whitney topology and H_c(M) the subgroup of H(M) consisting of homeomorphisms with compact support. It is shown that the group H_c(M) is locally contractible and the identity component H_0(M) of H(M) is an open normal subgr…

Characterizes isolated compact subgroups in Lie groups.

It is given a topological pinching for the injectivity radius of a compact embedded surface either in the sphere or in the hyperbolic space

Study shows certain surface homeomorphisms groups can't be precompact.

New compact manifolds with closed G2 structures found.

Counting geodesics on compact symmetric spaces using orbit dimensions and topological data.

We prove that the automorphism group of a topological parallelism on real projective 3-space is compact. In a preceding article it was proved that at least the connected component of the identity is compact. The present proof does not depend on that earlier result.

The goal of this article is to establish estimates involving the Yamabe minimal volume, mixed minimal volume and some topological invariants on compact 4-manifolds. In addition, we provide topological sphere theorems for compact submanifolds of spheres and Euclidean spaces, provided that the full norm of the second fun…

Let X be a locally compact Polish space and G a non-discrete Polish ANR group. By C(X,G), we denote the topological group of all continuous maps f:X \to G endowed with the Whitney (graph) topology and by C_c(X,G) the subgroup consisting of all maps with compact support. It is known that if X is compact and non-discrete…

The paper examines the geometry and topology of Sasaki-Ricci solitons, proving they are either connected at infinity or compact.

Study on compact hypersurfaces in spheres with Ricci curvature bounds.

Compact Special Weingarten surfaces with planar convex boundaries are disks.

The space of closed subgroups of a locally compact topological group is endowed with a natural topology, called the Chabauty topology. Let X be a symmetric space of noncompact type, and G be its group of isometries. The space X identifies with the subspace of maximal compact subgroups of G : taking the closure gives ri…

Study on scalar curvature decay on non-compact manifolds linked at infinity.

In this paper we study the topology of conformally compact Einstein 4-manifolds. When the conformal infinity has positive Yamabe invariant and the renormalized volume is also positive we show that the conformally compact Einstein 4-manifold will have at most finite fundamental group. Under the further assumption that t…

Every locally compact local group is locally isomorphic to a topological group.

Computational method approximates homology groups of compact metric spaces.

Improved proofs for topological and smooth pseudo-isotopies of simply connected 4-manifolds.

Restrictions are obtained on the topology of a compact divergence-free null hypersurface in a four-dimensional Lorentzian manifold whose Ricci tensor is zero or satisfies some weaker conditions. This is done by showing that each null hypersurface of this type can be used to construct a family of three-dimensional Riema…

Study shows only two topological configurations for Spin(7)-manifold fibrations, ruling out smooth Cayley fibrations.

The paper studies dynamical properties in semigroups modulo ideals.

We prove that for any non-compact connected surface $M$ the group $H_c(M)$ of compactly suported homeomorphisms of $M$ endowed with the Whitney topology is homeomorphic to $R^\infty\times l_2$ or $Z\times R^\infty\times l_2$.

In this note we show that compact self shrinkers in $\mathbb{R}^3$ are "topologically standard" in that any genus $g$ compact self shrinker is ambiently isotopic to the standard genus $g$ embedded surface in $\mathbb{R}^3$. As a consequence self shrinking tori are unknotted.

The paper explores the topology of compact Einstein manifolds, proving conditions for them to be homological spheres or spherical space forms.

We prove that the group D^r(R) of C^r diffeomorphisms of the real line, endowed with the compact-open and Whitney C^r topologies, is bihomeomorphic to the group H(R) of homeomorphisms of the real line endowed with the compact-open and Whitney topologies. This implies that the diffeomorphism group D^r(R) endowed with th…

Let $X$ be a Hausdorff topological group and $G$ a locally compact subgroup of $X$. We show that $X$ admits a locally finite $σ$-discrete $G$-functionally open cover each member of which is $G$-homeomorphic to a twisted product $G\times_H S_i$, where $H$ is a compact large subgroup of $G$ (i.e., the quotient $G/H$ is a…

Study identifies topologies of 3D spaces with boundary.