arXiv research

Research shows RCD* spaces are semi-locally simply connected.

Ricci limit spaces are semi-locally simply connected.

Extends foliation results to singular cases.

Let $X$ be a non-collapsing Ricci limit space and let $x\in X$. We show that for any $ε>0$, there is $r>0$ such that every loop in $B_t(x)$ is contractible in $B_{(1+ε)t}(x)$, where $t\in(0,r]$. In particular, $X$ is semi-locally simply connected.

This paper classifies fundamental groups of perforated surfaces.

Study the topology of Ricci limit spaces using Gromov-Hausdorff limits.

We describe a semi-local canonical form for Legendrian foliations on contact manifolds in the neighbourhood of a Legendrian submanifold. This result generalizes local results by Libermann and Pang on Legendrian foliations on contact manifolds, and is analogeous to a semi-local result by Weinstein in the symplectic case…

We extend the cobordism based categorification of the virtual Jones polynomial to virtual tangles. This extension is combinatorial and has semi-local properties. We use the semi-local property to prove an applications, i.e. we give a discussion of Lee's degeneration of virtual homology.

Symplectic classification for a specific type of singularity in integrable systems.

This paper examines limits of Riemannian 2-manifolds with bounded curvature.

New Alexander invariants for knot groups computed using $K_1$-groups.

We give a classification of generic bifurcations of intersections of wavefronts generated by different points of a hypersurface with or without boundaries.

Algorithm classifies saddle-focus singularities in Hamiltonian systems.

Simply-connected shrinking Kähler-Ricci solitons are proven.

We compute all the simply connected homogeneous and infinitesimally homogeneous surfaces admitting one or more invariant affine connections. We find exactly six non equivalent simply connected homogeneous surfaces admitting more than one invariant connections and four classes of simply connected homogeneous surfaces ad…

The paper characterizes simply connected quandles using cocycles with prime values.

We prove that an open 3-manifold proper homotopy equivalent to a geometrically simply connected polyhedron is simply connected at infinity, generalizing a theorem of V.Poenaru.

We use an extension of Sunada's theorem to construct a nonisometric pair of isospectral simply connected domains in the Euclidean plane, thus answering negatively Kac's question, ``can one hear the shape of a drum?'' In order to construct simply connected examples, we exploit the observation that an orbifold whose unde…

Simply connected spaces of tight frames identified.

Minimal graphs over simply connected domains grow at most exponentially.

Defines spectral varieties for non-simply connected manifolds and constructs conformal invariants.

Complex surfaces show non-simply connected diffeomorphism groups with non-homotopic loops.

We give two constructions of surfaces in simply-connected 4-manifolds with non simply-connected complements. One is an iteration of the twisted rim surgery introduced by the first author. We also construct, for any group G satisfying some simple conditions, a simply-connected symplectic manifold containing a symplectic…

Using the topologist sine curve we present a new functorial construction of cone-like spaces, starting in the category of all path-connected topological spaces with a base point and continuous maps, and ending in the subcategory of all simply connected spaces. If one starts by a noncontractible n-dimensional Peano cont…

Classifies 3D spaces using specific invariants.

Researchers create earring spaces from metric spaces to study fundamental groups.

We extend Bar-Natan's cobordism based categorification of the Jones polynomial to virtual links. Our topological complex allows a direct extension of the classical Khovanov complex ($h=t=0$), the variant of Lee ($h=0,t=1$) and other classical link homologies. We show that our construction allows, over rings of characte…

Simply-connected 4-manifolds are covered by products with a 2-torus.

Simplified proof of foliation closure theorem for linear foliations.

Several authors have recently attempted to show that the intersection of three simply connected subcontinua of the plane is simply connected provided it is non-empty and the intersection of each two of the continua is path connected. In this note we give a very short complete proof of this fact. We also confirm a relat…

We completely determine which simply connected rational homology 5-spheres admit Sasaki-Einstein metrics.

New invariant P helps classify simply-connected 8-manifolds.

We show that the topological classification and the smooth classification are generically the same for certain families of plane curves in a semi-local case(the double local case). Especially we give the normal form of transversely jointed two families of plane curves with second order contact at the envelope.

Circle's metric is at least π/4 away from any simply connected geodesic space.

This paper computes the obstruction to the existence of equivariant extensions of basic gerbes over non-simply connected compact simple Lie groups. By modifying a (finite dimensional) construction of Gawȩdzki-Reis [J. Geom. Phys. 50(1):28-55, 2004], we exhibit basic equivariant bundle gerbes over non-simply connected c…

The study refines known counterexamples in 4D to satisfy certain inequalities.

We construct examples of non-formal simply connected and compact oriented manifolds of any dimension bigger or equal to 7.

To any connected and simply connected nilpotent Lie group N, one can associate its group of affine transformations Aff(N). In this paper, we study simply transitive actions of a given nilpotent Lie group G on another nilpotent Lie group N, via such affine transformations. We succeed in translating the existence questio…

The paper proves conditions for Kähler and Riemannian manifolds to be simply connected.

Proves Rudyak's conjecture for low-dimensional simply connected spin manifolds.

We show that a singular Riemannian foliation of codimension two on a compact simply-connected Riemannian $(n+2)$-manifold, with regular leaves homeomorphic to the $n$-torus, is given by a smooth effective $n$-torus action. This solves in the negative for the codimension $2$ case a question about the existence of foliat…

We compute the equivariant $K$-theory $K_G^*(G)$ for a simply connected Lie group $G$ (acting on itself by conjugation). We prove that $K_G^*(G)$ is isomorphic to the algebra of Grothendieck differentials on the representation ring. We also study a special example of a non-simply connected Lie group $G$, namely PSU(3),…

This paper shows the reduced characteristic group of Lie LCP manifolds is simply connected.

Study on spheres in simply-connected 4-manifolds with abelian complements.

We construct a compact simply-connected 7-dimensional manifold admitting a K-contact structure but not a Sasakian structure. We also study rational homotopy properties of such manifolds, proving in particular that a simply-connected 7-dimensional Sasakian manifold has vanishing cup-product on the second cohomology and …

In this paper, we construct simply connected symplectic Calabi-Yau 6-manifolds by applying Gompf's symplectic fiber sum operation along $T^4$. Using our construction, we also produce symplectic non-Kähler Calabi-Yau 6-manifolds with fundamental group $\Z$. In this paper, we also produce the first examples of simply con…

We produce infinite families of exotic actions of finite cyclic groups on simply connected smooth 4-manifolds with nontrivial Seiberg-Witten invariants.

We construct a group (an HNN extension of a free group) with polynomial isoperimetric function, linear isodiametric function and non-simply connected asymptotic cones.