Shy maps preserve path-connectedness in continuous functions.
problem Understanding properties of continuous functions in topological spaces.
method Defining shy maps as continuous functions with specific path-connected inverse images.
result Every shy map onto a semilocally simply connected space induces a surjection of fundamental groups.
The purpose of this paper is: (i) to construct a space which is semilocally simply connected in the sense of Spanier even though its Spanier group is non-trivial; (ii) to propose a modification of the notion of a Spanier group so that via the modified Spanier group semilocal simple connectivity can be characterized; an…
The aim of this paper is to introduce the concepts of homotopical smallness and closeness. These are the properties of homotopical classes of maps that are related to recent developments in homotopy theory and to the construction of universal covering spaces for non-semilocally simply connected spaces, in particular to…
Defines new geodesic semilocal E-preinvex functions and studies their properties.
problem Defines new functions to generalize existing convex and preinvex concepts.
method Introduces geodesic semilocal E-preinvex functions and proves their properties.
result Establishes sufficient optimality conditions for nonlinear fractional multiobjective programming.
Unimodularity criteria for Poisson structures on foliated manifolds are established.
problem Understanding unimodularity in Poisson structures on foliated manifolds.
method Explicit formula for bigraded decomposition of modular vector fields and analysis of the Reeb class.
result Unimodularity of transverse Poisson structure is a necessary condition for semilocal unimodularity.
We generalize and strengthen the theorem of Gromov that every compact Riemannian manifold of diameter at most D has a set of generators g_1,...,g_k of length at most 2D and relators of the form g_ig_m = g_j . In particular, we obtain an explicit bound for the number k of generators in terms of the number "short loops" …
Study finds six homogeneous surfaces with multiple invariant connections.
problem Characterizing homogeneous surfaces with invariant connections.
method Computed all simply connected homogeneous and infinitesimally homogeneous surfaces.
result Found six non-equivalent surfaces with multiple invariant connections.
Researchers found all simply connected rational 5-spheres with Sasaki-Einstein metrics.
problem Identifying rational homology 5-spheres with Sasaki-Einstein metrics.
method Comprehensive analysis of simply connected rational homology 5-spheres.
result Determination of which simply connected rational homology 5-spheres admit Sasaki-Einstein metrics.
Simply-connected shrinking Kähler-Ricci solitons are proven.
problem Simply-connectedness of shrinking Kähler-Ricci solitons.
method Using results by Sun-Zhang and Wylie.
result Shrinking Kähler-Ricci solitons are simply-connected.
Research shows RCD* spaces are semi-locally simply connected.
problem Understanding the topological properties of RCD* spaces.
method Proving semi-locally simply connected property for any point and radius.
result RCD* spaces are semi-locally simply connected.
Study shows codimension 2 foliations on simply-connected manifolds are smooth.
problem Existence of foliations by exotic tori on simply-connected manifolds.
method Analyzes singular Riemannian foliations on compact simply-connected manifolds.
result Foliations of codimension 2 on compact simply-connected manifolds are smooth.
The paper characterizes simply connected quandles using cocycles with prime values.
problem Characterizing simply connected quandles.
method Using cocycles with values in abelian groups of prime size.
result Classification of simply connected quandles of sizes p2 and p3 for p>3. 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.
problem Understanding the connectivity of spaces of tight frames.
method Viewing tight frames as elements of Stiefel manifolds and identifying simply connected spaces.
result Spaces of tight frames, including finite unit-norm tight frames, are simply connected.
Minimal graphs over simply connected domains grow at most exponentially.
problem Growth of minimal graphs over simply connected domains with boundary values 0.
method Analyzing solutions to the minimal surface equation.
result Minimal graphs have at most exponential growth.
Defines spectral varieties for non-simply connected manifolds and constructs conformal invariants.
problem Analyzing spectra of magnetic Laplacians on non-simply connected manifolds.
method Definition of spectral varieties and construction of conformal invariants.
result New conformal invariants of immersions of surfaces into 3- and 4-dimensional spaces.
The paper studies equivariant gerbes on complex Lie groups.
problem Obstruction to equivariant extensions of basic gerbes.
method Modified construction of Gawȩdzki-Reis [2004]
result Exhibit basic equivariant bundle gerbes over non-simply connected Lie groups.
Complex surfaces show non-simply connected diffeomorphism groups with non-homotopic loops.
problem Complex surfaces with non-simply connected diffeomorphism groups and non-homotopic loops.
method Exhibited examples of complex surfaces.
result Diffeomorphism groups of complex surfaces are not simply-connected and contain 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…
Abstract: Geometrically describes Poisson cohomology groups around symplectic leaves.
problem Understanding the first Poisson cohomology groups around symplectic leaves.
method Splitting theorems for infinitesimal automorphisms of coupling Poisson structures.
result Derives criteria for vanishing of first Poisson cohomology groups.
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…
Researchers create earring spaces from metric spaces to study fundamental groups.
problem Understanding fundamental groups of constructed spaces from metric spaces.
method Attach copies of the circle to a dense subset of a separable metric space to form an earring space, then analyze its fundamental group.
result The fundamental group of the earring space is isomorphic to a subgroup of the Hawaiian earring group under specific conditions.
Ricci limit spaces are semi-locally simply connected.
problem Understanding the topological properties of Ricci limit spaces.
method Demonstrating that for any loop in a specified radius, it can be contracted within a larger radius.
result Ricci limit spaces are semi-locally simply connected.
Simply-connected 4-manifolds are covered by products with a 2-torus.
problem Proving that any closed simply-connected smooth 4-manifold is branched covered by a product of an orientable surface and a 2-torus.
method Natural construction with respect to spin structures, solving Problem 4.113(C) in Kirby's list.
result Closed simply-connected smooth 4-manifolds are 16-fold branched covered by a product of an orientable surface and a 2-torus.
Classifies 3D spaces using specific invariants.
problem Classifying 3D simply connected Mori fibre spaces.
method Uses finitely many numerical invariants.
result Finds finitely many invariants to classify diffeomorphism types.
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…
New invariant P helps classify simply-connected 8-manifolds.
problem Classifying simply-connected 8-manifolds.
method Introducing a rational homotopy invariant P and using it to classify manifolds.
result Simply-connected 8-manifolds are classified by the value of P.
Circle's metric is at least π/4 away from any simply connected geodesic space.
problem Comparing simply connected geodesic spaces to the circle.
method Using Gromov-Hausdorff distance and topological properties.
result The Gromov-Hausdorff distance between circle and any simply connected geodesic space is at least π/4.
The study refines known counterexamples in 4D to satisfy certain inequalities.
problem Addressing counterexamples in Gromov's and Rosenberg's conjectures.
method Analyzing simply connected and non-simply connected four manifolds up to homeomorphism.
result Gromov's and Rosenberg's conjectures hold for simply connected four manifolds up to homeomorphism.
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…
We construct examples of non-formal simply connected and compact oriented manifolds of any dimension bigger or equal to 7.
The paper proves conditions for Kähler and Riemannian manifolds to be simply connected.
problem Conditions for Kähler and Riemannian manifolds to be simply connected.
method Spectral positivity assumptions for Kähler manifolds and a specific spectral positivity assumption for Riemannian manifolds.
result Compact Kähler manifolds and Riemannian manifolds under the specified spectral positivity assumptions are simply connected.
Proves Rudyak's conjecture for low-dimensional simply connected spin manifolds.
problem Rudyak's conjecture on the relationship between the Lusternik-Schnirelmann category of manifolds.
method Analyzes simply connected spin manifolds of dimensions up to 8.
result Proves the conjecture for n-dimensional simply connected spin manifolds for n≤8. This paper shows the reduced characteristic group of Lie LCP manifolds is simply connected.
problem Understanding the structure of Lie LCP manifolds and their characteristic groups.
method Restricting the action of the fundamental group to the non-flat factor of the universal cover and taking the connected component of the identity in the closure of this restriction.
result The reduced characteristic group of any Lie LCP manifold is simply connected.
Classifies Alexandrov spaces in low dimensions with cohomogeneity one.
problem Classifying Alexandrov spaces with specific cohomogeneity.
method Analyzing simply-connected spaces with cohomogeneity one in dimensions 5, 6, and 7.
result Every closed, simply-connected smooth n-orbifold with cohomogeneity one is equivariantly homeomorphic to a smooth good orbifold of cohomogeneity one.
We compute the equivariant K-theory KG∗(G) for a simply connected Lie group G (acting on itself by conjugation). We prove that KG∗(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),…
A novel density regularizer improves data interpolation on non-simply-connected manifolds.
problem Topological differences between model-defined simply-connected manifolds and dataset's non-simply-connected regions.
method Density regularizer to circumvent low-probability-density regions (holes).
result Consistently better interpolation results on real-world image datasets.
Study on spheres in simply-connected 4-manifolds with abelian complements.
problem Classifying locally flat spheres in simply-connected 4-manifolds.
method Focus on complements with abelian fundamental groups.
result New classifications of spheres in specific 4-manifolds.
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 T4. 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…
Monodromy and vanishing cycles computed for ample linear systems on simply connected surfaces.
problem Characterizing curves that can be vanishing cycles in degenerations of linear systems.
method Computing mapping class group-valued monodromy and identifying it with r-spin mapping class groups.
result Identifies simple closed curves as vanishing cycles and provides characterizations of discriminants and Lefschetz fibrations.
We will describe some results regarding the algorithmic nature of homeomorphism problems for manifolds; in particular, the following theorem. Theorem 1: Every PL or smooth simply connected manifold M^n of dimension n at least 5 can be recognized among simply connected manifolds. That is, there is an algorithm to decide…
3D Schoenflies theorem for simply-connected 2-complexes.
problem Embedding simply-connected 2-complexes in 3-space uniquely.
method Proving a 3-dimensional Schoenflies theorem for 3-connected link graphs.
result Essentially unique locally flat embedding into 3-sphere.
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.
Study determines homotopy types of specific 6-manifolds.
problem Identify homotopy types of triple suspended 6-manifolds.
method Assumes trivial action of certain cohomology operation and uses triple suspension.
result Determines homotopy types of triple suspended 6-manifolds.
The boundary of hyperbolic groups is locally simply connected.
problem Topology of hyperbolic group boundaries
method Proving local simple connectedness in terms of global topology
result Boundary is locally simply connected if and only if complement of any point is simply connected