Paper compares topological and pro-étale fundamental groups.
problem No specific problem stated; comparing two fundamental groups.
method Constructs a comparison map between topological and pro-étale fundamental groups.
result Establishes a map between fundamental groups.
Study calculates fundamental groups of torus knots using algebraic topology.
problem Calculating the fundamental group of torus knots.
method Algebraic topology and group theory.
result Computed fundamental groups of torus knots.
The study finds conditions for nonmaximal topological complexity of manifolds with abelian fundamental groups.
problem Conditions for nonmaximal topological complexity of manifolds with abelian fundamental groups.
method Analyzes conditions and examples to generalize results on topological complexity and Lusternik-Schnirelmann category.
result Generalizes results on topological complexity and Lusternik-Schnirelmann category for manifolds with abelian fundamental groups.
Investigates fundamental groups and path lifting for algebraic varieties.
problem Understand fundamental groups and path lifting properties of algebraic varieties.
method Analyzes three fundamental questions about fundamental groups of algebraic varieties.
result Identifies conditions for surjectivity on fundamental groups and path lifting properties.
Authors compute fundamental groups for a specific topological group.
problem Computing fundamental groups of a specific topological group.
method Direct computation of fundamental groups.
result Fundamental groups π1Diff∂(D4k) for k≥3 are computed. Study on the topology of leaves in singular Riemannian foliations.
problem Characterizing the topology of leaves in singular Riemannian foliations.
method Analyzing the fundamental groups and using nilpotent spaces.
result Leaves of singular Riemannian foliations are finitely covered by nilpotent spaces when M is simply connected. New examples of knotting phenomena in 4-manifolds with specific fundamental groups.
problem Finding 2-spheres in simply connected 4-manifolds with prescribed fundamental groups.
method Construction of infinite sets of pairwise smoothly inequivalent 2-spheres that are topologically isotopic.
result First known examples of knotting phenomena in 4-manifolds with specific properties.
Study surfaces in 4-manifolds with cyclic fundamental group.
problem Characterize surfaces with infinite cyclic knot group.
method Algebraic topological criteria and homeomorphism proofs.
result Ambiently isotopic surfaces with the same genus have a homeomorphism.
A simple method to create new 4-manifolds by altering fundamental groups.
problem Creating new 4-manifolds with specific fundamental groups.
method Cut-and-paste mechanism to alter fundamental groups.
result Constructs new 4-manifolds with fixed-point free involutions.
The topological fundamental group π1top is a topological invariant that assigns to each space a quasi-topological group and is discrete on spaces which are well behaved locally. For a totally path-disconnected, Hausdorff, unbased space X, we compute the topological fundamental group of the "hoop earring" spac…
The goal of these lectures is to give an introduction to the study of the fundamental group of a Klein surface. We start by reviewing the topological classification of Klein surfaces and by explaining the relation with real algebraic curves. Then we introduce the fundamental group of a Klein surface and present its mai…
Paper finds conditions for two geodesics on complex manifolds.
problem Existence of two distinct closed geodesics on manifolds with infinite fundamental group.
method Topological and metric conditions for existence of geodesics in Riemannian and Finsler metrics.
result Generic Finsler metrics have two distinct closed geodesics.
The paper explores conditions for topological rigidity in quotients of the Davis complex.
problem Understanding when quotients of the Davis complex are topologically rigid.
method Analyzing quotients of the Davis complex of right-angled Coxeter groups and conditions on defining graphs.
result Introduction of infinitely many infinite topologically rigid subclasses.
New classification of nonorientable 4-manifolds with specific fundamental groups.
problem Classifying nonorientable 4-manifolds with cyclic fundamental groups.
method Simple cut-and-paste construction, using results from Hambleton-Kreck-Teichner and Khan.
result Plausible classification of a large set of nonorientable 4-manifolds with cyclic fundamental groups of order 2p.
Study shows how to create 2-links with any group in 4-manifolds.
problem Creating 2-links with any given group in 4-manifolds.
method Constructing simply connected 4-manifolds containing 2-links with specific properties.
result 2-links can have any group as their 2-link group.
Study topological 4-manifolds with specific fundamental groups.
problem Classify topological 4-manifolds with 4-dimensional fundamental group.
method Use algebraic topology, Poincaré duality, and Kirby-Siebenmann invariant.
result Two manifolds are homeomorphic if they are s-cobordant and have same Kirby-Siebenmann invariant.
Study loop ensembles on graphs, linking group theory and topology.
problem Understanding loop homotopy classes and homologies on graphs.
method Determined distributions of loop homotopy classes and homologies using the lower central series of the fundamental group.
result Distributions of loop homotopy classes and homologies defined by the lower central series of the fundamental group.
Study of circle configurations in the plane, proving aspherical space and computing fundamental groups.
problem Understanding the space of configurations of circles in the plane.
method Proved the space is aspherical and computed fundamental groups of its components.
result Fundamental groups are iterated semidirect products of braid groups, with structure dictated by a finite rooted tree.
A new isomorphism connects fundamental group ring quotients to cohomology.
problem Understanding the structure of fundamental group rings and their quotients.
method Constructing a simple isomorphism between I/In+1 and Hn(Xn,X(n,∗);Z). result Expressed Hopf algebra structure on fundamental group ring quotients in terms of cohomology.
Classifies 4-manifolds with specific properties and fundamental group.
problem Classifying 4-manifolds with fundamental group Z and boundary. method Topological and smooth classification methods, including Hermitian forms and 2-handlebody constructions. result Every Hermitian form over Z[t±1] arises as the equivariant intersection form of exotic smooth 4-manifolds. Infinitesimal calculations link fundamental groups to Lie algebras.
problem Calculating logarithm maps in fundamental groups.
method Hopf invariants defined by Harrison cohomology of commutative cochains.
result Zeroth Harrison cohomology is a universal dual to Malcev Lie algebra.
We classify closed, topological spin+ 4-manifolds with fundamental group π of cohomological dimension ≤3 (up to s-cobordism), after stabilization by connected sum with at most b3(π) copies of S2×S2. In general we must also assume that π also satisfies certain K-theory and assembly map conditio…
The space of closed subgroups of a locally compact topological group is endowed with a natural topology, called the Chabauty topology. We completely describe the space of closed sugroups of the group RxZ, which is not trivial : for example, its fundamental group is uncountable.
Study uses graph techniques to understand meromorphic quadratic differential strata.
problem Understanding the topology of meromorphic quadratic differential strata.
method Exchange graph techniques to study fundamental groups; generalizes relations for mixed-angulations.
result Explicit presentations of fundamental groups in genus-zero case with four singularities.
New bounds on Euler characteristics for certain manifolds with finite groups.
problem Estimating Euler characteristics of specific topological manifolds.
method Defining a new invariant and using cohomological invariants of fundamental groups.
result Established new bounds on minimal Euler characteristics for 4-manifolds.
In the early 1980's Mike Freedman showed that all knots with trivial Alexander polynomial are topologically slice (with fundamental group Z). This paper contains the first new examples of topologically slice knots. In fact, we give a sufficient homological condition under which a knot is slice with fundamental group Z …
Study reveals fundamental group properties of manifolds with specific curvature and growth.
problem Understanding the fundamental groups of manifolds with nonnegative Ricci curvature and linear volume growth.
method Analysis of covering spaces and rigidity results for RCD spaces.
result Fundamental groups of manifolds contain subgroups of finite index or are finite.
String topology coproduct and Turaev cobracket computed for surfaces.
problem Computing string topology coproduct and cobracket for surfaces.
method Algorithm to compute coproduct of cyclic words in terms of generators of the fundamental group.
result String cobracket is the negative of Turaev cobracket.
Two approaches for understanding and visualizing 3D surgery on manifolds.
problem Understanding and visualizing the effects of 3D surgery on manifolds.
method Two approaches: fundamental group alterations and visualizations of rotations.
result New visualizations and methods for understanding 3D surgery.
In this paper we address the relation between the orbifold fundamental group and the topology of the underlying space. In particular, under the assumption that the orbifold fundamental group is equal to the fundamental group of the underlying space, we prove Poincaré Duality for orbifolds of dimension 4 and 5.
We define new topological invariants for Anosov representations and study them in detail for maximal representations of the fundamental group of a closed oriented surface into the symplectic group.
Study mapping class group actions on surface fundamental groups.
problem Understanding group actions on surface fundamental groups.
method Computing image of mapping class group generators as outer automorphisms.
result Computed images of mapping class group generators on fundamental groups.
Endowed with quotient topology inherited from the space of based loops, the fundamental group of the Hawaiian earring fails to be metrizable. The fundamental group of any space which retracts to the Hawaiian earring is also nonmetrizable.
Study conic-line arrangements of degree 7, finding their topology and components.
problem Understanding the topology and components of conic-line arrangements of degree 7.
method Examined the embedded topology using SageMath and Sirocco package.
result Determined the number of connected components and fundamental groups.
Some generalizations and variations of the Fintushel-Stern rim surgery are known to produce smoothly knotted surfaces. We show that if the fundamental groups of their complements are cyclic, then these surfaces are topologically unknotted. Using a twist-spinning construction from high-dimensional knot theory, we constr…
Classifies 4-manifolds with elementary amenable groups and their boundaries.
problem Characterizing compact aspherical 4-manifolds with elementary amenable fundamental groups.
method Classification based on fundamental group properties and Farrell-Jones Conjecture.
result Such manifolds are either polycyclic or solvable Baumslag-Solitar groups.
Proves 3-manifold groups uniquely identify hyperbolic bundles.
problem Identifying hyperbolic 3-manifold groups from their finite quotients.
method Upgraded Liu's result to detect fiber type via profinite completion.
result Proves hyperbolic bundles are distinguished by their profinite completions.
Topological 4-dimensional surgery is conjectured to fail, in general, for free fundamental groups. M. Freedman and P. Teichner have shown that surgery problems with an arbitrary fundamental group have a solution, provided they satisfy a certain condition on Dwyer's filtration on second homology. We give a new geometric…
The study examines conditions for minimal volume entropy of simplicial complexes.
problem Conditions for minimal volume entropy of simplicial complexes.
method Topological conditions and growth of fundamental groups.
result Examples of simplicial complexes with zero simplicial volume and large minimal volume entropy.
The premier exhibition of the following phenomenon: The fundamental group of any Peano continuum constructed in similar fashion to the Hawaiian earring admits two natural distinct topological group structures. However despite being uncountable and regular, neither group is a Baire space and hence neither group admits a…
This paper classifies fundamental groups of perforated surfaces.
problem Classifying the fundamental groups of perforated surfaces.
method Using the classification theorem for surfaces and covering spaces.
result Fundamental groups of perforated surfaces are large and not Hopfian.
Theorem proves topological censorship for universes with positive cosmological constant.
problem Proving topological censorship for spacetimes with positive cosmological constant.
method Developed a new theorem assuming eventual isolation of black hole collections.
result Regions near black hole collections have trivial fundamental group.
Closed oriented 4-manifolds with the same geometrically 2-dimensional fundamental group (satisfying certain properties) are classified up to s-cobordism by their w2-type, equivariant intersection form and the Kirby-Siebenmann invariant. As an application, we obtain a complete homeomorphism classification of closed…
The paper develops a new theory of orbit braids using topological spaces.
problem Developing a new theory of orbit braids in topological spaces.
method Introducing orbit braid groups and analyzing their properties using orbit configuration spaces.
result The orbit braid group is isomorphic to a group with additional operations and contains various braid groups as subgroups.
The study explores smooth structures on specific four-manifolds with cyclic groups, finding many admit infinitely many smooth structures.
problem Exploring smooth structures on four-manifolds with finite cyclic fundamental groups.
method Analyzes topological four-manifolds with odd intersection forms and diverse fundamental groups.
result Many four-manifolds with cyclic fundamental groups admit infinitely many distinct smooth structures.
The disk embedding lemma is a technique underlying the topological classification results in 4-manifold topology for good fundamental groups. The purpose of this paper is to develop new tools for disk embedding that work up to s-cobordism, without restrictions on the fundamental group. As an application we show that a …
Endowed with natural topologies, the fundamental group of the Hawaiian earring continuously injects into the inverse limit of free groups. This note shows the injection fails to have a continuous inverse. Such a phenomenon was unexpected and appears to contradict results of another author.
Algorithm decides homeomorphism of 4-manifolds.
problem Deciding homeomorphism of closed, simply connected 4-manifolds.
method Represent 4-manifolds by Kirby diagrams, compute Kirby-Siebenmann invariant, and use algorithms for Kirby diagrams and intersection forms.
result Algorithmic decision for homeomorphism of 4-manifolds.