Paper proves homotopy braid group properties over integers and three strands.
problem Understanding homotopy braid group properties.
method Proved linearity over integers and torsion freeness for three strands.
result Homotopy braid group on three strands is torsion free.
By considering homotopies that preserve the stratification, one obtains a natural notion of homotopy for stratified spaces. In this short note, we introduce invariants of stratified homotopy, the stratified homotopy groups. We show that they satisify a stratified version of Whitehead's theorem. As an example, we introd…
Classifies non-linear Fredholm maps linking to stable homotopy groups of spheres.
problem Classifying non-linear proper Fredholm maps between Hilbert spaces.
method Using stable homotopy groups of spheres to classify maps up to proper homotopy.
result Determines the non-trivial kernel of the map from stable homotopy groups to non-linear proper Fredholm maps.
Finite type and finitely generated homotopy groups for manifold automorphisms.
problem Finite type and homotopy group properties of manifold automorphism spaces.
method Analyzing the classifying space of diffeomorphism groups and using simple homotopy theory.
result The classifying space of diffeomorphism groups has finitely generated homotopy groups.
In this paper, we investigate some applications of commutator subgroups to homotopy groups and geometric groups. In particular, we show that the intersection subgroups of some canonical subgroups in certain link groups modulo their symmetric commutator subgroups are isomorphic to the (higher) homotopy groups. This give…
Smooth actions of infinite groups linked to homotopy theory.
problem Connecting infinite-dimensional smooth groups to homotopy theory.
method Two computations: diffeological homotopy groups and localization of a strict category.
result Natural constructions yield homotopically coherent group actions of G.
Study homotopy groups of spaces of long links and knots, finding new generators.
problem Understanding homotopy groups of spaces of long links and knots.
method Graphing map increases dimensions, split injections from homotopy groups of spheres, and analyzing knotting effects.
result Generators for homotopy groups in a new degree for spaces of equidimensional long links.
Characterizes Stein surfaces with finite homotopy rank-sum.
problem Finite homotopy rank-sum in Stein spaces.
method Rational homotopy theory, classification of Stein surfaces.
result Affine Stein surfaces with finite fundamental group are either simply connected or of order 2.
Study contact 3-manifolds using sub-Riemannian geometry, proving new properties of their Lipschitz homotopy groups.
problem Characterize the Lipschitz homotopy groups of contact 3-manifolds.
method Sub-Riemannian geometry, geometric measure theory, biLipschitz equivalence, purely unrectifiable sets.
result Contact 3-manifolds are K(π,1) spaces with uncountably generated first homotopy groups. In this note on coarse geometry we revisit coarse homotopy. We prove that coarse homotopy indeed is an equivalence relation, and this in the most general context of abstract coarse structures. We introduce (in a geometric way) coarse homotopy groups. The main result is that the coarse homotopy groups of cone of a compa…
New examples of manifolds with similar homotopy but different simple homotopy types.
problem Characterizing groups for which high-dimensional manifolds can be homotopy equivalent but not simple homotopy equivalent.
method Construction of doubles of thickenings and use of a formula for Whitehead torsion.
result Examples of high-dimensional manifolds exist for any finitely presented group with a nontrivial Whitehead group involution.
Groups of homotopy equivalences of graphs help realize compact subgroups.
problem Realizing compact subgroups of homotopy equivalences of graphs.
method Introduced a Polish group topology on the group of proper homotopy equivalences and proved the Nielsen Realization theorem.
result Compact subgroups of homotopy equivalences can be realized by simplicial isomorphisms of graphs.
Homotopy types of 4-manifolds tied to their fundamental groups.
problem Determining the homotopy type of 4-manifolds based on their fundamental groups.
method Uses the fundamental group, second homotopy group, first Stiefel-Whitney class, and equivariant intersection pairing.
result Homotopy type of 4-manifolds is determined by given group properties.
The paper classifies homotopy ribbon discs with specific groups.
problem Classifying homotopy ribbon discs with given fundamental groups.
method Using geometric and algebraic properties of groups, particularly Farrell-Jones conjecture.
result Classification of homotopy ribbon discs for specific knot groups and Baumslag-Solitar groups.
Quasi-holomorphic homotopies of immersions of 3-manifolds into 5-manifolds
problem The study of homotopies of immersions of 3-manifolds into 5-manifolds
method Describing the local form of quasi-holomorphic homotopies and connections with holomorphic map germs
result A complete description of how the fundamental group of the complement of the image of an immersion changes under a quasi-holomorphic homotopy
4-manifolds with specific groups have unique homotopy types.
problem Classifying 4-manifolds with finite abelian 2-generator fundamental groups.
method Showed homotopy type is determined by quadratic 2-type.
result Homotopy type of 4-manifolds is determined by their quadratic 2-type.
Some properties of [L]-homotopy group for finite complex L are investigated. It is proved that for complex L whose extension type lying between Sn and Sn+1 n-th [L]-homotopy group of Sn is isomorphic to Z.
We establish an interesting connection between Morin singularities and stable homotopy groups of spheres. We apply this connection to computations of cobordism groups of certain singular maps. The differentials of the spectral sequence computing these cobordism groups are given by the composition multiplication in the …
The study finds infinite geodesics on manifolds with specific homotopy group properties.
problem Existence of non-contractible closed geodesics on manifolds with certain homotopy group properties.
method Analyzes the homotopy groups and their action on the fundamental group to prove the existence of infinitely many closed geodesics.
result Infinitely many geometrically distinct closed geodesics exist under specific conditions on homotopy groups.
We show that if M and N have the same homotopy type of simply connected closed smooth m-manifolds such that the integral and mod-2 cohomologies of M vanish in odd degrees, then their homotopy inertia groups are equal. Let M2n be a closed (n−1)-connected 2n-dimensional smooth manifold. We show that, f…
We introduce the (general) homotopy groups of spheres as link invariants for Brunnian-type links through the investigations on the intersection subgroup of the normal closures of the meridians of strongly nonsplittable links. The homotopy groups measure the difference between the intersection subgroup and symmetric com…
Study simplifies homotopy groups of 6D manifolds.
problem Homotopy groups of 6D manifolds.
method Double suspension homotopy decomposition.
result Easily determines K- and KO-groups. The homotopy theory of gauge groups has received considerable attention in recent decades. In this work, we study the homotopy theory of gauge groups over some high dimensional manifolds. To be more specific, we study gauge groups of bundles over (n−1)-connected closed 2n-manifolds, the classification of which was …
Study automorphisms of pure braid groups on sphere homotopy groups.
problem Understanding automorphisms' effect on sphere homotopy groups.
method Examined Delta-group structure, proved invariance of cycle and boundary groups, computed action for few strands.
result Induced action of all automorphisms of pure braid groups on sphere homotopy groups.
Study shows equivariant Khovanov homotopy types are equivalent.
problem Understanding equivariant structures in Khovanov homotopy types.
method Investigates group actions on homotopy coherent diagrams to prove equivalence.
result Equivariant Khovanov homotopy types are equivariantly stably homotopy equivalent.
The paper proves infinite-dimensional rational homotopy groups for a specific embedding space.
problem The study of rational homotopy groups of the space of long embeddings of codimension 2.
method Utilizing hairy graphs, the authors construct elements in the homotopy groups and prove their nontriviality.
result The rational homotopy groups of the space of long embeddings are infinite-dimensional in infinitely many degrees.
Homotopy classification for certain 4-manifolds with dihedral fundamental groups.
problem Classifying the homotopy types of specific 4-manifolds with dihedral fundamental groups.
method Using quadratic 2-type and combining with results from Hambleton-Kreck and Bauer.
result Homotopy types of finite oriented Poincaré 4-complexes are determined by their quadratic 2-type when fundamental group is dihedral.
We explore homotopies in quantum field theory formalism.
problem Constructing homotopies in Batalin-Vilkovisky formalism.
method Review and construction of homotopies from renormalization group flow and gauge fixing changes.
result Constructing spans of quantum master actions with isomorphic effective actions using homotopies.
The study shows that certain manifolds with positive curvature cannot contain specific geometric structures.
problem The existence of certain geometric structures in manifolds with positive curvature.
method Careful study of the stability of minimal two spheres in manifolds of positive curvature.
result No element of the fundamental group reverses the orientation of a class in the second homotopy group.
Study shows a copy of 7D real projective space in the space of almost complex structures on 6-sphere.
problem Understanding the topology of almost complex structures on the 6-sphere.
method Analyzes the fundamental and rational homotopy groups, computes homotopy fiber and groups, and generalizes to 6-manifolds.
result Induces isomorphism on fundamental and rational homotopy groups of the 6-sphere.
K. Orr defined a Milnor-type invariant of links that lies in the third homotopy group of a certain space Kω. The problem of non-triviality of this third homotopy group has been open. We show that it is an infinitely generated group. The question of realization of its elements as links remains open.
Characterizes compact complex surfaces with finite homotopy rank-sum.
problem Compact complex surfaces with finite homotopy rank-sum.
method Characterization and proof of Steinness of universal cover.
result Smooth compact complex Kaehler surfaces with finite homotopy rank-sum.
In this paper a geometric approach toward stable homotopy groups of spheres, based on the Pontrjagin-Thom construction is proposed. From this approach a new proof of Hopf Invariant One Theorem by J.F.Adams for all dimensions except 15,31,63,127 is obtained. It is proved that for n>127 in the stable homotopy group o…
The purpose of this article is to describe connections between the loop space of the 2-sphere, Artin's braid groups, a choice of simplicial group whose homotopy groups are given by modules called Lie(n), as well as work of Milnor, and Habegger-Lin on "homotopy string links". The current article exploits Lie algebras as…
For a complex projective space the inertia group, the homotopy inertia group and the concordance inertia group are isomorphic. In complex dimension 4n+1, these groups are related to computations in stable cohomotopy. Using stable homotopy theory, we make explicit computations to show that the inertia group is non-trivi…
Homotopy braid groups are shown to be torsion-free.
problem The existence of non-abelian torsion-free quotients of the braid group.
method Diagrammatic theory of welded braids and Artin representation.
result Homotopy braid groups are torsion-free.
Study homotopy groups of open books and their pages, pages, and bindings.
problem Homotopy groups of open books and their components.
method Homotopy theoretic conditions on monodromy, integral and rational loop space decompositions.
result Integral and rational loop space decompositions for open books under specific conditions.
Constructing manifold bundles from orbifolds and proving the existence of free subgroups in second homotopy groups.
problem Constructing manifold bundles from orbifolds and proving the existence of free subgroups in second homotopy groups.
method Resolving singularities of orbifolds via twisted families of blow-ups and using tools from real homotopy theory.
result Proving the existence of free subgroups in second homotopy groups of moduli spaces of torsion-free G2 structures.
The paper defines when surfaces are homotopy equivalent to graphs and explores their mapping class groups.
problem Understanding when surfaces are homotopy equivalent to graphs.
method Analyzes second-countable orientable surfaces with noncompact boundary.
result Defines a necessary and sufficient condition for surfaces to be homotopy equivalent to graphs.
A study on the relation between the smooth structure of a symplectic homotopy K3 surface and its symplectic symmetries is initiated. A measurement of exoticness of a symplectic homotopy K3 surface is introduced, and the influence of an effective action of a K3 group via symplectic symmetries is investigated. It is show…
Study calculates homotopy groups and derivatives for disc diffeomorphisms.
problem Understanding the homotopy groups of diffeomorphisms of discs.
method Computes rational homotopy groups and uses Weiss' orthogonal calculus.
result Determines optimal rational concordance stable range for high-dimensional discs.
Study the space of embeddings of split links in 3D and 4D.
problem Homotopy type of embedding spaces of split links.
method Semi-simplicial space of separating systems, homotopy equivalence.
result Simple description of fundamental group of embedding space.
Study on high-dimensional solid tori reveals infinite generation in their diffeomorphism groups.
problem Infinite generation in the homotopy groups of high-dimensional solid tori diffeomorphisms.
method Analysis of homotopy fibre of a linearisation map from the plus-construction of the classifying space of certain space of self-embeddings of stabilisations of the manifold to a form of Hermitian K-theory of the integral group ring of π1(S1).
result Homotopy groups of diffeomorphisms of high-dimensional solid tori are infinite in certain degrees.
We give an alternative to Postnikov's homotopy classification of maps from 3-dimensional CW-complexes to homogeneous spaces G/H of Lie groups. It describes homotopy classes in terms of lifts to the group G and is suitable for extending the notion of homotopy to Sobolev maps. This is required for applications to variati…
Added examples of S^1-manifolds with finite 2nd homotopy group and non-zero A-genus.
problem Constructing examples of S^1-manifolds with finite 2nd homotopy group and non-zero A-genus.
method Explicit equivariant surgeries to construct examples.
result Construction of new examples with finite 2nd homotopy group and non-zero A-genus.
Proves a connectivity conjecture for free groups, showing homotopy type of spheres.
problem Establishing a connectivity conjecture for free groups.
method Provided homotopy-equivalent models of the common basis complex using free factors and sphere systems.
result The common basis complex of a free group of rank n has the homotopy type of a wedge of spheres of dimension 2n-3.
Survey on finite group actions on CW-complexes homotopy to spheres.
problem Understanding finite group actions on CW-complexes homotopy equivalent to spheres.
method Survey of extensive literature on finite G-CW-complexes homotopy equivalent to spheres. result Finite G-CW-complexes homotopy equivalent to spheres have finite group actions. We consider a natural question: "Is it true that each homotopy domination of a polyhedron over itself is a homotopy equivalence?" and a strongly related problem of K. Borsuk (1967): "Is it true that two ANR's homotopy dominating each other have the same homotopy type?" The answer was earlier known to be positive for ma…