Study on link-homotopy of pretzel links using quasi-trivial quandles and biquandles.
problem Link-homotopy of pretzel links.
method Investigation of quasi-trivial quandles and biquandles, cocycle enhancements.
result Necessary and sufficient condition for pretzel links to be trivial under link-homotopy.
This paper improves bounds on how many Delta-moves are needed to trivialize a link.
problem Counting the minimum number of Delta-moves to make a link homotopy trivial.
method Classification of link homotopy and extremal graph theory.
result Quadratic and cubic upper bounds on the homotopy trivializing numbers of links.
We introduce the notion of quasi-triviality of quandles and define homology of quasi-trivial quandles. Quandle cocycle invariants are invariant under link-homotopy if they are associated with 2-cocycles of quasi-trivial quandles. We thus obtain a lot of numerical link-homotopy invariants.
The study creates symplectic examples with non-trivial homotopy groups.
problem Constructing symplectic manifolds with specific homotopy groups.
method Examples of symplectic manifolds with non-trivial second homotopy groups in various dimensions.
result Examples of symplectic manifolds with non-trivial homotopy groups in dimensions 4 and greater than 6.
The paper defines new homotopy relations on knot projections and classifies certain knot types.
problem Defining and classifying knot homotopy relations.
method Introducing cross chord numbers and using them to define strong and weak (1, 3) homotopies.
result Complete classification of knot projections with trivializing number two.
We show that there exist non-trivial piecewise-linear (PL) knots with isolated singularities Sn−2⊂Sn, n≥5, whose complements have the homotopy type of a circle. This is in contrast to the case of smooth, PL locally-flat, and topological locally-flat knots, for which it is known that if the complement…
The study finds non-trivial elements in moduli spaces of curvature metrics.
problem Identifying non-trivial elements in moduli spaces of curvature metrics.
method Constructing elements in homotopy groups of moduli spaces of positive curvature metrics.
result Non-trivial elements found in moduli spaces of positive curvature metrics.
Study shows non-trivial higher homotopy groups in moduli space of metrics.
problem Understanding invariant metrics with positive scalar curvature.
method Analysis of higher homotopy groups on quasitoric manifolds.
result Non-trivial higher homotopy groups found in moduli space.
Link-homotopy and self Delta-equivalence are equivalence relations on links. It was shown by J. Milnor (resp. the last author) that Milnor invariants determine whether or not a link is link-homotopic (resp. self Delta-equivalent) to a trivial link. We study link-homotopy and self Delta-equivalence on a certain componen…
The aim of this paper is to show the rigidity of homologically trivial actions of prime order on K3 surfaces. To be precise, we show that homotopy K3 surfaces do not admit a periodic diffeomorphism of odd prime order 3 acting trivially on cohomology. Moreover, we give an obstruction in terms of the rationality and sign…
Paper proves existence of Dirac-harmonic maps with trivial index.
problem Finding Dirac-harmonic maps with trivial index.
method Defining a new quantity and proving its homotopy invariance.
result Existence of Dirac-harmonic maps from closed Riemann surfaces to Kähler manifolds.
New tribrackets defined to count link homotopy invariants.
problem Counting invariants of link homotopy.
method Defined Δ-tribrackets and showed their invariants. result Counting invariants for certain tribrackets are trivial.
Paper defines weak (1, 3) homotopy for knot projections and classifies trivial knots.
problem Classifying knot projections under weak (1, 3) homotopy.
method Defines weak (1, 3) homotopy, introduces a map to knot isotopy classes, and determines trivial knots.
result Determines which knot projections are trivial under weak (1, 3) homotopy.
Proofs contractibility of geodesic triangulations spaces and non-trivial homotopy groups.
problem Contractibility and homotopy groups of geodesic triangulations.
method Short proofs and existence proofs for specific cases.
result Existence of polygon triangulations with non-trivial nth 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…
New homotopy types and invariants defined for knots.
problem Defining and characterizing different homotopy types of knot projections.
method Introducing strong and weak (1, 2) homotopies and defining new invariants.
result New necessary and sufficient conditions for homotopy equivalence of knot projections.
Sphere bundles over 4-manifolds are trivial after looping, except for two cases.
problem Understanding the triviality of sphere bundles over 4-manifolds.
method Analyzing the splitting of sphere bundles after looping.
result The loop spaces of total manifolds of sphere bundles are homotopy equivalent, except for two special cases.
Non-trivial elements in homotopy groups lead to metrics of positive curvature.
problem Constructing non-trivial elements in homotopy groups to understand metrics of positive curvature.
method Using Gromoll filtration, Toda brackets, and Morlet's homotopy equivalence.
result Non-trivial elements in homotopy groups act on spaces of metrics of positive curvature.
Paper shows maps from m-sphere to n-sphere are trivial cobordant.
problem Understanding cobordism classes of maps and covers for spheres.
method Analyzes cobordism classes of maps from m-sphere to n-sphere.
result For m>n, cobordism classes of maps from m-sphere to n-sphere are trivial.
We show that the n-homotopy category of connected (n+1)-dimensional Menger manifolds is isomorphic to the homotopy category of connected Hilbert cube manifolds whose k-dimensional homotopy groups are trivial for each k > n.
Edge-homotopy and vertex-homotopy are equivalence relations on spatial graphs which are generalizations of Milnor's link-homotopy. We introduce some edge (resp. vertex)-homotopy invariants of spatial graphs by applying the Sato-Levine invariant for the 2-component constituent algebraically split links and show examples…
The paper studies smooth structures on quaternionic projective spaces using inertia groups.
problem Understanding smooth structures on quaternionic projective spaces.
method Computation of inertia groups and their analogues using stable homotopy theory.
result The concordance inertia group is trivial in dimension 20 but non-trivial in higher dimensions.
We show that Orr's space's third homotopy group is infinitely generated.
problem Non-triviality of Orr's space's third homotopy group.
method Analyzing the structure of the third homotopy group of Orr's space.
result The third homotopy group is infinitely generated.
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.
Study on homotopy type of Sullivan diagrams space.
problem Homotopy type of Sullivan diagrams space.
method Vanishing and non-vanishing results, connectivity increase with components, genus stabilization maps, fundamental group computation, homology class construction.
result Non-trivial fundamental group and homology classes of harmonic compactification.
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.
The paper finds exotic non-trivial elements in the rational homotopy groups of Diff(S^4).
problem Identifying exotic non-trivial elements in the rational homotopy groups of Diff(S^4).
method Utilizing Kontsevich's characteristic classes for smooth disk bundles and a version of clasper surgery for families.
result The 4-dimensional Smale conjecture is disproved, finding many exotic non-trivial elements in π_*Diff(S^4)⊗Q.
The paper constructs infinite rank summands in diffeomorphism groups via Seiberg-Witten theory.
problem Constructing infinite rank summands in diffeomorphism groups.
method Using Seiberg-Witten theory, the paper constructs spherical families and computes invariants.
result Infinite rank summands in homotopy and homology groups of diffeomorphism groups.
Derivative map for disk diffeomorphisms induces nontrivial homotopy groups.
problem Proving nontriviality of homomorphisms induced by derivative maps.
method Combining recent results on homotopy spheres, plumbing approach, and explicit constructions.
result Non-zero homomorphism between specific homotopy groups.
The study determines fiber homotopy trivial bundles and their impact on curvature.
problem Understanding fiber homotopy trivial bundles and their effect on curvature.
method Classical approach via block bundles and surgery theory.
result Existence of elements of infinite order in homotopy groups of spaces of positive curvature.
Authors prove de Rham cohomology of Poisson and Jacobi manifolds is trivial.
problem Understanding algebraic structures on de Rham cohomology of Poisson and Jacobi manifolds.
method Using DG operads and quasi-isomorphisms, they show the de Rham cohomology structure is trivial.
result The de Rham cohomology of Poisson and Jacobi manifolds has no higher structure beyond commutativity.
Study shows differences between smooth and topological almost concordance of knots.
problem Disparity between smooth and topological almost concordance of knots.
method Defined and analyzed almost concordance in 3-manifolds, proving results through infinite families and specific examples.
result Infinite families of knots exist that are topologically concordant but not smoothly almost concordant.
Invariant ω is shown to be equivalent to σ and weaker.
problem Determining the complete obstruction to link homotopy in S4. method Defined on the kernel of σ and shown to be equivalent to σ. result Invariant ω is equivalent to σ and strictly weaker. The study constructs bundles with non-multiplicative A-genus and finds non-trivial homotopy groups in spaces of metrics.
problem Locating non-trivial rational homotopy groups in spaces of metrics with lower curvature bounds.
method Constructs smooth bundles with non-vanishing A-genus and uses them to find homotopy groups.
result Non-trivial homotopy groups in spaces of metrics with lower curvature bounds.
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.
Study shows non-trivial moduli spaces for certain highly connected manifolds.
problem Understanding moduli spaces of positive scalar curvature metrics on highly connected manifolds.
method Analyzing higher homotopy groups and fundamental groups of moduli spaces.
result Fundamental groups of moduli spaces are non-trivial for certain manifolds.
We prove that for many degrees in a stable range the homotopy groups of the moduli space of metrics of positive scalar curvature on S^n and on other manifolds are non-trivial. This is achieved by further developing and then applying a family version of the surgery construction of Gromov-Lawson to an exotic smooth famil…
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…
New constructions in Legendrian embeddings space yield novel invariants.
problem Understanding higher-order homotopy in Legendrian embeddings.
method Introduced parametric satellite and connected-sum constructions.
result Constructed new infinite families of Legendrian embeddings.
Study pochette surgery on 4-manifolds, focusing on 4-spheres.
problem Understanding pochette surgery on 4-spheres and its effects.
method Using linking number of pochette embeddings, compute homology and analyze surgeries.
result Pochette surgery on any homology 4-sphere can be computed via homology, and trivial cord surgeries do not change diffeomorphism type.
New invariants for handlebody-links using Milnor's invariants.
problem Classifying HL-homotopy classes of handlebody-links.
method Constructing invariants using Milnor's link-homotopy invariants.
result A bijection between HL-homotopy classes and tensor product space.
Study of area minimizing surfaces in homotopy classes of maps.
problem Existence and regularity of area minimizing surfaces in metric spaces.
method Introducing relative 1-homotopy type for Sobolev maps, using local quadratic isoperimetric inequality, and analog for closed surfaces.
result Existence and local Hölder regularity of area minimizing surfaces in proper geodesic metric spaces.
In this paper we study the set of projective maps between compact proper convex real projective manifolds. We show that this set contains only finitely many distinct homotopy classes and each homotopy class has the structure of a real projective manifold. When the target manifold is strictly convex, our results imply t…
This study examines the space of initial values strictly satisfying the dominant energy condition.
problem The dominant energy condition on initial value pairs in spacetime.
method Introduced an index difference for initial value pairs and compared it to Riemannian metrics.
result The space of initial values has non-trivial homotopy groups.
The 61-stem in stable homotopy groups of spheres is trivial.
problem Proving the triviality of the 61-stem in stable homotopy groups of spheres.
method Computation of homotopy groups of spheres, introducing a new technique based on Kahn-Priddy theorems.
result The 2-primary π61 is zero. Authors prove existence of exotic surfaces and invariants not detecting self-diffeomorphisms.
problem Existence and properties of exotic surfaces and diffeomorphisms.
method Vanishing theorem of family Bauer--Furuta invariant for diffeomorphisms on spin 4-manifolds.
result Family Bauer--Furuta invariants do not detect exotic self-diffeomorphisms on S4 or S2imesS2. Study on the topology of ordered disc configurations, revealing nontrivial homotopy classes.
problem Topology of ordered disc configurations and their homotopy types.
method Analysis of ordered configuration spaces of hard discs, focusing on homotopy types and nontrivial classes.
result Exhibit nontrivial classes in π_{n-3} for all n, and their persistence in deformed ambient discs.
Study detects non-trivial elements in diffeomorphism groups via trivalent graphs.
problem Detecting non-trivial elements in homotopy groups of diffeomorphism spaces.
method Using Kontsevich classes and trivalent graphs, we lift elements from one moduli space to another.
result Non-trivial elements in π∗(BDiff∂(Dd))⊗Q are lifted to π∗(BDiff⊔(DdimesI))⊗Q and π∗(M∂psc(Dd)h0)⊗Q.