Derives geometrically a description of a 3-manifold's second homotopy group.
problem Describing the second homotopy group of a 3-manifold using kernels of homomorphisms.
method Geometric derivation of Lei and Wu's result.
result Geometric derivation of the second homotopy group description.
Study shows solutions of differential inclusions are homotopy equivalent in W1,p-topology.
problem Homotopy properties of solutions in differential inclusions.
method Analyzes differential inclusion with specific assumptions on corank one distribution.
result Solutions are homotopy equivalent to loop spaces in W1,p-topology. 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.
This thesis extends contact structures to differentiable stacks using line bundle-valued 1-forms.
problem Extending classical contact structures to differentiable stacks.
method Introducing 0 and +1-shifted contact structures on Lie groupoids, using line bundle-valued 1-forms and homotopy kernels. result Definition and examples of 0 and +1-shifted contact structures on Lie groupoids. J.H.C. Whitehead defined a map Jr:πr(SO)→πrs from the homotopy of the special orthogonal group to the stable homotopy of spheres. Within a toy model we show how the known computation for kernel(J) leads to nonlinear σ-models with spherical source (space) and spherical target which admit false vacua…
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.
Cobordism groups of cooriented fold maps of codimension 1 are computed completely. Namely their odd torsion part coincides with that of the stable homotopy group of spheres in the same dimension, while the 2-primary part is the kernel of the Kahn-Priddy map. (The Kahn-Priddy map is an epimorhism of the stable homotopy …
The purpose of this paper is twofold. On one hand, we introduce a modification of the dual canonical basis for invariant tensors of the 3-dimensional irreducible representation of Uq(sl2), given in terms of Jacobi diagrams, a central tool in quantum topology. On the other hand, we use this modified basis to study t…
In this paper, I introduce weak representations of a Lie groupoid G. I also show that there is an equivalence of categories between the categories of 2-term representations up to homotopy and weak representations of G. Furthermore, I show that any VB-groupoid is isomorphic to an action groupoid associated to a weak…
A toric arrangement is a finite set of hypersurfaces in a complex torus, every hypersurface being the kernel of a character. In the present paper we build a CW-complex homotopy equivalent to the arrangement complement, with a combinatorial description similar to that of the well-known Salvetti complex. If the toric arr…
For any Engel 4-fold, we show that the scanning map from the space of Engel knots to the space of formal Engel knots is a weak homotopy equivalence when restricted to the complement of the orbits of the Engel kernel. This is a relative, parametric and close h-principle.
In this paper, we discuss the general existence theory of Dirac-harmonic maps from closed surfaces via the heat flow for α-Dirac-harmonic maps and blow-up analysis. More precisely, given any initial map along which the Dirac operator has nontrivial minimal kernel, we first prove the short time existence of the heat f…
Defines Whitehead torsion for topological spaces via K-theory.
problem Classical Whitehead torsion for topological spaces.
method Introduces Whitehead categories and torsion cosheaves.
result Establishes a connection between Whitehead torsion and K-theory.
We consider a parallelizable 2n-manifold F which has the homotopy type of the wedge product of n-spheres and show that the group of pseudo-isotopy classes of orientation preserving diffeomorphisms that keep the boundary ∂F pointwise fixed and induce the trivial variation operator is a central extension …
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.
Graph potentials link to topological QFTs, with computational methods.
problem Defining a topological quantum field theory using graph potentials.
method Using colored trivalent graphs and birational type to define a topological QFT.
result Graph potentials' birational type depends on the graph's homotopy type.
Non-injectivity proven for trace map on character varieties.
problem Proving non-injectivity of trace map on character varieties.
method Using Amitsur-Levitzki identity and free homotopy classes.
result Explicit construction of nonzero elements in kernel of trace map.
It is an open problem whether Kirk's σ invariant is the complete obstruction to a link map S2∪S2→S4 being link homotopically trivial. With the objective of constructing counterexamples, Li proposed a link homotopy invariant ω that is defined on the kernel of σ and also obstructs link nullhomotopy. We …
We give an explicit handy (and cocycle-free) description of the groupoid of weak maps between two crossed-modules in terms of certain digrams of groups which we we call a {\em butterflies}. We define composition of butterflies and this way find a bicategory that is naturally biequivalent to the 2-category of pointed ho…
The first author's geometric Hopf invariant of a stable map F:Σ∞X→Σ∞Y is a stable Z2-equivariant map h(F):Σ∞X→Σ∞(Y∧Y) constructed by an explicit difference construction applied to (F∧F)ΔX−ΔYF. The stable Z2-equivariant homotopy c…
New examples of knots with infinitely many inequivalent slice disks.
problem Existence of knots with infinitely many pairwise nonisotopic slice disks.
method Classification of fibered, homotopy-ribbon disks and use of satellite operations.
result Examples of fibered, hyperbolic knots bounding infinitely many fibered, ribbon disks that are pairwise nonisotopic modulo local knotting.
New results show metrics with positive scalar curvature can cancel on certain 4-manifolds.
problem Understanding cancellation of metrics with positive scalar curvature.
method Cobordism theory and rigidity properties of diffeomorphism groups.
result Metrics from Ruberman's family map to the same path component on product manifolds.
Homotopy on nanophrases is an equivalence relation defined using some data called a homotopy data triple. We define a product on homotopy data triples. We show that any homotopy data triple can be factorized into a product of prime homotopy data triples and this factorization is unique up to isomorphism and order. If a…
New examples of manifolds that are homotopy but not simple homotopy equivalent.
problem Characterizing simple homotopy types of even dimensional manifolds.
method Using algebraic K-theory, surgery obstruction map, and homotopy automorphisms.
result Construction of infinite families of manifolds that are homotopy equivalent but not simple homotopy equivalent.
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 colored links and spatial graphs up to colored link-homotopy.
problem Classifying colored links and spatial graphs up to colored link-homotopy.
method Using Habegger-Lin theory for colored string links, and extending to colored links and spatial graphs.
result Classification of colored links and spatial graphs up to colored link-homotopy.
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.
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.
V. Turaev introduced the theory of topology of words and phrases in 2005. This is a combinatorialy extension of the theory of virtual knots and links. In this paper we generalize the notion of homotopy of words and phrases and we give geometric meanings of the generalized homotopy of words. Moreover using the generaliz…
The study shows how stabilizing manifolds with projective spaces affects their homotopy structure.
problem Understanding the homotopy of manifolds stabilized by projective spaces.
method Trace the effect of surgery on product manifolds, showing a loop homotopy decomposition after localization.
result A loop homotopy decomposition of a manifold after stabilization by a projective space is provided.
New polynomials detect non-rotatable knotoid shapes.
problem Detecting non-rotatable knotoid shapes.
method Defined homotopy index polynomials for knotoids.
result Homotopy polynomials detect non-rotatable spherical knotoids.
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…
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.
Two approaches study the homotopy of blow ups in algebraic and symplectic geometry.
problem Investigate the homotopy of blow ups in algebraic and symplectic geometry.
method Develops fibrewise surgery theory and a purely homotopy theoretic approach.
result Obtained homotopy decompositions of the based loop space on blow ups.
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.
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.
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.
Introduces homotopy momentum sections on multisymplectic manifolds.
problem No specific problem stated; focuses on introducing a new concept.
method Introduces a new concept of homotopy momentum sections on multisymplectic manifolds.
result Shows that a gauged nonlinear sigma model with Wess-Zumino term has homotopy momentum section structure.
Homotopy commutativity in quasitoric manifolds depends on polytope structure and characteristic matrix type.
problem Conditions for homotopy commutativity in quasitoric manifolds.
method Analyzing characteristic matrices and polytope structures.
result Homotopy commutativity is determined by specific polytope and matrix conditions.
Simplified proofs for splitting homotopy idempotents.
problem Understanding the splitting of homotopy idempotents.
method Simplified proofs for both pointed and unpointed cases.
result Homotopy idempotents split in specific categories.
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
Link homotopy has been an active area of research for knot theorists since its introduction by Milnor in the 1950s. We introduce a new equivalence relation on spatial graphs called component homotopy, which reduces to link homotopy in the classical case. Unlike previous attempts at generalizing link homotopy to spatial…
This paper describes a method to construct standard 4-balls from homotopy 4-balls in C2.
problem The problem is whether every homotopy 4-ball in S4 is standard. method The approach is to use Stein surfaces and pseudoconvex domains to construct a diffeomorphic domain that is the union of three pseudoconvex domains, ensuring it is a standard 4-ball.
result The construction method ensures that the domain is a standard 4-ball, providing a compelling reimbedding construction for homotopy 4-balls in C2. Study rational homotopy types of embedding spaces of manifolds.
problem Understanding the rational homotopy types of embedding spaces of manifolds.
method Express rational homotopy types through combinatorially defined L-infinity algebras of diagrams.
result Expressed the rational homotopy type of connected components of embedding spaces.
Edge-homotopy and vertex-homotopy are equivalence relations on spatial graphs which are generalizations of Milnor's link-homotopy. Fleming and the author introduced some edge (resp. vertex)-homotopy invariants of spatial graphs by applying the Sato-Levine invariant for the constituent 2-component algebraically split li…
The paper classifies links up to link-homotopy using claspers.
problem Classifying links up to link-homotopy.
method Using Habiro's clasper calculus, defining a linear representation of the homotopy braid group, and providing a geometric proof.
result Geometric proof of Levine's classification of 4-component links and further classification of 5-component links in the algebraically split case.
New homotopy 4-spheres and real projective 4-spaces created.
problem Creating new homotopy 4-spheres and real projective 4-spaces.
method Extending Cappell-Shaneson's construction to produce new sets of smooth 4-manifolds.
result Produces new collections of homotopy 4-spheres and real projective 4-spaces.
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…