New algebraic structures for Hermitian geometry cohomologies.
problem Understanding cohomologies of Hermitian manifolds.
method Introducing BV-algebras and homotopy BV-algebras.
result Cohomologies of Hermitian manifolds are endowed with homotopy hypercommutative algebra structures.
Characterizes a specific type of Courant algebroid with a Calabi-Yau structure.
problem Understanding specific types of Courant algebroids with Calabi-Yau structures.
method Explains how a homotopy BV algebra with certain properties characterizes these algebroids.
result A Courant algebroid with a Calabi-Yau structure is a homotopy BV algebra with specific properties.
We define and study the degeneration property for BV-infinity algebras and show that it implies that the underlying L-infinity algebras are homotopy abelian. The proof is based on a generalisation of the well-known identity Δ(e^x)=e^x(Δ(x)+[x,x]/2) which holds in all BV algebras. As an application we show that the high…
We introduce higher Kirillov brackets and algebroids for supermanifolds.
problem Understanding higher structures on supermanifolds.
method Introducing homotopy Kirillov algebras and algebroids.
result Construction of homotopy versions of Kirillov's theorems.
The homotopy fiber of the inclusion from the long embedding space to the long immersion space is known to be an iterated based loop space (if the codimension is greater than two). In this paper we deloop the homotopy fiber to obtain the topological Stiefel manifold, combining results of Lashof and of Lees. We also give…
A BV algebra is a formal framework within which the BV quantization algorithm is implemented. In addition to the gauge symmetry, encoded in the BV master equation, the master action often exhibits further global symmetries, which may be in turn gauged. We show how to carry this out in a BV algebraic set up. Depending o…
Let A=A0+A1+A2+... be a Gerstenhaber algebra generated by A0 and A1. Given a degree -1 operator D on A0+A1, we find the condition on D that makes A a BV-algebra. Subsequently, we apply it to the Gerstenhaber or BV algebra associated to a Lie algebroid and obtain a global proof of the corresponden…
Several topological and homological operads based on families of projectively weighted arcs in bounded surfaces are introduced and studied. The spaces underlying the basic operad are identified with open subsets of a compactification due to Penner of a space closely related to Riemann's moduli space. Algebras over thes…
The purpose of this paper is to establish an explicit correspondence between various geometric structures on a vector bundle with some well-known algebraic structures such as Gerstenhaber algebras and BV-algebras. Some applications are discussed. In particular, we found an explicit connection between the Koszul-Brylins…
Defines a simplicial operad related to Fulton-MacPherson.
problem Equipping the symplectic cochain complex with BV algebra structure.
method Constructs an operad in Top with CW and simplicial decompositions.
result Expected isomorphism to the 2-dimensional Fulton-MacPherson operad.
This paper is an exposition of the new subject of String Topology. We present an introduction to this exciting new area, as well as a survey of some of the latest developments, and our views about future directions of research. We begin with reviewing the seminal paper of Chas and Sullivan, which started String Topolog…
The aim of this paper is to define a chain level refinement of the Batalin-Vilkovisky (BV) algebra structure on the homology of the free loop space of a closed, oriented C∞-manifold. For this purpose, we define a (nonsymmetric) cyclic dg operad which consists of "de Rham chains" of free loops with marked points…
We show that the Gerstenhaber algebra of the 1-jet Lie algebroid of a Jacobi manifold has a canonical exact generator, and discuss duality between its homology and the Lie algebroid cohomology. We also discuss a new example of a Lie bialgebroid on Poisson manifolds.
New algebraic tools solve Poisson and Lie bialgebra problems.
problem Modular class and intrinsic biderivation in Poisson geometry.
method Algebraic tools from differential Gerstenhaber algebras and Batalin-Vilkobisky algebras.
result Applications to Lie bialgebra and Poisson cohomology.
For almost any compact connected Lie group G and any field F_p, we compute the Batalin-Vilkoviskyalgebra H∗+dim G(LBG;F_p) on the loop cohomology of the classifying space introduced byChataur and the second author.In particular, if p is odd or p=0, this Batalin-Vilkovisky algebra…
We introduce BV-algebra structures on the homology of the space of framed long knots in Rn in two ways. The first one is given in a similar fashion to Chas-Sullivan's string topology. The second one is defined on the Hochschild homology associated with a cyclic, multiplicative operad of graded modules. The …
The paper introduces knot invariants using stratified homotopy groups.
problem Defining knot invariants in stratified spaces.
method Introducing stratified homotopy groups and proving their properties.
result Stratified Whitehead's theorem holds for these groups.
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.
Study on coarse homotopy groups, proving equivalence and matching with usual homotopy groups.
problem Understanding coarse homotopy groups in abstract coarse structures.
method Developed geometric triangulation techniques for cones to prove the equivalence and matching of coarse homotopy groups with usual homotopy groups.
result Coarse homotopy groups of the cone of a compact simplicial complex coincide with the usual homotopy groups of the underlying compact simplicial complex.
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 shadow homotopy invariant defined for links.
problem Homotopy invariants for classical links.
method Defined extended quandle spaces and constructed shadow homotopy invariant.
result Shadow homotopy invariant equals quandle homotopy invariant times quandle order.
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.
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.
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…
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.
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.
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.
The paper studies homotopy inertia groups and tangential structures of manifolds.
problem Understanding the homotopy inertia groups and tangential structures of manifolds.
method Analyzing the homotopy type and cohomology of manifolds to determine homotopy inertia groups and tangential structures.
result The homotopy inertia groups of certain manifolds are shown to be trivial under specific conditions.
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.
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.
Constructs odd Khovanov homotopy types for links, linking them to even types.
problem Understanding and constructing odd Khovanov homotopy types for links.
method Constructs stable homotopy types X^j_o(L) for links L, with cohomology matching odd Khovanov homology.
result Odd Khovanov homotopy types carry a Z/2 action whose fixed points are related to even Khovanov homotopy types.
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
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. 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…
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.
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…
Surveying A1-homotopy theory and contractible varieties.
problem Understanding the relationship between A1-homotopy theory and contractible varieties. method Exploring the interplay between A1-homotopy theory and affine algebraic geometry. result Highlighting the connection between A1-homotopy theory and contractible varieties. 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.