Study 2-loop part of Johnson cokernel using trace map.
problem Identify components of Johnson cokernel in degree 6.
method Use 2-loop trace map to capture Johnson cokernels.
result Capture all components of Johnson cokernels in degree 6.
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48 results for “Johnson cokernels”
Paper compares and intersects classes in Johnson cokernels of surface mapping groups.
problem Comparing and intersecting classes in Johnson cokernels of mapping class groups of surfaces.
method Representation theoretic and topological approaches.
result Anti-Morita obstructions and hook-type components appear in the Johnson cokernels.
Study Hopf algebras to understand invariants of the Johnson cokernel.
problem Understanding invariants of the Johnson cokernel using Hopf algebras.
method Analyzing actions of automorphisms and outer automorphisms on tensor powers of Hopf algebras.
result The Johnson cokernel's invariant projects to the top dimensional cohomology of Out(F_n).
In the present paper, we study the Sp-module structure of the cokernel of the Johnson homomorphism of the mapping class groups of surfaces. We detect the Sp-irreducible components with highest weight [1^k] (and [k]) in the cokernel. We also show that the multiplicities of them is equal to one.
The paper finds torsion in Johnson homomorphisms' cokernels for large genus surfaces.
problem Existence of torsion in the cokernels of Johnson homomorphisms.
method Defined a map on Ker(Tr)∩D(H) whose image is 2-torsion and vanishes on the image of τ.
result Found 2-torsion in the cokernels of Johnson homomorphisms for large genus surfaces.
In this paper, we survey recent works on the structure of the mapping class groups of surfaces mainly from the point of view of topology. We then discuss several possible directions for future research. These include the relation between the structure of the mapping class group and invariants of 3-manifolds, the unstab…
Higher hairy graph homologymath.AT
We study the hairy graph homology of a cyclic operad; in particular we show how to assemble corresponding hairy graph cohomology classes to form cocycles for ordinary graph homology, as defined by Kontsevich. We identify the part of hairy graph homology coming from graphs with cyclic fundamental group as the dihedral h…
Study shows non-locally-flat concordance elements in knot theory.
problem Understanding non-locally-flat concordance elements in knot theory.
method Utilized Heegaard Floer homology.
result Cokernel of the map from smooth knot concordance group to homology concordance group is infinitely generated and contains elements of infinite order.
New parametrization of 3-spheres using Johnson subgroups.
problem Constructing integral homology 3-spheres.
method Intrinsic description of equivalence relation on fourth Johnson subgroup.
result Intrinsic description of equivalence relation using fourth Johnson handlebody subgroups.
The Johnson kernel is the subgroup of the mapping class group of a surface generated by Dehn twists along bounding simple closed curves, and has the second Johnson homomorphism as a free abelian quotient. In terms of the representation theory of the symplectic group, we give a complete description of cup products of tw…
The paper extends Johnson homomorphisms to groups with extended N-series.
problem Johnson homomorphisms for mapping class groups.
method Developed a theory of Johnson homomorphisms for groups with extended N-series.
result Many known and new variants of Johnson homomorphisms.
We consider two mod-p central series of the free group given by Stallings and Zassenhaus. Applying these series to definitions of Dennis Johnson's filtration of the mapping class group we obtain two mod-p Johnson filtrations. Further, we adapt the definition of the Johnson homomorphisms to obtain mod-p Johnson homomorp…
Study of handlebody group intersections with Torelli and Johnson kernels.
problem Intersection of handlebody group with Torelli and Johnson kernels.
method Use Birman--Craggs--Johnson (BCJ) homomorphism to study intersections and compute cup products.
result Determine images of BCJ homomorphism restricted to handlebody group intersections and compute cup products.
Johnson Homomorphismsmath.GT
Paper surveys Johnson homomorphisms and related tools.
problem Understanding generalized Johnson homomorphisms and their stable images.
method Surveying and unifying various related threads in literature using Hodge theory.
result Clarification of existing results and relationships among Johnson homomorphisms.
We give a new proof of a celebrated theorem of Dennis Johnson that asserts that the kernel of the Johnson homomorphism on the Torelli subgroup of the mapping class group is generated by separating twists. In fact, we prove a more general result that also applies to "subsurface Torelli groups". Using this, we extend Joh…
We present a version of the Penrose transform which relates compactly supported cohomology on a complex or CR manifold Z to kernels and cokernels of differential operators on a parameter space X of compact complex submanifolds of Z.
We propose an approach to study non-Abelian Iwasawa theory, using the idea of Johnson homomorphisms in low dimensional topology. We introduce arithmetic analogues of Johnson homomorphisms/maps, called the p-Johnson homomorphisms/maps, associated to the Zassenhaus filtration of a pro-p Galois group over a Z_p-extension …
Johnson filtrations of mapping class groups are finitely generated.
problem Finiteness of Johnson filtrations in mapping class groups.
method Proved linear stable range for lower central series and Johnson filtrations of Torelli subgroups.
result Every term of the filtrations is finitely generated.
The paper studies new filtrations and homomorphisms related to mapping class groups and 3-manifold invariants.
problem Exploring new filtrations and homomorphisms in mapping class groups.
method Investigates a new filtration introduced by Habiro and Massuyeau, compares it with existing filtrations, and connects it to the LMO functor.
result Alternative Johnson homomorphisms can be read in the tree reduction of the LMO functor.
Proves finite generation of Johnson filtrations terms in mapping class groups and automorphism groups.
problem Finite generation of Johnson filtrations terms in mapping class groups and automorphism groups.
method Proves finite generation of terms in lower central series and Johnson filtrations of Torelli subgroups.
result Proves every term of the lower central series and Johnson filtrations of Torelli subgroups are finitely generated in a stable range.
This paper has two main goals. First, we give a complete, explicit, and computable solution to the problem of when two simple closed curves on a surface are equivalent under the Johnson kernel. Second, we show that the Johnson filtration and the Johnson homomorphism can be defined intrinsically on subsurfaces and prove…
We prove the existence of homeomorphisms of a closed, orientable surface of genus 3 or greater that do not extend to any handlebody bounded by the surface. We show that such homeomorphisms exist arbitrarily deep in the Johnson filtration of the mapping class group. The second and third terms of the Johnson filtration a…
The paper develops a new theory of double Johnson filtrations for mapping class groups.
problem Understanding the structure of mapping class groups using filtrations.
method Developed a general theory of Johnson filtrations and homomorphisms for groups acting on filtered groups, specializing to mapping class groups.
result Obtained a theory of double Johnson filtrations and homomorphisms for mapping class groups of surfaces with one boundary component.
Lagrangian traces help understand Johnson filtration in handlebody groups.
problem Understanding the Johnson filtration in handlebody groups.
method Defining Lagrangian traces on derivations of free Lie algebra.
result Lagrangian traces vanish on Johnson filtration elements.
Paper proves non-triviality of Johnson kernel torsion subgroup.
problem Non-triviality of the torsion subgroup of the abelianized Johnson kernel.
method Action of mapping class group on Malcev Lie algebra, diagrammatic techniques.
result Proves non-triviality of the torsion subgroup with a purely 2-dimensional proof.
We survey a geometric approach to the Johnson homomorphisms using the Goldman-Turaev Lie bialgebra.
This paper interprets topologically the tree reduction of the LMO functor using Johnson-Levine homomorphisms.
problem Interpreting the LMO functor's tree reduction topologically.
method Topological interpretation of the LMO functor's tree reduction using Johnson-Levine homomorphisms.
result Topological interpretation of the LMO functor's tree reduction.
The paper proves the finiteness of a Johnson subgroup's rational homology.
problem Finiteness of the first rational homology of Johnson subgroups.
method Symplectic representation theory and augmentation ideal in rational group algebra.
result The completion of is finite-dimensional for all .
Johnson kernel generated by specific Dehn twists on surfaces.
problem Identifying the generating set of Johnson kernel for surfaces.
method Analyzing Dehn twists on specific types of surfaces.
result Johnson kernel generated by Dehn twists on specified surfaces.
Extended logarithm for solvable elements in mapping class groups.
problem Logarithm of Johnson map extension to solvable elements.
method Extension to exponential solvable elements in mapping class groups using solvable Lie groups.
result Solvability of extended logarithm.
New findings on Johnson kernel's top homology group.
problem Understanding the Johnson kernel's top homology group structure.
method Analyzing the Johnson filtration and using cohomological dimensions.
result The top homology group of Johnson kernel contains a free abelian subgroup of infinite rank.
Study on cohomological dimension of surface terms, answering Farb's question.
problem Determining the cohomological dimension of specific surface terms.
method Analyzing the Johnson filtration of closed, orientable surfaces of genus .
result The th term of the Johnson filtration has cohomological dimension for all and .
New invariants of homology spheres derived from Torelli group and Johnson kernel.
problem Determining new rational differences between filtrations of the Torelli group.
method Analyzing the Johnson filtration and the lower central series, proving no other rational difference up to degree 6.
result Explicit computation of and expression of finite type rational invariants of homology 3-spheres.
Johnson has defined a surjective homomorphism from the Torelli subgroup of the mapping class group of the surface of genus with one boundary component to , the third exterior product of the homology of the surface. Morita then extended Johnson's homomorphism to a homomorphism from the entire mapping cla…
Study subgroup of mapping class group related to handlebody, answering a question about Johnson homomorphism.
problem Understanding the image of the second Johnson homomorphism for a specific subgroup of mapping class groups.
method Introduced trace-like operators and used them to compute images of Johnson homomorphisms.
result Answered a question about algebraic description of the image of the second Johnson homomorphism.
Study on Casson invariant and its variants in mapping class groups.
problem Understanding finite type invariants and their vanishing properties in Johnson filtration.
method Analyzing Casson invariant, , , and their morphisms on Johnson filtration levels.
result Invariant vanishes on the fifth level of Johnson filtration, .
Answering a question of Farb-Leininger-Margalit, we give explicit lower bounds for the dilatations of pseudo-Anosov mapping classes lying in the kth term of the Johnson filtration of the mapping class group.
The paper extends Johnson's result on Torelli group homology.
problem Understanding the homology of the Torelli group and its subgroups.
method Analyzing the pushforward homomorphism on higher homology groups induced by Dehn twists.
result The pushforward homomorphism is injective for certain subgroups of higher homology groups.
We define new bordism and spin bordism invariants of certain subgroups of the mapping class group of a surface. In particular, they are invariants of the Johnson filtration of the mapping class group. The second and third terms of this filtration are the well-known Torelli group and Johnson subgroup, respectively. We i…
Two references added and the introduction slightly expanded. We show that the tree-level part of a recent theory of invariants of 3-manifolds (due, independently, to Goussarov and Habiro) is essentially given by classical algebraic topology in terms of the Johnson homomorphism and Massey products, for arbitrary 3-manif…
For k >= 1, let Torelli_g^1(k) be the k-th term in the Johnson filtration of the mapping class group of a genus g surface with one boundary component. We prove that for all k, there exists some G_k >= 0 such that Torelli_g^1(k) is generated by elements which are supported on subsurfaces whose genus is at most G_k. We a…
Paper defines and analyzes Chillingworth classes for subsurface Torelli groups.
problem Understanding Chillingworth classes for subsurface Torelli groups.
method Combinatorial description and proof of properties of the Chillingworth class.
result Relates Chillingworth class to partitioned Johnson homomorphism.
New group-theoretic Johnson classes applied to curves with torsion Ceresa classes.
problem Analyzing torsion in Ceresa classes of curves.
method Group-theoretic analogues of Johnson/Morita cocycles applied to pro-l etale fundamental groups of curves.
result Example of a non-hyperelliptic curve with torsion Ceresa class.
Study detects Hopf elements and Kervaire classes in sphere spectra.
problem Detecting Hopf elements and Kervaire classes in stable homotopy groups.
method Using Hurewicz map and Real Johnson-Wilson theories.
result Certain families of elements are detected by Real Johnson-Wilson theories.
New method constructs solution operators for PDEs with prescribed support properties.
problem Constructing solution operators for under/overdetermined PDEs with specific support properties.
method Using a recovery on curves condition and taking smooth averages over curves, we obtain integral solution operators and representation formulas.
result Our method leads to integral representation formulas for overdetermined PDEs and solution operators for underdetermined PDEs.
We examine the Johnson filtration of the (outer) automorphism group of a finitely generated group. In the case of a free group, we find a surprising result: the first Betti number of the second subgroup in the Johnson filtration is finite. Moreover, the corresponding Alexander invariant is a non-trivial module over the…
Non-trivial action on surface configuration homology.
problem Understanding the Johnson filtration's action on surface configuration homology.
method Action of mapping class group on configuration space homology, focusing on Johnson filtration stages.
result Non-trivial action on the stage of the Johnson filtration for all and .
The paper introduces a new complex analytic invariant called the pointed harmonic volume and its relation to the Johnson homomorphism.
problem Exploring new complex analytic invariants related to the complex structure of Riemann surfaces.
method Defining and computing the pointed harmonic volume as a natural extension of Chen's iterated integrals.
result Established a relationship between the harmonic volume and the first extended Johnson homomorphism.