The paper studies trace-preserving homomorphisms on SL(2,C) and their implications.
problem Characterizing trace-preserving homomorphisms in SL(2,C).
method Elementary methods to determine when a trace-invariant homomorphism is conjugation.
result Determines which traces of generators and products determine a group up to conjugation.
Characterizes a general range decreasing group homomorphism.
problem Understanding range decreasing group homomorphisms in the entire mapping group.
method Characterization of a general range decreasing group homomorphism.
result Computes a particular class of homomorphisms and identifies all range decreasing group homomorphisms on specific mapping groups.
We extend certain homomorphisms defined on the higher Torelli subgroups of the mapping class group to crossed homomorphisms defined on the entire mapping class group. In particular, for every k≥2, we construct a crossed homomorphism εk which extends Morita's homomorphism τ~k to the entire mapping clas…
Two crossing homomorphisms on braid groups are shown to be equivalent.
problem Comparing two definitions of crossing homomorphisms on braid groups.
method Diagrammatic and algebraic definitions of crossing homomorphisms compared and computed for simple braids.
result Diagrammatic and algebraic crossing homomorphisms are equivalent.
The study classifies homomorphisms from mapping class groups using finite subgroups.
problem Classifying homomorphisms from mapping class groups.
method Using finite subgroups to classify homomorphisms.
result Only finitely many mapping class groups have non-trivial homomorphisms into Homeo(S^n) for any n.
Study of Chebyshev-Frobenius homomorphism in 3-manifold skein modules.
problem Exploring the Chebyshev-Frobenius homomorphism in 3-manifold skein modules.
method Generalization of splitting homomorphism for stated skein modules of 3-manifolds.
result Existence and properties of Chebyshev-Frobenius homomorphism for 3-manifold skein modules.
Study on singularities of bundle homomorphisms induced by Morin maps.
problem Characterizing singular points of bundle homomorphisms.
method Analyzing conditions for singularities induced by Morin maps, using Hamilton vector fields for contact structures.
result Characterization of singularities in bundle homomorphisms induced by Morin maps.
Graph homomorphism numbers embed graphs for classification.
problem Graph classification using graph homomorphisms.
method Embed graphs into vectors using homomorphism numbers.
result Homomorphism vectors are universal for approximating graph invariants.
New conditions for weighted composition operators in group homomorphisms.
problem Conditions for weighted composition operators in group homomorphisms.
method Range decreasing group homomorphisms.
result New insights into weighted composition operators and their algebraic structure.
Study homomorphisms from groups to 3-manifold fundamental groups.
problem Understanding homomorphisms between groups and 3-manifold fundamental groups.
method Proves foundational results and answers specific questions.
result Answers questions posed by Reid-Wang-Zhou and Agol-Liu.
New homomorphisms from knot Floer homology help classify knots.
problem Classifying knots based on their concordance properties.
method Defined an infinite family of concordance homomorphisms using knot Floer complexes.
result Explicitly computable homomorphisms that are linearly independent.
New homomorphism from Khovanov homology for knot concordance.
problem Understanding smooth concordance of knots.
method Modulo equivalence relation on Khovanov chain complex.
result Strictly stronger than Rasmussen invariants.
Stability of Lie group homomorphisms and subgroups via Moser type argument.
problem When a deformation of Lie group homomorphisms and subgroups is trivial.
method Moser type argument for compact groups.
result Stability results for compact Lie groups.
Classifies homomorphisms from braid groups, proving their extensions to automorphisms.
problem Classifying homomorphisms from commutator subgroups of braid groups.
method Theory of totally symmetric sets.
result Each nontrivial homomorphism extends to an automorphism of the braid group.
Classifies homomorphisms between specific braid groups.
problem Classifying homomorphisms between braid groups.
method Complete classification through recursive approach.
result Recursive classification of homomorphisms between braid groups.
We introduce the notion of tight homomorphism into a locally compact group with nonvanishing bounded cohomology and study these homomorphisms in detail when the target is a Lie group of Hermitian type. Tight homomorphisms between Lie groups of Hermitian type give rise to tight totally geodesic maps of Hermitian symmetr…
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.
A chord index homomorphism for knots in thickened surfaces is constructed.
problem Knot invariants in thickened surfaces.
method Constructing a chord index homomorphism from a subgroup of H1(Σ,Z) to chord indices of a knot K in ΣimesI. result Derived knot invariants from the homomorphism.
Study classifies biharmonic and harmonic homomorphisms between specific Lie groups.
problem Classifying biharmonic and harmonic homomorphisms between Riemannian three-dimensional unimodular Lie groups.
method Classification based on left invariant Riemannian metrics.
result Classification of biharmonic and harmonic homomorphisms between specific Lie groups.
Study on Euler class and flux homomorphisms for non-orientable surfaces.
problem Investigate Euler class and flux homomorphisms for non-orientable surfaces.
method Analyze Euler class and flux homomorphisms for non-orientable compact surfaces with one boundary component.
result Prove the simplicity of the kernel of the flux homomorphisms, implying the non-existence of invariants analogous to the Calabi invariant.
We examine functorial and homotopy properties of the exotic characteristic homomorphism in the category of Lie algebroids which was lastly obtained by the authors in [4]. This homomorphism depends on a triple (A,B,∇) where B ⊂ A are regular Lie algebroids, both over the same regular foliated manifold (M,…
We extend each higher Johnson homomorphism to a crossed homomorphism from the automorphism group of a finite-rank free group to a finite-rank abelian group. We also extend each Morita homomorphism to a crossed homomorphism from the mapping class group of once-bounded surface to a finite-rank abelian group. This improve…
A new homomorphism connects group actions on circles to Euler classes.
problem Understanding group actions on circles and their implications.
method Using crossed homomorphisms and Poincaré translation numbers.
result Relates the Euler class of actions to a specific homomorphism.
The paper disproves a conjecture about satellite maps not inducing homomorphisms.
problem Satellite maps and their impact on knot concordance groups.
method Casson-Gordon signatures and n-solvable filtration analysis. result Examples of satellite maps that act like homomorphisms but do not induce them.
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 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 …
Satellite operations with winding number ≠ 1 are not homomorphisms.
problem Characterizing homomorphisms in satellite operations.
method Using d-invariants of branched covers and Torelli group properties. result Satellite operations with winding number ≠ 1 are not homomorphisms.
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.
Homomorphism from braid groups to Steinberg groups defined.
problem Understanding the relationship between braid groups and Steinberg groups.
method Construction of a homomorphism from braid groups to Steinberg groups.
result Description of the image and kernel of the homomorphism.
Johnson and Livingston have characterized peripheral structures in homomorphs of knot groups. We extend their approach to the case of links. The main result is an algebraic characterization of all possible peripheral structures in certain homomorphic images of link groups.
Study new invariants for string links using Milnor and Morita-Milnor homomorphisms.
problem Developing invariants for string links.
method Define and study total Milnor and infinitesimal Morita-Milnor homomorphisms.
result Introduced new invariants for string links.
The paper studies symmetries in quandles and their relative versions.
problem Understanding symmetries in quandle structures and their transformations.
method Introducing relative versions of inner automorphism and transvection groups, and using them to characterize and classify surjective homomorphisms.
result Characterization of connected homomorphisms and classification of quandle structures under certain symmetry assumptions.
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.
If phi: G-->G' is a surjective homomorphism, we prove that the twisted Alexander polynomial of G is divisible by the twisted Alexander polynomial of G'. As an application, we show non-existence of surjective homomorphism between certain knot groups.
Study proves equality of LS-category and cohomological dimension for specific group homomorphisms.
problem Proving equality of LS-category and cohomological dimension for specific group homomorphisms.
method Analyzing epimorphisms and homomorphisms between specific types of almost nilpotent and virtually nilpotent groups.
result Equality of LS-category and cohomological dimension for specified group homomorphisms.
The paper shows how surjective homomorphisms between surface braid groups factor and computes their automorphism groups.
problem Characterizing surjective homomorphisms and automorphisms of surface braid groups.
method Analyzing the structure of surface braid groups and their homomorphisms.
result Surjective homomorphisms factor through forgetful maps and automorphisms are geometric.
Study de Rham homomorphism for Lipschitz cohomologies on metric simplicial complexes.
problem Triviality of de Rham homomorphism kernel and non-increasing monotonicity of parameters.
method Regularization in Lipschitz de Rham calculus on metric simplicial complexes with bounded geometry.
result Explicit specification of non-trivial cohomology classes for a sequence of parameters.
New homomorphism from Khovanov homology gives slice genus bounds.
problem Understanding slice genus of knots.
method A 1-parameter family of concordance homomorphisms from Khovanov homology.
result Can prove linear independence of certain knot families.
Let K be the kernel of an epimorphism G -> Z, where G is a finitely presented group. If K has infinitely many subgroups of index 2, 3, or 4, then it has uncountably many. Moreover, if K is the commutator subgroup of a classical knot group G, then any homomorphism from K onto the symmetric group S_2 lifts to a homomorph…
Johnson has defined a surjective homomorphism from the Torelli subgroup of the mapping class group of the surface of genus g with one boundary component to ∧3H, the third exterior product of the homology of the surface. Morita then extended Johnson's homomorphism to a homomorphism from the entire mapping cla…
New homomorphisms from homology cylinders to torsion modules via LMO functor.
problem Understanding the abelian quotients of homology cylinders.
method Constructing homomorphisms via LMO functor and analyzing their properties.
result Abelianization of Johnson kernel has torsion elements.
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…
Let M and N be two closed C∞ manifolds and let Diffc(M) denote the group of C∞ diffeomorphisms isotopic to the identity. We prove that any (discrete) group homomorphism between Diffc(M) and Diffc(N) is continuous. We also show that a non-trivial group homomorphism $…
On a closed symplectic surface Sigma of genus two or more, we give a new construction of an extended flux map (a crossed homomorphism from the symplectomorphism group Symp(Sigma) to the cohomology group H^1(Sigma;R) that extends the flux homomorphism). This construction uses the topology of the Jacobian of the surface …
New algorithms sample random graph homomorphisms for network analysis.
problem Sampling random graph homomorphisms from a graph into a large network.
method Proposed two MCMC algorithms with bounds on mixing times and concentration.
result Network observables are stable under renormalized cut distance.
New quasi-homomorphisms on surface groups vanish on handlebody groups.
problem Understanding quasi-homomorphisms on mapping class groups and their behavior on handlebody subgroups.
method Constructing infinitely many linearly independent quasi-homomorphisms that vanish on a handlebody subgroup.
result Disproved a conjecture about bounded width in Heegaard splittings and found infinitely many quasi-invariants on Heegaard splittings.
This is partly a survey and partly a research article. Some known results and open problems about Kaehler groups (fundamental groups of compact Kaehler manifolds) are discussed. A new notion of Kaehler homomorphism is introduced. This is a homomorphism induced by a holomorphic map between these types of manifolds. Some…
Paper classifies totally symmetric sets in groups and bounds their sizes.
problem Understanding homomorphisms between groups using totally symmetric sets.
method Full classifications and size bounds for totally symmetric sets in various groups.
result Derives restrictions on homomorphisms between certain groups.