arXiv research

Extended logarithm for solvable elements in mapping class groups.

Study new invariants for string links using Milnor and Morita-Milnor homomorphisms.

New parametrization of 3-spheres using Johnson subgroups.

Study 2-loop part of Johnson cokernel using trace map.

Let S be a compact connected oriented surface with one boundary component, and let P be the fundamental group of S. The Johnson filtration is a decreasing sequence of subgroups of the Torelli group of S, whose k-th term consists of the self-homeomorphisms of S that act trivially at the level of the k-th nilpotent quoti…

Johnson filtrations of mapping class groups are finitely generated.

Proves finite generation of Johnson filtrations terms in mapping class groups and automorphism groups.

Let $Σ_g \to E \to Σ_h$ be a surface bundle over a surface with monodromy representation $ρ: π_1 Σ_h \to \operatorname{Mod}(Σ_g)$ contained in the Torelli group $\mathcal{I}_g$. In this paper we express the cup product structure in $H^*(E, \mathbb{Z})$ in terms of the Johnson homomorphism $τ: \mathcal{I}_g \to \wedge^3…

The paper extends Johnson homomorphisms to groups with extended N-series.

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 …

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…

The paper develops a new theory of double Johnson filtrations for mapping class groups.

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 $\wedge^3 H$, the third exterior product of the homology of the surface. Morita then extended Johnson's homomorphism to a homomorphism from the entire mapping cla…

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 studies new filtrations and homomorphisms related to mapping class groups and 3-manifold invariants.

This paper interprets topologically the tree reduction of the LMO functor using Johnson-Levine homomorphisms.

Paper compares and intersects classes in Johnson cokernels of surface mapping groups.

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.

Study on Casson invariant and its variants in mapping class groups.

Paper proves non-triviality of Johnson kernel torsion subgroup.

The paper finds torsion in Johnson homomorphisms' cokernels for large genus surfaces.

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…

Johnson kernel generated by specific Dehn twists on surfaces.

Study subgroup of mapping class group related to handlebody, answering a question about Johnson homomorphism.

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…

New descriptions of a subgroup in mapping class groups.

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…

Study detects Hopf elements and Kervaire classes in sphere spectra.

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 $k\geq 2$, we construct a crossed homomorphism $ε_k$ which extends Morita's homomorphism $\tilde τ_k$ to the entire mapping clas…

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.

Study shows mapping class group actions on configuration spaces are trivial for specific stages.

Study mapping class group actions on cohomology of configuration spaces.

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…

Non-trivial action on surface configuration homology.

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…

Proves a conjecture about matrix orders for pseudo-Anosov maps.

The study proves properties of subgroups in automorphism groups and mapping class groups.

We generalize the notion of a Magnus expansion of a free group in order to extend each of the Johnson homomorphisms defined on a decreasing filtration of the Torelli group for a surface with one boundary component to the whole of the automorphism group of a free group $\operatorname{Aut}(F_{n})$. The extended ones are …

For all but finitely many compact orientable surfaces, we show that any superinjective map from the complex of separating curves into itself is induced by an element of the extended mapping class group. We apply this result to proving that any finite index subgroup of the Johnson kernel is co-Hopfian. Analogous propert…

The paper connects cup products in surface bundles to cohomology of mapping class groups.

Develops Johnson-Morita theory for 3D handlebody groups.

The main goal of this note is to provide evidence that the first rational homology of the Johnson subgroup $K_{g,1}$ of the mapping class group of a genus g surface with one marked point is finite-dimensional. Building on work of Dimca-Papadima, we use symplectic representation theory to show that, for all $g > 3$, the…

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 …

The paper introduces a new complex analytic invariant called the pointed harmonic volume and its relation to the Johnson homomorphism.

Paper defines and analyzes Chillingworth classes for subsurface Torelli groups.

Introduces $\mathbb{K}$-framings for surfaces, generalizing quadratic forms.

Study abelian cycles in Torelli group homology, proving new results in stable rational homology.

New findings on mapping class groups and their subgroups in homological representations.