arXiv research

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.

The paper studies symmetries in quandles and their relative versions.

The paper introduces two-tone colorings for links and shows conditions for surjective dihedral representations.

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…

Let $K$ be a prime knot in $S^3$ and $G(K)=π_1(S^3-K)$ the knot group. We write $K_1 \geq K_2$ if there exists a surjective homomorphism from $G(K_1)$ onto $G(K_2)$. In this paper, we determine this partial order on the set of prime knots with up to 11 crossings. There exist such 801 prime knots and then $640,800$ shou…

Let $PB_n(S_{g,p})$ be the pure braid group of a genus $g>1$ surface with $p$ punctures. In this paper we prove that any surjective homomorphism $PB_n(S_{g,p})\to PB_m(S_{g,p})$ factors through one of the forgetful homomorphisms. We then compute the automorphism group of $PB_n(S_{g,p})$, extending Irmak, Ivanov and McC…

This paper studies the relationship between fundamental groups of manifolds and their effective regular sets.

Maps between classifying spaces for certain groups are studied, with rational cohomology results.

This paper extends Thurston and Tsuboi's work on foliations of $S^3$.

The paper calculates Alexander polynomials for knots using finite group representations.

We give an exposition of Delzant's ideas extending the notion of Scott complexity of finitely generated groups to surjective homomorphisms of finitely presented groups to finitely generated groups.

Study fundamental quandle of ribbon concordances, proving homomorphisms.

Study surjections between braid groups on surfaces, focusing on lower central series.

Extends Jones' construction to map Thompson group to pointed links.

The paper extends Johnson's characterization of amenable groups to homomorphisms and acyclicity in bounded cohomology.

Researchers solve a question about embedding knots into Legendrian structures.

We construct an action of the braid group B_{2g+2} on the free group F_{2g} extending an action of B_4 on F_2 introduced earlier by Reutenauer and the author. Our action induces a homomorphism from B_{2g+2} into the symplectic modular group Sp_{2g}(Z). In the special case g=2 we show that the latter homomorphism is sur…

Study big mapping class groups and their co-Hopfian property, finding new examples and proving injective homomorphisms results.

By introducing a refinement of the Goldman-Turaev Lie bialgebra, we interpret the divergence cocycle in the Kashiwara-Vergne problem and the Enomoto-Satoh obstructions for the surjectivity of the Johnson homomorphisms as some part of a regular homotopy version of the Turaev cobracket.

We study a model situation in which direct limit ($\text{colim}$) and inverse limit ($\lim$) do not commute, and offer some computations of their "commutator". The homology of a separable metrizable space $X$ has two well-known approximants: $qH_n(X)$ ("Čech homology") and $pH_n(X)$ ("Čech homology with compact support…

The paper explores actions of surface mapping class groups on 3-manifolds.

In this paper, we undertake the study of the Tannaka duality construction for the ordinary representations of a proper Lie groupoid on vector bundles. We show that for each proper Lie groupoid G, the canonical homomorphism of G into the reconstructed groupoid T(G) is surjective, although, contrary to what happens in th…

A map from a circle to a graph splits if pre-image diameters are small.

We study a natural Lie algebra structure on the free vector space generated by all rooted planar trees as the associated Lie algebra of the nonsymmetric operad (non-$Σ$ operad, preoperad) of rooted planar trees. We determine whether the Lie algebra and some related Lie algebras are finitely generated or not, and prove …

We show the problem of counting homomorphisms from the fundamental group of a homology $3$-sphere $M$ to a finite, non-abelian simple group $G$ is #P-complete, in the case that $G$ is fixed and $M$ is the computational input. Similarly, deciding if there is a non-trivial homomorphism is NP-complete. In both reductions,…

We show that the commutator subgroup G' of a classical knot group G need not have subgroups of every finite index, but it will if G' has a surjective homomorphism to the integers and we give an exact criterion for that to happen. We also give an example of a smoothly knotted n-sphere in the (n+2)-sphere for all n at le…

The paper connects two skein algebras and characterizes their representations.

In the 1970s, Birman-Craggs-Johnson used Rochlin's invariant for homology 3-spheres to construct a remarkable surjective homomorphism sigma:I_{g,1}->B_3, where I_{g,1} is the Torelli group and B_3 is a certain F_2-vector space of Boolean (square-free) polynomials. By pulling back cohomology classes and evaluating them …

Main Theorem (3.3): Let $M$ be a compact four-dimensional manifold either with curvature, positive on complex isotropic two-planes, or self-dual of positive scalar curvature. If $π_1 (M)$ admits a nontrivial unitary representation, and $M$ is orientable, then there exists a surjective homomorphism from $π_1 (M)$ on $\b…

Investigates neural codes and their embeddings, proving conjectures and introducing new code types.

A group is known as `large' if some finite index subgroup admits a surjective homomorphism onto a non-abelian free group. In this paper, we give a necessary and sufficient condition for a finitely presented group to be large, in terms of the existence of a normal series where successive quotients are finite abelian gro…

Generators of the 4-sphere's smooth mapping class group via diffeomorphisms of Montesinos twins.

New hyperbolic groups exhibit unusual finiteness properties.

Establishes a connection between skein modules and algebraic sets.

An isometric compact group action $G \times (M,g) \rightarrow (M,g)$ is called polar if there exists a closed embedded submanifold $Σ\subseteq M$ which meets all orbits orthogonally. Let $Π$ be the associated generalized Weyl group. We study the properties of the lifting action $G$ on the cotangent bundle $T^*M$. In pa…

The study bounds Dehn functions of coabelian subgroups using a second BNSR invariant.

The paper studies bundles over surfaces and constructs invariants.

The paper studies mapping class groups of nontrivial $S^2$ fiber bundles.

Proves a theorem about groups and 3-manifolds.

A new approach to symbol calculus on filtered manifolds using $C^{*}$-algebras.

A group is known as `large' if some finite index subgroup admits a surjective homomorphism onto a non-abelian free group. The main theorem of the paper is as follows. Let G be a finitely generated, large group and let g_1,...,g_r be a collection of elements of G. Then G/<<g_1^n,...,g_r^n>> is also large, for infinitely…

Proves Schoen's conjecture on tori with specific conditions.

Assume that $X$ is a homogeneous toric bundle of the form $G^{\mathbb{C}}\times_{P,τ} F$ and is Fano, where $G$ is a compact semisimple Lie group with complexification $G^\mathbb{C}$, $P$ a parabolic subgroup of $G^\mathbb{C}$, $τ:P\rightarrow (T^m)^\mathbb{C}$ is a surjective homomorphism from $P$ to the algebraic tor…

Let $S$ be a closed connected oriented surface of genus $g>0$. We study a Poisson subalgebra $W_1(g)$ of $C^{\infty}(\mathrm{Hom}(π_1(S), \mathrm{GL}(1, \mathbb{R}))/\mathrm{GL}(1, \mathbb{R}))$, the smooth functions on the moduli space of flat $\mathrm{GL}(1, \mathbb{R})$-bundles over $S$. There is a surjective Lie al…

Extended symmetric union with multiple tangle regions and Alexander polynomial properties.

Study Higgs bundles on smooth projective varieties and their restrictions to curves.

We show that the existence of an embedded compact, boundaryless hypersurface S of strictly positive mean curvature in a noncompact, connected, complete Riemannian n-manifold N of nonnegative Ricci curvature implies that the homomorphism between the fundamental groups of S and N induced by the inclusion is surjective, p…

New methods reveal rare epimorphisms linking 3-manifold groups to free groups.