arXiv research

Complex projective manifolds without rational curves are quotients of Abelian varieties.

Study homology groups of mapping and Torelli groups for surfaces with abelian covers.

Homology of abelian differentials stabilizes with more zeros.

It was shown in my earlier article that the splice diagram of a rational homology sphere graph manifold determines the manifolds universal abelian cover. In this article we use the proof of this to give a condition on the splice diagram to determine when the universal abelian cover itself is a rational homology sphere.

The paper disproves the existence of certain subgroups with nontrivial rational abelianization.

Researchers determine the rational abelianization of a subgroup of mapping class groups.

The rational cohomology ring of A_3, the moduli space of abelian 3-folds is computed. This is isomorphic to the the rational cohomology ring of the group Sp_3(Z) of 6x6 integral symplectic matrices. The main ingredients in the computation are (1) Looijenga's computation of the rational cohomology ring of M_3, the modul…

Study shows compact mapping class groups of infinite type surfaces are never perfect.

We compute the asymptotic dimension of the rationals given with an invariant proper metric. Also, we show that a countable torsion abelian group taken with an invariant proper metric has asymptotic dimension zero.

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

Extends Chern character to non-abelian cohomology, linking to physics.

This paper gives an explicit formula for the SL_2(C)-non-abelian Reidemeister torsion as defined in [Dub06] in the case of twist knots. For hyperbolic twist knots, we also prove that the non-abelian Reidemeister torsion at the holonomy representation can be expressed as a rational function evaluated at the cusp shape o…

To a rational homology sphere graph manifold one can associate a weighted tree invariant called splice diagram. It was shown earlier that the splice diagram determines the universal abelian cover of the manifold. We will in this article turn the proof of this in to an algorithm to explicitly construct the universal abe…

Study the monodromy and center-focus problems for rational maps defined by products of generic lines.

Study Torelli subgroups of handlebody groups and their cohomology.

In this paper, we consider the formal power series whose n-th coefficient is the number of copies of a given finite graph in the ball of radius n centred at the identity element in the Cayley graph of a finitely generated group and call it the growth function. Epstein, Iano-Fletcher and Uri Zwick proved that the growth…

By a Cantor group we mean a topological group homeomorphic to the Cantor set. The author earlier proved that every compact metric space of rational cohomological dimension n can be obtained as the orbit space of a Cantor group action on a metric compact space of covering dimension n. In this paper, we consider actions …

Study orbits in right triangles, deducing periodic billiard paths and classifying orbit closures.

Controlled $K$-theory is used to show that algebraic $K$-theory of virtually abelian groups is described by an assembly map defined using possibly-infinite hyperelementary subgroups. The Farrell-Jones summand (coming from infinite subgroups) is parameterized by the rational projective space of the group, and a reduced …

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 structure theorem to projective klt varieties with specific tangent sheaf properties.

We compute the rational stable homology of the automorphism groups of free nilpotent groups. These groups interpolate between the general linear groups over the ring of integers and the automorphism groups of free groups, and we employ functor homology to reduce to the abelian case. As an application, we also compute t…

Paper proves non-triviality of Johnson kernel torsion subgroup.

New insights into knot surgeries via instanton 2-torsion.

A nice trick for studying the billiard flow in a rational polygon is to unfold the polygon along the trajectories. This gives rise to a translation or half-translation surface tiled by the original polygon, or equivalently an Abelian or quadratic differential. Veech surfaces are a special class of translation surfaces …

To a rational homology sphere graph manifold one can associate a weighted tree invariant called splice diagram. In this article we prove a sufficient numerical condition on the splice diagram for a graph manifold to be a singularity link. We also show that if two manifolds have the same splice diagram, then their unive…

We calculate the first homology group of the mapping class group with coefficients in the first rational homology group of the universal abelian $\Z / L \Z$-cover of the surface. If the surface has one marked point, then the answer is $\Q^{τ(L)}$, where $τ(L)$ is the number of positive divisors of $L$. If the surface i…

We establish a close connection between stable commutator length in free groups and the geometry of sails (roughly, the boundary of the convex hull of the set of integer lattice points) in integral polyhedral cones. This connection allows us to show that the scl norm is piecewise rational linear in free products of Abe…

We prove that the space $M(K(x,y))$ of $\mathbb R$-places of the field $K(x,y)$ of rational functions of two variables with coefficients in a totally Archimedean field $K$ has covering and integral dimensions $\dim M(K(x,y))=\dim_\IZ M(K(x,y))=2$ and the cohomological dimension $\dim_G M(K(x,y))=1$ for any Abelian 2-di…

The paper studies webs formed by rational curves on moduli spaces and their abelian relations.

The paper studies orbifold splice quotients and log covers of surface pairs.

The operation of (untwisted) Whitehead doubling trivializes the Alexander module of a knot (and consequently, all known abelian invariants), and converts knots to topologically slice ones. In this note we show that Whitehead doubling does not trivialize the rational function that equals to the 2-loop part of the Kontse…

M5-branes' flux quantization linked to non-abelian cohomology.

The paper resolves fundamental groups for three exceptional surface singularity families.

We analyze the degree-two part of the Torelli group's associated graded.

Given a rational homology sphere M, whose splice diagram satisfy the semigroup condition, Neumann and Wahl were able to define a complete intersection surface singularity called splice diagram singularity from the splice diagram of M. They were also able to show that under an additional hypothesis on M called the congr…

In this paper, it is explained that a topological invariant for 3-manifold $M$ with $b_1(M)=1$ can be constructed by applying Fukaya's Morse homotopy theoretic approach for Chern--Simons perturbation theory to a local system on $M$ of rational functions associated to the free abelian covering of $M$. Our invariant take…

We determine the abelianizations of the following three kinds of graded Lie algebras in certain stable ranges: derivations of the free associative algebra, derivations of the free Lie algebra and symplectic derivations of the free associative algebra. In each case, we consider both the whole derivation Lie algebra and …

Recently L. Nicolaescu and the author formulated a conjecture which relates the geometric genus of a complex analytic normal surface singularity (whose link $M$ is a rational homology sphere) with the Seiberg-Witten invariant of $M$ associated with the ``canonical'' $spin^c$ structure of $M$. Since the Seiberg-Witten t…

It has long been known that every quasi-homogeneous normal complex surface singularity with Q-homology sphere link has universal abelian cover a Brieskorn complete intersection singularity. We describe a broad generalization: First, one has a class of complete intersection normal complex surface singularities called "s…

For $4 \nmid L$ and $g$ large, we calculate the integral Picard groups of the moduli spaces of curves and principally polarized abelian varieties with level $L$ structures. In particular, we determine the divisibility properties of the standard line bundles over these moduli spaces and we calculate the second integral …

We prove an additivity property for the normalized Seiberg-Witten invariants with respect to the universal abelian cover of those 3-manifolds, which are obtained via negative rational Dehn surgeries along connected sum of algebraic knots. Although the statement is purely topological, we use the theory of complex singul…

Over the moduli space of rank $n$ semi-stable lattices is a universal family of tori. Along the fibers, there are natural differential operators and differential equations, particularly, the heat equations and the Fokker-Planck equations in statistical mechanics. In this paper, we explain why, by taking averages over t…

Let $M$ be a symplectic manifold, equipped with a Hamiltonian action of a torus $T$. We give an explicit formula for the rational cohomology ring of the symplectic quotient $M//T$ in terms of the cohomology ring of $M$ and fixed point data. Under some restrictions, our formulas apply to integral cohomology. In certain …

In this paper we prove trace formulae for the Reidemeister number of a group endomorphism. This result implies the rationality of the Reidemeister zeta function in the following cases: the group is a direct product of a finite group and a finitely generated Abelian group; the group is finitely generated, nilpotent and …

Let M be an orientable topological manifold of dimension m, m greater or equal to 5, with fundamental group $Γ$. Let S(M) be the topological structure set, endowed with the group structure induced by its identification with Ranicki's algebraic structure set. We prove that the (rationalized) rho map $ρ_Γ: S(M)\rightarro…

Study invariants of $\mathbb{Z}/p$-homology 3-spheres from abelianization of mapping class groups.

If $M$ is a finite volume complete hyperbolic 3-manifold with one cusp and no 2-torsion, the geometric component $X_M$ of its $\SL(2,\BC)$-character variety is an affine complex curve, which is smooth at the discrete faithful representation $ρ_0$. Porti defined a non-abelian Reidemeister torsion in a neighborhood of $ρ…