arXiv research

3-manifold groups are rigid under certain conditions.

Proves Grothendieck-Teichmüller group acts on specific mapping class groups.

3-manifold groups are uniquely identifiable via their profinite completions.

Study automorphism groups of braid groups with 4 or more strings.

Study on string links invariant under associator choice and Grothendieck--Teichmüller group action.

Polytopes in R^n with integral vertices form a monoid under the Minkowski sum, and the Grothendieck construction gives rise to a group. We show that every symmetric polytope is a norm in this group for every n.

Develops equivariant Chern characters for coherent sheaves with group actions.

We identify the Grothendieck group of the tangle Floer dg algebra with a tensor product of certain $U_q(gl(1|1))$ representations. Under this identification, up to a scalar factor, the map on the Grothendieck group induced by the tangle Floer dg bimodule associated to a tangle agrees with the Reshetikhin-Turaev homomor…

Shows natural quasi-Poisson structure on multiplicative Grothendieck-Springer space.

The aim of this note is to take benefit of the foam nature of the Khovanov-Kuperberg algebras to compute the Grothendieck groups of their categories of finitely generated projective modules. The computation relies on the Hattori-Stallings trace and some geometrical properties of foams in a solid torus.

The paper studies the commutator subgroups of twin groups and their properties.

This paper attempts to relate some ideas of Grothendieck in his Esquisse d'un programme and some of the recent results on 2-dimensional topology and geometry. Especially, we shall discuss the Teichmüller theory, the mapping class groups, $SL(2, \bold C)$ representation variety of surface groups, and Thurston's theory o…

Computes second homology groups of orbifold groups, proving profinite rigidity and Grothendieck pairs.

Study of Fukaya category on surfaces with mapping class group action.

Grothendieck's Esquisse d'un programme is often referred to for the ideas it contains on dessins d'enfants, the Teichm{ü}ller tower, and the actions of the absolute Galois group on these objects or their etale fundamental groups. But this program contains several other important ideas. In particular, motivated by surfa…

The study explores rigidity in groups and their products, finding uncountable families of non-isomorphic subgroups.

We use the Grauert--Grothendieck complex on differentiable spaces to study basic relative forms on the inertia space of a compact Lie group action on a manifold. We prove that the sheaf complex of basic relative forms on the inertia space is a fine resolution of Bryliski's sheaf of functions on the inertia space.

The paper shows plentiful non-homotopy finite Poincaré duality spaces.

We review some ideas of Grothendieck and others on actions of the absolute Galois group Γ Q of Q (the automorphism group of the tower of finite extensions of Q), related to the geometry and topology of surfaces (mapping class groups, Teichm{ü}ller spaces and moduli spaces of Riemann surfaces). Grothendieck's motivation…

Introduces arithmetic analogues of Orr invariants and spaces for absolute Galois groups.

This paper characterizes Kashiwara-Vergne groups using algebraic structures of knotted tubes.

We give a necessary and sufficient condition for lagrangians in a symplectic vector bundle to be deformed stably into transversal lagrangians. In the case of three lagrangians, we show that the associated Grothendieck group can be identified with a Hermitian K-theory group.

The study explores how different Grothendieck topologies and functors between categories preserve locality.

The paper proves stabilization in hypersurface sections using Grothendieck rings and probabilistic methods.

We construct the Fukaya category of a closed surface equipped with an area form using only elementary (essentially combinatorial) methods. We also compute the Grothendieck group of its derived category.

Affine actions of Lie groups are studied and their K-theory is related to standard K-theory.

The paper introduces pseudo-quotients for algebraic actions and applies them to character varieties.

Proves a theorem for complex flat vector bundles using differential forms.

We give a direct proof that the Freed-Lott differential analytic index is well defined and a condensed proof of the differential Grothendieck-Riemann-Roch theorem. As a byproduct we also obtain a direct proof that the R/Z analytic index is well defined and a condensed proof of the R/Z Grothendieck-Riemann-Roch theorem.

We show that the Grothendieck group associated to integral polytopes in $\mathbb{R}^n$ is free-abelian by providing an explicit basis. Moreover, we identify the involution on this polytope group given by reflection about the origin as a sum of Euler characteristic type. We also compute the kernel of the norm map sendin…

We compute the equivariant $K$-theory $K_G^*(G)$ for a simply connected Lie group $G$ (acting on itself by conjugation). We prove that $K_G^*(G)$ is isomorphic to the algebra of Grothendieck differentials on the representation ring. We also study a special example of a non-simply connected Lie group $G$, namely PSU(3),…

Develops a new index theory for odd Z/kZ K-theory.

Our aim of this and subsequent papers is to enlighten (a part of, presumably) arithmetic structures of knots. This paper introduces a notion of profinite knots which extends topological knots and shows its various basic properties. Particularly an action of the absolute Galois group of the rational number field on prof…

Study of colored triangulations linked to symmetric groups.

We give a combinatorial description of closed curves on oriented surfaces in terms of certain permutations, called charts. We describe automorphisms of curves in terms of charts and compute the total number of curves counted with appropriate weights. We also discuss relations between curves, Grothendieck dessins d'enfa…

We show that for complex analytic K3 surfaces any torsion class in H^2(X,O_X^*) comes from an Azumaya algebra. In other words, the Brauer group equals the cohomological Brauer group. For algebraic surfaces, such results go back to Grothendieck. In our situation, we use twistor spaces to deform a given analytic K3 surfa…

The monoids l_{2q+1}(Z[π]) detect s-cobordisms amongst certain bordisms between stably diffeomorphic 2q-dimensional manifolds and generalise the Wall simple surgery obstruction groups, L_{2q+1}^s(Z[π]) \subset l_{2q+1}(Z[π]). In this paper we give exact sequences which completely describe l_{2q+1}(Z[π]) as a set and wh…

Study of orbifold Chern character using superconnections.

In this paper we give a simplified proof of the flat Grothendieck-Riemann-Roch theorem. The proof makes use of the local family index theorem and basic computations of the Chern-Simons form. In particular, it does not involve any adiabatic limit computation of the reduced eta-invariant.

In his 1944 paper Veränderliche Riemannsche Flächen , Teichmüller defined a structure of complex manifold on the set of isomorphism classes of marked closed Riemann surfaces of genus g. The complex manifold he obtained is the space called today Teichmüller space. In the same paper, Teichmüller introduced the so-called …

Explains historical connections between vector bundle splitting and Riemann-Hilbert problems.

Study automorphisms of profinite mapping class groups for surfaces with negative Euler characteristic.

The geometry of cosets in the subgroups H of the two-generator free group G =\textless{} a, b \textgreater{} nicely fits, via Grothendieck's dessins d'enfants, the geometry of commutation for quantum observables. Dessins stabilize point-line incidence geometries that reflect the commutation of (generalized) Pauli opera…

The paper constructs families of algebraic and hyperbolic manifolds sharing invariants.

The study explores infinitesimal automorphisms of principal bundles and their implications.

Paper constructs Chern character for coherent sheaves.

Researchers prove Lie algebras of differential operators and Grothendieck constructions coincide.

New method solves SDPs for MaxCut and synchronization problems efficiently.