Related papers: Studying complex manifolds by using groups $G_{n}^…

Recently the first named author defined a 2-parametric family of groups $G_n^k$. Those groups may be regarded as analogues of braid groups. Study of the connection between the groups $G_n^k$ and dynamical systems led to the discovery of the…

We construct a map from fundamental groups of complements to some plane configurations to the groups $G_{n}^{k}$ for large $k$. We discuss connection between the groups $G_{n}^{k}$ for different $G_{n}^{k}$ and their geometric realization.

Take a bounded symmetric domain $D$ and an arithmetic subgroup $\Gamma$ of ${\rm Aut}(D)$. Take the quotient $D/\Gamma$, compactify and resolve the singularities. We study the fundamental group of the compact complex manifolds that result…

In \cite{HigherGnk}, the author has constructed natural maps from fundamental groups of topological spaces (restricted configuration spaces) to the groups $G_{n}^{k}$. In the present paper, we show that in the case of $n=k+1$, the group…

The spaces of triangulations of a given manifold have been widely studied. The celebrated theorem of Pachner~\cite{Pachner} says that any two triangulations of a given manifold can be connected by a sequence of bistellar moves, or Pachner…

We introduce a family of groups $\Gamma_n^k$ for integer parameters $n>k$. These groups originate from discussion of braid groups on $2$-surfaces. On the other hand, they turn out to be related to 3-manifolds (in particular, they lead to…

We describe moduli spaces of invariant generalized complex structures and moduli spaces of invariant generalized K\"ahler structures on maximal flag manifolds under $B$-transformations. We give an alternative description of the moduli space…

In the first part of this paper we study geometric formality for generalized flag manifolds, including full flag manifolds of exceptional Lie groups. In the second part we deal with the problem of the classification of invariant almost…

In the first part of this article, which contains three of them, we have identified the notion of level $N$ strong modular unit. It enabled us to structure the modular forms family $(M_{2k}(\Gamma_0(N)))_{k\in \mathbb{N}^*}$ and to propose…

The present paper mainly presents, for example, explicit classifications of compact smooth manifolds having non-empty boundaries and simple structures where the dimensions are general. Studies of this type is fundamental and important. They…

We obtain some new results on classical solutions of two dimensional Euclidean sigma models. From earlier work of Din-Zakrzewski, Glaser-Stora, and numerous differential geometers, one knows explicit solutions in the case of the…

We study fundamental groups of non compact Riemannian manifolds. We find conditions which ensure that the fundamental group is trivial, finite or finitely generated.

Let $C_n$ denote a cyclic group of order $n$. In this paper we investigate modules and chain complexes over the constant integral Mackey functor $\underline{\mathbb{Z}}$ and perform some related homological calculations. Along the way we…

This is the first in a series of papers in which we describe explicit structural properties of spaces of diagonal rectangular harmonic polynomials in $k$ sets of $n$ variables, both as $GL_k$-modules and $S_n$-modules, as well as some of…

The main goal of our paper is the study of several classes of submanifolds of generalized complex manifolds. Along with the generalized complex submanifolds defined by Gualtieri and Hitchin (we call these ``generalized Lagrangian…

Based on the notion of strong modular form developed in Part I, we propose to structure the family of cuspidal modular form spaces $(S_{2k}(\Gamma_0(N)))_{k\in \mathbb{N}^*}$ and to determine bases for each of these spaces, once known bases…

Let S be a surface of genus g with n points removed, G a connected Lie group, and X(G) the moduli space of representations of the fundamental group of S into G. We compute the fundamental group of X(G) when n>0 and G is a real or complex…

We define a certain compactifiction of the general linear group and give a modular description for its points with values in arbitrary schemes. This is a first step in the construction of a higher rank generalization of Gieseker's…

We study fundamental groups of algebraic stacks. We show that these fundamental groups carry an additional structure coming from the inertia groups. Then use this additional structure to analyze geometric/ topological properties of stacks.…

In this paper, the authors apply a stratification of moduli spaces of complex Lie algebras to analyzing the moduli spaces of nxn matrices under scalar similarity and bilinear forms under the cogredient action. For similar matrices, we give…