Related papers: Relative homological representations of framed map…

Given a manifold with corners $X$, we associates to it the corner structure simplicial complex $\Sigma_X$. Its reduced K-homology is isomorphic to the K-theory of the $C^*$-algebra $\mathcal{K}_b(X)$ of b-compact operators on $X$. Moreover,…

Let $p:\Sigma'\to\Sigma$ be a finite Galois cover, possibly branched, with Galois group $G$. We are interested in the structure of the cohomology of $\Sigma'$ as a module over $G$. We treat the cases of branched and unbranched covers…

We compare the marked length spectra of isometric actions of groups with non-positively curved features. Inspired by the recent works of Butt we study approximate versions of marked length spectrum rigidity. We show that for pairs of…

Let $G$ be a finite group acting on a connected compact surface $\Sigma$, and $M$ be an integer homology 3-sphere. We show that if each element of $G$ is extendable over $M$ with respect to a fixed embedding $\Sigma\rightarrow M$, then $G$…

We study configuration spaces of framed points on oriented closed smooth manifolds. Such configuration spaces admit natural actions of the framed little discs operads, that play an important role in the study of embedding spaces of…

A Riemannian orbifold is a mildly singular generalization of a Riemannian manifold that is locally modeled on $R^n$ modulo the action of a finite group. Orbifolds have proven interesting in a variety of settings. Spectral geometers have…

Complex tight frames can be canonically viewed as elements of a complex Stiefel manifold. We present a class of spaces of such frames which are simply connected relative to the subspace topology. To this class belongs the space of finite…

We address the question of determining which mapping class groups of infinite-type surfaces admit nonelementary continuous actions on hyperbolic spaces. More precisely, let $\Sigma$ be a connected, orientable surface of infinite type with…

This is a survey of Magnus representations with particular emphasis on their applications to mapping class groups and monoids (groups) of homology cobordisms of surfaces. In the first half, we begin by recalling the basics of the Fox…

We construct a graph complex calculating the integral ho- mology of the bordered mapping class groups. We compute the ho- mology of the bordered mapping class groups of various surfaces. Using the circle action on this graph complex, we…

Let $S_{g,1,p}$ be an orientable surface of genus $g$ with one boundary component and $p$ punctures. Let $\mathcal{M}_{g,1,p}$ be the mapping-class group of $S_{g,1,p}$ relative to the boundary. We construct homomorphisms…

Symmetric homology is an analog of cyclic homology in which the cyclic groups are replaced by symmetric groups. The foundations for the theory of symmetric homology of algebras are developed in the context of crossed simplicial groups using…

We classify, up to few exceptions, the orbit closures of the $\mathrm{Mod}(\Sigma)$-action on the affine character variety $\chi(\mathrm{Aff}(\mathbb{C}))$. We obtain from this classification that the only obstruction for a non-abelian…

We construct membrane homology groups $\h(M)$ associated with each compact connected oriented smooth manifold, and show that $\h(M)$ is matrix graded algebra.

This article is dedicated to the computation of an explicit presentation of some asymptotically rigid mapping class groups, namely the braided Higman-Thompson groups. To do so, we use the action of these groups on the spine complex, a…

The mapping class group of a surface with one boundary component admits numerous interesting representations including as a group of automorphisms of a free group and as a group of symplectic transformations. Insofar as the mapping class…

It has long been known that the combinatorial properties of a graph $\Gamma$ are closely related to the group theoretic properties of its right angled artin group (raag). It's natural to ask if the graph homomorphisms are similarly related…

The closure of a braid in a closed orientable surface $\Sigma$ is a link in $\Sigma\times S^1$. We classify such closed surface braids up to isotopy and homeomorphism (with a small indeterminacy for isotopy of closed sphere braids),…

This paper concerns rigidity of the mapping class groups. We show that any homomorphism $\phi:{\rm Mod}_g\to {\rm Mod}_h$ between mapping class groups of closed orientable surfaces with distinct genera $g>h$ is trivial if $g\geq 3$ and has…

Let ${\mathcal M}_g$ be the moduli space of compact connected Riemann surfaces of genus $g\geq 2$ and let $\widehat{{\mathcal M}_g}$ be its Deligne-Mumford compactification, which is stratified by the topological type of the stable Riemann…