Related papers: Mapping class groups and function spaces: a survey

Consider a compact surface of genus at least two. We prove that the first cohomology group of the mapping class group with coefficients in the space of algebraic functions on the SL(2, C) moduli space vanishes.

We prove that the mapping class group is not an $h$-cobordism invariant of high-dimensional manifolds by exhibiting $h$-cobordant manifolds whose mapping class groups have different cardinalities. In order to do so, we introduce a moduli…

We define orbifold mapping class groups (with marked points) and study them using their action on certain orbifold analogs of arcs and simple closed curves. Moreover, we establish a Birman exact sequence for suitable subgroups of orbifold…

We compute the cohomology ring of a generalised type of configuration space of points in $\mathbb{R}^r$. This configuration space is indexed by a graph. In the case the graph is complete the result is known and it is due to Arnold and…

We study some relation between some geometrically defined classes of diffeomorphisms between manifolds and the $L_{q,p}$-cohomology of these manifolds. Some applications to vanishing and non vanishing results in $L_{q,p}$-cohomology are…

We construct a topological space to study contextuality in quantum mechanics. The resulting space is a classifying space in the sense of algebraic topology. Cohomological invariants of our space correspond to physical quantities relevant to…

Fold maps are smooth maps at each singular point of which it is represented as the product map of a Morse function and the identity map. Round fold maps are, in short, such maps the sets of all singular points of which are embedded…

The aim of this paper is to survey some aspects of mapping class groups with focus on their finite dimensional representations arising in topological quantum field theory.

In this paper, we review a method for computing and parameterizing the set of homotopy classes of chain maps between two chain complexes. This is then applied to finding topologically meaningful maps between simplicial complexes, which in…

We compute small rational models for configuration spaces of points on oriented surfaces, as right modules over the framed little disks operad. We do this by splitting these surfaces in unions of several handles. We first describe rational…

The article is devoted to a structure of topological spaces related with topological quasigroups. Regular and complete spaces over topological quasigroups are studied. Separations and embeddings are also investigated for them. Their…

We classify generic coadjoint orbits for symplectomorphism groups of compact symplectic surfaces with or without boundary. We also classify simple Morse functions on such surfaces up to a symplectomorphism.

We study the interrelation of space functions of groups and the space complexity of the algorithmic word problem in groups.

We compute the stable cohomology groups of the mapping class groups of compact orientable surfaces with one boundary, with twisted coefficients given by the homology of the unit tangent bundle of the surface. This stable twisted cohomology…

In this paper we first survey some basic results in the cohomology of finite groups, and then discuss recent work on constructing free actions of finite groups on products of spheres.

Let $M$ be a smooth connected orientable closed surface and $f_0\in C^\infty(M)$ a function having only critical points of the $A_\mu$-types, $\mu\in\mathbb N$. Let ${\mathcal F}={\mathcal F}(f_0)$ be the set of functions $f\in C^\infty(M)$…

We determine the first homology group of the mapping class group M(N) of a nonorientable surface N with coefficients in H_1(N;Z).

We define a kind of moduli space of nested surfaces and mappings, which we call a comparison moduli space. We review examples of such spaces in geometric function theory and modern Teichmueller theory, and illustrate how a wide range of…

We study the action of (big) mapping class groups on the first homology of the corresponding surface. We give a precise characterization of the image of the induced homology representation.

This expository article describes applications of topological configuration spaces to the control of robotic systems. In particular, we review recent work by the authors on configuration spaces of graphs. These are lovely spaces: we show…