Related papers: Configuration spaces and braid groups on graphs in…
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…
The paper surveys topological problems relevant to the motion planning problem of robotics and includes some new results and constructions. First we analyse the notion of topological complexity of configuration spaces which is responsible…
We consider the space of all configurations of finitely many (potentially nested) circles in the plane. We prove that this space is aspherical, and compute the fundamental group of each of its connected components. It turns out these…
Configuration spaces form a rich class of topological objects which are not usually presented to an undergraduate audience. Our aim is to present configuration spaces in a manner accessible to the advanced undergraduate. We begin with a…
We show that the homology of ordered configuration spaces of finite trees with loops is torsion free. We introduce configuration spaces with sinks, which allow for taking quotients of the base space. Furthermore, we give a concrete…
Let G be a graph. The (unlabeled) configuration space of n points on G is the space of all n-element subsets of G. The fundamental group of such a configuration space is called a graph braid group. We use a version of discrete Morse theory…
This paper is concerned with problems relevant to motion planning in robotics. Configuration spaces are of practical relevance in designing safe control schemes for robots moving on a track. The topological complexity of a configuration…
A topological theory initiated recently by the author uses methods of algebraic topology to estimate numerically the character of instabilities arising in motion planning algorithms. The present paper studies random motion planning…
We investigate an algebraic problem related to the determination of the fundamental group of a class of spaces of configurations on surfaces. The configuration spaces are spaces of points grouped into colors. Whether two points are allowed…
In this paper we determine the topological complexity of configuration spaces of graphs which are not necessarily trees, which is a crucial assumption in previous results. We do this for two very different classes of graphs: fully…
If $G$ is a graph with vertex set $V$, let Conf$_n^{\text{sink}}(G,V)$ be the space of $n$-tuples of points on $G$, which are only allowed to overlap on elements of $V$. We think of Conf$_n^{\text{sink}}(G,V)$ as a configuration space of…
We present a mathematical framework for describing the topology of configuration spaces for particles on one-connected graphs. In particular, we compute the homology groups over integers for different classes of one-connected graphs. Our…
It is a classical result that configuration spaces of labelled particles in $\mathbb{R}^d$ are free $E_d$-algebras and that their $d$-fold bar construction is equivalent to the $d$-fold suspension of the labelling space. In this paper, we…
We discuss various aspects of "braid spaces'' or configuration spaces of unordered points on manifolds. First we describe how the homology of these spaces is affected by puncturing the underlying manifold, hence extending some results of…
We are concerned with orderable groups and particularly those with orderings invariant not only under multiplication, but also under a given automorphism or family of automorphisms. Several applications to topology are given: we prove that…
We study the problem of computing the homology of the configuration spaces of a finite cell complex $X$. We proceed by viewing $X$, together with its subdivisions, as a subdivisional space--a kind of diagram object in a category of cell…
Given a finite graph G and a topological space Z, the graphical configuration space Conf(G, Z) is the space of functions V(G) -> Z so that adjacent vertices map to distinct points. We provide a homotopy decomposition of Conf(G, X x Y) in…
This paper is intended to provide concrete examples of concepts discussed elsewhere in this volume, especially splittings of groups and non-positively curved cube complexes but also other things. The idea of the construction (configuration…
In this paper we consider the generalized anchored configuration spaces on $n$ labeled points on a~graph. These are the spaces of all configurations of $n$ points on a~fixed graph $G$, subject to the condition that at least $q$ vertices in…
For a given bundle $\xi \colon E \to M$ over a manifold, configuration-section spaces on $\xi$ parametrise finite subsets $z \subseteq M$ equipped with a section of $\xi$ defined on $M \smallsetminus z$, with prescribed "charge" in a…