id: 05911869
dt: j
an: 05911869
au: Deeley, Kenneth
ti: Configuration spaces of thick particles on a metric graph.
so: Algebr. Geom. Topol. 11, No. 4, 1861-1892 (2011).
py: 2011
pu: Geometry \& Topology Publications, Mathematics Institute, University of
    Warwick, Coventry; Mathematical Sciences Publishers, Berkeley, CA
la: EN
cc: 
ut: topology of configuration spaces; metric graph; PL topology; topological
    robotics
ci: 
li: doi:10.2140/agt.2011.11.1861
ab: From the abstract: “We study the topology of configuration spaces
    $F_r(Γ,2)$ of two thick particles (robots) of radius $r>0$ moving on a
    metric graph $Γ$. As the size of robots increases, the topology of
    $F_r(Γ,2)$ varies. Given $Γ$ and $r$, we provide an algorithm for
    computing the number of path-connected components of $F_r(Γ,2)$. Using
    our main tool of $\mathrm{PL}$ Morse-Bott theory, we show that there
    are finitely many critical values of $r$ where the homotopy type of
    $F_r(Γ,2)$ changes. We study the transition across a critical value
    $R\in (a,b)$ by computing the ranks of the relative homology group of
    $(F_a(Γ,2), F_b(Γ,2))$.”  Some definitions: Let $X$ be a space. The
    $n$th ordered configuration space $F(X,n)$ is defined by setting
    $$F(X,n)=\{(x_1,\ldots,x_n) \in X^n  |  x_i\not=x_j \text{ for }
    i\not=j\}$$ with the subspace topology of the Cartesian product $X^n$.
    In this article, the space $X$ is a metric graph $Γ$; the thick
    configuration space of radius $r>0$ is defined by setting
    $$F_r(Γ,2)=\{(x,y)\inΓ |  d(x,y)\geq 2r\}$$ with the subspace
    topology of $F(Γ,2)$.
rv: Jie Wu (Singapore)