Motion groupoids and mapping class groupoids

Abstract

Here $\underline{M}$ denotes a pair $(M,A)$ of a manifold and a subset (e.g. $A=\partial M$ or $A=\emptyset$). We construct for each $\underline{M}$ its motion groupoid $\mathrm{Mot}_{\underline{M}}$, whose object set is the power set $ {\mathcal P} M$ of $M$, and whose morphisms are certain equivalence classes of continuous flows of the `ambient space' $M$, that fix $A$, acting on ${\mathcal P} M$. These groupoids generalise the classical definition of a motion group associated to a manifold $M$ and a submanifold $N$, which can be recovered by considering the automorphisms in $\mathrm{Mot}_{\underline{M}}$ of $N\in {\mathcal P} M$. We also construct the mapping class groupoid $\mathrm{MCG}_{\underline{M}}$ associated to a pair $\underline{M}$ with the same object class, whose morphisms are now equivalence classes of homeomorphisms of $M$, that fix $A$. We recover the classical definition of the mapping class group of a pair by taking automorphisms at the appropriate object. For each pair $\underline{M}$ we explicitly construct a functor $\mathsf{F}\colon \mathrm{Mot}_{\underline{M}} \to \mathrm{MCG}_{\underline{M}}$, which is the identity on objects, and prove that this is full and faithful, and hence an isomorphism, if $\pi_0$ and $\pi_1$ of the appropriate space of self-homeomorphisms of $M$ are trivial. In particular, we have an isomorphism in the physically important case $\underline{M}=([0,1]^n, \partial [0,1]^n)$, for any $n\in \mathbb{N}$. We show that the congruence relation used in the construction $\mathrm{Mot}_{\underline{M}}$ can be formulated entirely in terms of a level preserving isotopy relation on the trajectories of objects under flows -- worldlines (e.g. monotonic `tangles'). We examine several explicit examples of $\mathrm{Mot}_{\underline{M}}$ and $\mathrm{MCG}_{\underline{M}}$ demonstrating the utility of the constructions.