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 $A=\varnothing$). 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 $\mathsf{F}:{\mathrm{Mot}}_{\underline{M}}\to {\mathrm{MCG}}_{\underline{M}}$, 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.

arXiv commentVersion accepted for publication in CMP. 75 pages, 14 figures

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