Motion Groupoids and Mapping Class Groupoids

Abstract

Here $${\underline{M}}$$ M ̲ denotes a pair (M, A) of a manifold and a subset (e.g. $$A=\partial M$$ A = ∂ M or $$A=\varnothing $$ A = ∅ ). We construct for each $${\underline{M}}$$ M ̲ its motion groupoid$$\textrm{Mot}_{{\underline{M}}}$$ Mot M ̲ , whose object set is the power set $$ {{\mathcal {P}}}M$$ P M of M, and whose morphisms are certain equivalence classes of continuous flows of the ‘ambient space’ M, that fix A, acting on $${{\mathcal {P}}}M$$ 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 $$\textrm{Mot}_{{\underline{M}}}$$ Mot M ̲ of $$N\in {{\mathcal {P}}}M$$ N ∈ P M . We also construct the mapping class groupoid$$\textrm{MCG}_{{\underline{M}}}$$ MCG M ̲ associated to a pair $${\underline{M}}$$ 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 $${\underline{M}}$$ M ̲ we explicitly construct a functor $${\textsf{F}}:\textrm{Mot}_{{\underline{M}}} \rightarrow \textrm{MCG}_{{\underline{M}}}$$ F : Mot M ̲ → MCG M ̲ , which is the identity on objects, and prove that this is full and faithful, and hence an isomorphism, if $$\pi _0$$ π 0 and $$\pi _1$$ π 1 of the appropriate space of self-homeomorphisms of M are trivial. In particular, we have an isomorphism in the physically important case $${\underline{M}}=([0,1]^n, \partial [0,1]^n)$$ M ̲ = ( [ 0 , 1 ] n , ∂ [ 0 , 1 ] n ) , for any $$n\in {\mathbb {N}}$$ n ∈ N . We show that the congruence relation used in the construction $$\textrm{Mot}_{{\underline{M}}}$$ Mot 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 $$\textrm{Mot}_{{\underline{M}}}$$ Mot M ̲ and $$\textrm{MCG}_{{\underline{M}}}$$ MCG M ̲ demonstrating the utility of the constructions.