Motion groupoids and mapping class groupoids

Here M denotes a pair (M, A) of a manifold and a subset (e.g. A = ∂M or A = ∅). We

construct for each M its motion groupoid MotM, whose object set is the power set PM of

M, and whose morphisms are certain equivalence classes of continuous flows of the ‘ambient

space’ M, that fix A, acting on PM. 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 MotM of N ∈ PM.

We also construct the mapping class groupoid MCGM associated to a pair 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 M we explicitly construct a functor F∶MotM → MCGM, which is the identity

on objects, and prove that this is full and faithful, and hence an isomorphism, if π0 and π1

of the appropriate space of self-homeomorphisms of M are trivial. In particular, we have an

isomorphism in the physically important case M = ([0, 1]

n

, ∂[0, 1]

n

), for any n ∈ N.

We show that the congruence relation used in the construction MotM 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 MotM and MCGM demonstrating the utility of

the constructions.