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 setP M of M, and whose morphisms are certain equivalence classes of continuous flows of the ‘ambient space’ M, that fix A, acting on 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 MotM of N ∈ P M. 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.