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 groupoidtextrm{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 textrm{Mot}_{{underline{M}}} of Nin {{mathcal {P}}}M. We also construct the mapping class groupoidtextrm{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 {textsf{F}}:textrm{Mot}_{{underline{M}}} rightarrow textrm{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 nin {mathbb {N}}. We show that the congruence relation used in the construction textrm{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 textrm{Mot}_{{underline{M}}} and textrm{MCG}_{{underline{M}}} demonstrating the utility of the constructions.