Motion Groupoids and Mapping Class Groupoids

Abstract Here $${\underline{M}}$$ M ̲ denotes a pair ( M , A ) of manifold and subset (e.g. $$A=\partial M$$ A = ∂ or $$A=\varnothing $$ ∅ ). We construct for each its motion groupoid $$\textrm{Mot}_{{\underline{M}}}$$ Mot whose object set is the power {{\mathcal {P}}}M$$ P morphisms are certain equivalence classes continuous flows ‘ambient space’ that fix acting on $${{\mathcal . These groupoids generalise classical definition group associated to submanifold N which can be recovered by considering automorphisms in $$N\in N ∈ also mapping class $$\textrm{MCG}_{{\underline{M}}}$$ MCG with same class, now homeomorphisms recover taking at appropriate object. For we explicitly functor $${\textsf{F}}:\textrm{Mot}_{{\underline{M}}} \rightarrow \textrm{MCG}_{{\underline{M}}}$$ F : → identity objects, prove this full faithful, hence an isomorphism, if $$\pi _0$$ π 0 _1$$ 1 space self-homeomorphisms trivial. In particular, have isomorphism physically important case $${\underline{M}}=([0,1]^n, \partial [0,1]^n)$$ ( [ , ] n ) any $$n\in {\mathbb {N}}$$ show congruence relation used construction formulated entirely terms level preserving isotopy trajectories objects under flows—worldlines monotonic ‘tangles’). examine several explicit examples demonstrating utility constructions.