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• W3138589915 abstract "Abstract Here $${underline{M}}$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:munder> <mml:mi>M</mml:mi> <mml:mo>̲</mml:mo> </mml:munder> </mml:math> denotes a pair ( M , A ) of a manifold and a subset (e.g. $$A=partial M$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>=</mml:mo> <mml:mi>∂</mml:mi> <mml:mi>M</mml:mi> </mml:mrow> </mml:math> or $$A=varnothing $$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>=</mml:mo> <mml:mi>∅</mml:mi> </mml:mrow> </mml:math> ). We construct for each $${underline{M}}$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:munder> <mml:mi>M</mml:mi> <mml:mo>̲</mml:mo> </mml:munder> </mml:math> its motion groupoid $$textrm{Mot}_{{underline{M}}}$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:msub> <mml:mtext>Mot</mml:mtext> <mml:munder> <mml:mi>M</mml:mi> <mml:mo>̲</mml:mo> </mml:munder> </mml:msub> </mml:math> , whose object set is the power set $$ {{mathcal {P}}}M$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>P</mml:mi> <mml:mi>M</mml:mi> </mml:mrow> </mml:math> of M , and whose morphisms are certain equivalence classes of continuous flows of the ‘ambient space’ M , that fix A , acting on $${{mathcal {P}}}M$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>P</mml:mi> <mml:mi>M</mml:mi> </mml:mrow> </mml:math> . 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 $$textrm{Mot}_{{underline{M}}}$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:msub> <mml:mtext>Mot</mml:mtext> <mml:munder> <mml:mi>M</mml:mi> <mml:mo>̲</mml:mo> </mml:munder> </mml:msub> </mml:math> of $$Nin {{mathcal {P}}}M$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>P</mml:mi> <mml:mi>M</mml:mi> </mml:mrow> </mml:math> . We also construct the mapping class groupoid $$textrm{MCG}_{{underline{M}}}$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:msub> <mml:mtext>MCG</mml:mtext> <mml:munder> <mml:mi>M</mml:mi> <mml:mo>̲</mml:mo> </mml:munder> </mml:msub> </mml:math> associated to a pair $${underline{M}}$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:munder> <mml:mi>M</mml:mi> <mml:mo>̲</mml:mo> </mml:munder> </mml:math> 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 $${underline{M}}$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:munder> <mml:mi>M</mml:mi> <mml:mo>̲</mml:mo> </mml:munder> </mml:math> we explicitly construct a functor $${textsf{F}}:textrm{Mot}_{{underline{M}}} rightarrow textrm{MCG}_{{underline{M}}}$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>F</mml:mi> <mml:mo>:</mml:mo> <mml:msub> <mml:mtext>Mot</mml:mtext> <mml:munder> <mml:mi>M</mml:mi> <mml:mo>̲</mml:mo> </mml:munder> </mml:msub> <mml:mo>→</mml:mo> <mml:msub> <mml:mtext>MCG</mml:mtext> <mml:munder> <mml:mi>M</mml:mi> <mml:mo>̲</mml:mo> </mml:munder> </mml:msub> </mml:mrow> </mml:math> , which is the identity on objects, and prove that this is full and faithful, and hence an isomorphism, if $$pi _0$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:msub> <mml:mi>π</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> and $$pi _1$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:msub> <mml:mi>π</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:math> of the appropriate space of self-homeomorphisms of M are trivial. In particular, we have an isomorphism in the physically important case $${underline{M}}=([0,1]^n, partial [0,1]^n)$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:munder> <mml:mi>M</mml:mi> <mml:mo>̲</mml:mo> </mml:munder> <mml:mo>=</mml:mo> <mml:mrow> <mml:mo>(</mml:mo> <mml:msup> <mml:mrow> <mml:mo>[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>]</mml:mo> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> <mml:mo>,</mml:mo> <mml:mi>∂</mml:mi> <mml:msup> <mml:mrow> <mml:mo>[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>]</mml:mo> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> , for any $$nin {mathbb {N}}$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>N</mml:mi> </mml:mrow> </mml:math> . We show that the congruence relation used in the construction $$textrm{Mot}_{{underline{M}}}$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:msub> <mml:mtext>Mot</mml:mtext> <mml:munder> <mml:mi>M</mml:mi> <mml:mo>̲</mml:mo> </mml:munder> </mml:msub> </mml:math> 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 $$textrm{Mot}_{{underline{M}}}$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:msub> <mml:mtext>Mot</mml:mtext> <mml:munder> <mml:mi>M</mml:mi> <mml:mo>̲</mml:mo> </mml:munder> </mml:msub> </mml:math> and $$textrm{MCG}_{{underline{M}}}$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:msub> <mml:mtext>MCG</mml:mtext> <mml:munder> <mml:mi>M</mml:mi> <mml:mo>̲</mml:mo> </mml:munder> </mml:msub> </mml:math> demonstrating the utility of the constructions." @default.
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• W3138589915 date "2023-08-04" @default.
• W3138589915 modified "2023-09-27" @default.
• W3138589915 title "Motion Groupoids and Mapping Class Groupoids" @default.
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