Motion groupoids and mapping class groupoids

Here $\underline{M}$ denotes a pair $(M,A)$ of a manifold and a subset (e.g. $A=\partial M$ or $A=\emptyset$). We construct for each $\underline{M}$ its motion groupoid $\mathrm{Mot}_{\underline{M}}$, 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 $\mathrm{Mot}_{\underline{M}}$ of $N\in {\mathcal P} M$. We also construct the mapping class groupoid $\mathrm{MCG}_{\underline{M}}$ 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 $\mathsf{F}\colon \mathrm{Mot}_{\underline{M}} \to \mathrm{MCG}_{\underline{M}}$, 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 $n\in \mathbb{N}$. We show that the congruence relation used in the construction $\mathrm{Mot}_{\underline{M}}$ 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 $\mathrm{Mot}_{\underline{M}}$ and $\mathrm{MCG}_{\underline{M}}$ demonstrating the utility of the constructions.


Introduction
Motion groups first appeared as a way to define the braid groups (rigorously studied in [14] although conjectured by Fox and Artin). Mapping class groups of a manifold and submanifold pair similarly have origins in the study of braid groups [7]. Their study has mainly focused on particular cases for which the two constructions yield the same group. The braid group can be equivalently defined as the mapping class group or as the motion group of finite sets of points in the 2-disk [7,8], and the loop braid group can be obtained as the mapping class group or as the motion group of unlinked, unknotted loops in the 3-disk [22,15,4]. (See also [12,16]).
In this paper we construct for each manifold M its motion groupoid M ot M , and its mapping class groupoid M CG M , the object set of both is the power set of M . We then study the relationship between our two constructions. In particular we construct a functor F ∶ M ot M → M CG M and give conditions for it to be an isomorphism. Here we work in the topological category following e.g [22]. However we expect that the theory holds in the smooth category as is considered in [4,41].
There is also another topological definition of the braid group, as paths in the configuration space of finite sets of points in the 2-disk [26]. We expect our groupoids to have interesting connections to configuration spaces (cf. [9]). Although we do not investigate this here.
We will present some key examples to demonstrate the properties of our construction. The category framework allows us to think about skeletons of these groupoids, note that homeomorphism between subspaces is not enough to ensure they are connected by a morphism in the motion groupoid. We also give examples for which the motion groupoid and mapping class groupoid are not isomorphic.
Another motivation for the groupoid setting is that it allows us to connect to relevant physical concepts, see 1.2 for more.
One of our motivating aims is to add a monoidal structure to the motion groupoid developed here. This will be addressed in a separate work. We also intend, in a future work, to prove a presentation for a full subcategory of the motion groupoid of points and unknotted, unlinked loops in the 3-disk, which is conjectured in 5.6.2.

Paper Overview
In this Section 1.1 we give an overview of the paper. In Section 1.2 we briefly review the physical contexts which inform our choices in these constructions (for example where motions are motions of particles in space [4,41]). In Sections 2-4 we begin by introducing notation. We also recall technology that will enable us to interpret our subsequent constructions of 'motion groupoids' so that morphisms are indeed 'motions of particles in space'.
In Section 2.1 we recall the compact-open topology on morphism sets in the category of topological spaces. It is intuitive that if there is a topology on the set of continuous maps from a space X to itself we would like it to be true that a homotopy is the same as a path into said space. (As with the internal hom adjunction in the category of sets.) With some conditions on X the compact-open topology achieves this. In Section 2.1 we recall this topology as well as giving some lemmas to aide understanding. We end this section by giving the product-hom adjunction Lemma 2.10 which gives conditions under which the usual product-hom adjunction in the category of sets becomes an adjunction in the category of topological spaces.
In Section 3 we recall the fundamental groupoid of a topological space X -Proposition 3.7. Here the aforementioned adjunction says that a homotopy of paths is the same as a path into path space. We also develop notation that will be useful later.
In Section 4 we recall Theorem 4.5 which gives conditions under which the space of selfhomeomorphisms of a space is a topological group [2]. This theorem together with the product-hom adjunction allows us to make the bridge from our mathematical construction to motions of particles in space.
In Section 4 we also, in Theorem 4.10, construct of a groupoid of self-homeomorphisms Homeo M corresponding to a manifold M , with object class the power set P(M ). In general this category is too large to be an interesting object of study itself but it is a natural first step in the construction that follows.
In Section 5 the main theorem is Theorem 5.33, the construction of the motion groupoid M ot M of a manifold M , whose object class is again the power set P(M ). We do this by first defining 'motions' in M and then imposing an equivalence. On route we have Theorem 5.23 which highlights the type of equivalence required to obtain a category. We also have a useful version of Theorem 5.33 which fixes a distinguished subset of M (e.g. ∂M ), this is Theorem 5.41. Picking a single set in P(M ) and looking at the group of automorphisms we get back the motion group constructed by Dahm [14] and developed by Goldsmith [21]. We also have Theorem 5.14 which uses the product-hom adjunction and Theorem 4.5 to say that our motions, defined as paths in the space of homeomorphisms of a manifold M , are equivalent to homeomorphisms M × [0, 1] → M × [0, 1] subject to some conditions. In Section 5.6 we have some examples with manifolds in dimensions 1 and 3. In general the object sets of our motion groupoids are very large so we look at restricting the object sets in various ways. In particular we take M = I 3 and consider objects that are certain configurations of loops and points. Looking at the automorphism groups at appropriate objects we can find well studied groups such as the familiar braid groups (see [8]) and the slightly less familiar loop braid groups (see [15], also [4,9]). We can also see a benefit of working with a motion groupoid: we can often write a more simple presentation.
The next Theorem is Theorem 6.5. Here we also start from motions in a manifold M but we use a different composition and equivalence used in the motion groupoid. In Theorem 6.9 we prove that this groupoid is isomorphic to the motion groupoid. This reformulation will be important in Section 8.
In Section 7 we have Theorem 7.3, the mapping class groupoid of a manifold M . We obtain this as a quotient of the category Homeo M . In Theorem 7.7 we have a corresponding subset-fixing version. The automorphism groups of a single object in this category is a mapping class group as described in [15].
In Section 8 we first have Theorem 8.11 in which we construct a functor from the motion groupoid of a manifold to its mapping class groupoid. We show the restriction of this functor to automorphism groups is part of the long exact sequence of homotopy groups, following the ideas used in the group case by [22]. This allows us to give, in Theorem 8.11, conditions on the space of homeomorphisms M to M under which we obtain an isomorphism between the motion groupoid of a manifold and its mapping class groupoid, and then in Theorem 8.12 the version relative to some distinguished subset. In Section 8.3 we give some examples demonstrating the use of the functor from Theorem 8.11. In particular we show that the boundary fixing motion groupoid and mapping class groupoid of D n are isomorphic for all n ∈ N and that for S 1 the mapping class groupoid and motion groupoid are not isomorphic.
Our approach will be motivated by physics so we start by discussing the relevant concepts.

Physical motivation
We are partly motivated by the particular challenge of develop mathematical models for materials useful in building quantum computers. Topological quantum field theories, and various generalisations thereof, are models potentially of this type (see for example [47,44] and cf. for example [5,6,13]). Thus we are interested in studying representations of (generalised) bordism categories and embedded bordism categories. And thus we are interested in studying generalisations of embedded bordism categories themselves (cf. for example [42,48,35,30,36,34,33,11,46,17]). However even ordinary bordism categories are not particularly well understood [31,29]. One way to approach this suite of categories is to restrict initially to the simpler subcategories of isomorphisms. If one does this with the tangle category one arrives at a category that can be shown to be isomorphic to the braid category (see [27,32,3]). From a representation theory perspective this is already a very interesting category in its own right, via the famous seven different realisations of the braid group with very different flavours, see [8]. In particular it can be realised as a motion group; and as a mapping class group. Our approach here then, to the motivating problem, is to consider what happens to (some of) the famous seven when we lift from the tangle category.
As with any lifting exercise, part of the challenge is to be not-too-narrow in the type of generalisation considered (keeping the aim in mind); and not-too-broad so as to avoid nebulous or wildly hard problems. Framing our choice of direction of generalisation is thus part of project. The 'laboratory principle' (for guiding generalisations of braids): a concrete morphism should be no more wild than can in principle be built and detected in a lab. For example, if our model is of some emergent behaviour of collections of molecules then there is no point in modelling features requiring length scales smaller than molecules, since this is beyond the range over which the emergent behaviour can possibly be defined. (Note that while mathematically smoothness, say, looks like a tough condition to be imposed and checked for, in the modelling sense it is simply an approximation to a finitary-ness/discreteness that is enforced by context. Strict smoothness requires an infinite amount of harmonic data -it just says that a lot of higher modes are exactly zero. This is no more meaningful in an emergent model than specifying infinitely many non-zero modes, as would be needed to build a non-tame knot for example. A good way of seeing what should be a relevant mode in a model is with a Hamiltonian -something well-approximated by smoothness arising because roughness requires a lot of energy.) Our next aim in this introduction is to show a little of the 'emergence' of emergent topological theories. This broadly embraces both theories relevant in topological quantum computing and in other areas of physics (cf. [38,40]). Both the emergence and the underlying systems are in general rather complex, and it is not easy to be rigorous or complete about this aspect. Nor is it necessary. The idea is simply to show the kind of mathematical objects that we considerembedded manifolds and bordisms, and the bases on which we consider them. (If a more complete picture is required then one could follow the trails started, for example, in [20,34,33].) With this aim in mind, we next show some Ising pictures, and so must also discuss caveats on Ising pictures. The pictures are of configurations of spins for the classical equilibrium Ising model on a square lattice. 'Equilibrium' means that this model does not consider dynamics. Expectation values of observables are time averaged -one considers ensembles of states rather than individual states, so there is no time evolution and so no 'motion'. A separation of physical dimensions is often spoken about (which in other settings might be space and time), but the 'transfer matrix' separation is a purely computational convenience. (Having said that, correlation functions, which probe physically useful properties, constructively resolve space dimensions into a direction of separation and the remaining perpendicular directions.) The fact that a physical state is modelled by an ensemble of configurations has baring in various ways, further complicating the story. So for our immediate purposes (towards statistical mechanics) the recasting of local configurations as emergent classes of global manifolds (with hybrid metric/topological properties) here is purely heuristic. Figures 1 and 2 show emergent topology in a low energy configuration of a lattice Ising model. Each square in this picture represents an atomic spin state. Each spin state can be in one of two configurations, represented by the black and white squares. The Ising model is a nearest neighbour interaction Hamiltonian, so adjacent spins being in different states is energetically penalised. The system certainly has a metric but at low energy it can be effectively described by a topological model. Here we have a small number of regions of black squares which we can describe by the one dimensional domain walls surrounding these regions. Movements of these manifolds preserving length do not have an energy penalty. As we increase the energy and have more regions of black squares, the topological properties disappear. We can obtain an Ising category taking objects to be boundary configurations encoding the points at which the spin changes, i.e. points in a line. Morphisms are then the one manifolds connecting them (see [33], [34] for more). The categorified form of such emergent (partly-)topological structures is hugely powerful in both computational Physics and representation theory (again see [33,47] and references therein). This is true in principle, but in practice significant technical obstructions arise. Here we set up a key special (groupoid) form of such categorifications in which all technicalities are completed.  Denote by Top the category with topological spaces (spaces) as objects and continuous maps as morphisms. Then we denote the set of morphisms from a space X to a space Y by Top(X, Y ).
We assume familiarity with Top (thus with product spaces, and so on, making it a monoidal category like the category Set). See e.g. Chapter 1 of [45] for this.
To construct our motion groupoids we will require a topology on the morphism sets in Top. For this we use the compact open topology.
In Section 2.1 we give the definition of the compact open topology and give some results demonstrating its intuitive naturality (Prop.2.6).
In addition to its intuitive naturality, the compact-open topology allows us to lift the classical product-hom adjunction in Set to an adjunction in Top (see Theorem 2.10).
We will show in Section 5.2 that this is required for our motions to have (in a suitable sense) a physically meaningful interpretation.
In Section 3 we will recall the construction of the fundamental groupoid and explore the use of the adjunction in this context.

Definition and examples
Definition 2.1. Given a set X, and a subset Y of the power set with ∪ A∈Y A = X, we write Y for the topology closure of Y . Hence the open sets in the topological space (X, Y ) are unions of finite intersections of elements in Y . We say that Y is a subbasis of (X, τ ) if Y = τ . (NB: τ = Y does not determine Y .) Definition 2.2. A neighbourhoods basis of (X, τ ) at x ∈ X is a subset B ⊂ τ , whose members are called basic neighbourhoods of x, such that every neighbourhood of x 1 contains an element of B. Definition 2.3. Let X and Y be topological spaces, then the compact open topology τ co XY on Top(X, Y ) has subbasis the set b XY of all subsets of the form Proposition 2.4. If X is the space with a single point then the τ co XY is the same in the obvious sense as the topology on Y .
Proof. The maps X → Y can be labelled by their image in Y . The only compact set K is the single point set X. The sets B(K, U ) are the sets of maps labelled by the elements of U . 1 Our conventions is that a neighbourhood of x is subset of X containing an open set containing x. 2 There are two conventions for compact-open topology. The one written here (which is the classical one) and the one where we additionally impose that each K in B(K, U ) be Hausdorff. For example [37,Chapter 5] takes the latter convention. This creates an a priori smaller set of open sets in the function space. However there will be no ambiguity issues in this paper as we will only be working with Hausdorff topological spaces. Proposition 2.5. If X is space of n points with discrete topology, τ co XY is the same in the obvious sense as the topology on Y n = Y × . . . × Y , the product of Y with itself n times.
Proof. Maps X → Y are tuples (y 1 , . . . , y n ) ∈ Y n where y i is the image of x i ∈ X and i ∈ {1, . . . , n}. All subsets of X are compact so we have Hence elements of the subbasis of τ co XY are open sets in the product topology. And basis elements in the topology on Y n are obtained from the compact open topology as follows. Let U n be a basic set in the topology on Y n , then U n is of the form U 1 × . . . × U n . Now Proposition 2.6. (A.13 in [24]) Let X be a compact space and Y a metric topological space with metric d.
is a metric on Top(X, Y ); and (ii) the compact open topology on Top(X, Y ) is the same as the one defined by the metric d ′ .
So, at least in any Euclidean setting, we have a notion of infinitesimal change in continuous maps.
Definition 2.7. We define topological space I as [0, 1] ⊂ R with the subset topology.
See Figure 3 for an example with Top(I, X) where X is a metric space.

The space TOP(X, Y ) and the product-hom adjunction
Here we construct a product hom adjunction in the category Top. (The adjunction holds subject to some conditions. These are not too restrictive for us.) In particular, the compact-open topology allows us to define a right-adjoint to the functor − × K∶ Top → Top, when K is a locally compact Hausdorff space, see Theorem 2.
It is clear that the product functor preserves the identity and respects the composition.
We will refer to this as the hom functor.
Proof. We must show that f ○ − is a continuous map. Open sets in the subbasis of τ co Y Z ′ are of the form B(K, U ) for some K a compact set in Y and U a compact set in Z ′ . The set is an open set in τ co Y Z and for any g ∈ B(K, f −1 (U )) we have f ○ g ∈ B(K, U ). Conversely suppose some h ∈ B(K, U ) can be written in the form f ○ g ′ for some g ′ ∈ TOP(Y, Z), then g ′ ∈ B(K, f −1 (U )). Lemma 2.10. Let Y be locally compact Hausdorff topological space. The product funtor − × Y ∶ Top → Top is left adjoint to the hom functor TOP(Y, −). In particular for objects X, Y, Z ∈ Top the usual hom-tensor correspondence from Set sending a set map

f is continuous); and this gives a set map
that is a bijection, natural in the variables X and Z. 3 Proof. That we have a bijection of sets is proved in Proposition A.14 of [24]. It remains to prove that this bijection is natural. Suppose we have a continuous map α∶ X ′ → X and β∶ Y → Y ′ then Looking first at the left hand vertical arrow, 3 I = [0, 1] and the path space TOP(I, X) In this section we spend some time focusing on TOP(I, X), the set of paths in X. We add a composition, and then an equivalence relation such that the composition is well defined on equivalence classes culminating in the construction of the fundamental groupoid (see Proposition 3.7). Some careful constructions can be found in the literature for example in [45] and [10], although we will use (more radical versions of) similar ideas repeatedly in later sections so we think this 'warm up' is worthwhile. This also allows us to give an often overlooked first example of the utility of the product hom adjunction from Theorem 2.10; paths in the fundamental groupoid are equivalent if and only if there is a path between them in the space of paths (Lemma 3.5). We will use path-equivalence alongside several other equivalence relations so we also introduce some careful notation here. called the path space of X. I.e. the underlying set is the set of all paths {γ∶ I → X γ is continuous} . Notation: We will use γ t for γ(t), and we say γ is a path from x to x ′ when γ 0 = x and γ 1 = x ′ . For x, x ′ ∈ X, let PX(x, x ′ ) denote the subset of paths in X with γ 0 = x and γ 1 = x ′ .
is an equivalence relation. Notation: If γ p ∼ γ ′ we say γ and γ ′ are path-equivalent . We use [γ] p for the path-equivalence class of γ.
Proof. We have that I is a locally compact Hausdorff topological space so Theorem 2.10 gives that there is a bijection between continuous maps I × I → X and continuous maps I → PX. We obtain the appropriate conditions by looking at the image of a path homotopy under this bijection. Lemma 3.6. Let X be a topological space. Path composition (see Proposition 3.2) in X passes to a well defined composition on path-equivalence classes, so for any x, x ′ , x ′′ ∈ X we have a map Proof. We check composition is well defined. Suppose γ,γ ∈ PX(x, x ′ ) are path-equivalent and so there exists a path homotopy, say H γ,γ from γ toγ. And suppose γ ′ ,γ ′ ∈ PX(x ′ , x ′′ ) are path-equivalent and so there exists a path homotopy, say H γ ′ ,γ ′ from γ ′ toγ ′ . Notice H γ,γ (1, s) = H γ ′ ,γ ′ (0, s) and so the function is a homotopy from γ ′ γ toγ ′γ .
Proposition 3.7. Let X be a topological space. There is a groupoid, π(X) such that • objects are points x in X, • morphisms from x to x ′ are path-equivalence classes in PX(x, x ′ ) p ∼ ; • composition is of morphisms [γ] p from x to x ′ and [γ ′ ] p from x ′ to x ′′ is given by [γ ′ γ] p where γγ ′ is as defined in Proposition 3.2 (note this is well defined by Lemma 3.6).
The identity morphism at each object x is the path-equivalence class of the constant path γ t = x for all t ∈ I, which we denote e x . The inverse of a morphism [γ] p from x to x ′ is the path-equivalence class ofγ t = γ 1−t . This π(X) is called the fundamental groupoid of X.
We take this opportunity to recall the definition of a groupoid before the proof (see e.g. [43]). • for each pair X, Y ∈ Ob(G) a collection G(X, Y ) of morphisms with source X and target Y (we use f ∶ X → Y to indicate that f is a morphism from X to Y ); such that: • each X ∈ Ob(X) has a designated identity morphism 1 X ∶ X → X; • for each triple of objects X, Y, Z ∈ Ob(G) there exists a function called composition; • for each pair X, Y ∈ Ob(G) a function called the inverse assigning function.
This data is subject to the following axioms: (G1) Identity law: for any morphism Proof. (Of Proposition 3.7) (G1) Suppose γ ∈ PX(x, x ′ ), the following function is a path homotopy from e x γ to γ: The following function is a homotopyγγ to e x : A similar homotopy gives γγ Definition 3.9. Let X be a topological space and A ⊂ X a subset. The fundamental groupoid of X with respect to A is the full subgroupoid of π(X) with object set A, denoted π(X, A).
Remark. Let X be a topological space and x ∈ X be a point, we have that π(X)(x, x) is the fundamental group based at x ∈ X. Here we show that TOP h (M, M ) becomes a topological group if M is a locally connected, locally compact Hausdorff space. Notice this means if we have a topological group then we also have that M satisfies the conditions of Theorem 2.10.
This section follows the proof of Theorem 4 in [2]. Recall that a space X is said to be locally compact if each x ∈ X has an open neighbourhood which is contained in a compact set. As discussed in [39], recalling that we take X to be Hausdorff, this implies that for each x ∈ X and open set U ⊂ X containing x, there exists an open set V containing x withV compact andV ⊂ U .
is a cover for K, and K is compact so there exists a finite subcover. Hence we have for some finite set {x 1 , . . . x n } ⊂ K. We can choose V = ⋃ i∈{1,...,n} V (x i ), noting that (since the union is finite) V = ⋃ i∈{1,...,n} V (x i ), and hence V is compact, since it is a finite union of compact subsets.
The previous Lemma can be stated in a more general case where the maps are not necessarily homeomorphisms, see Theorem 2.2 of [18].
where each L i is compact, connected and has non empty interior.
Since M is locally compact, for each x ∈ K we can find an open set and closed subsets of compact spaces are compact.
The V ′ (x) cover K and so there exists a finite subcover by . Taking complements and reversing the inclusion

Groupoid of self homeomorphisms Homeo M
For paths in TOP h (M, M ) to be phsyically meaningful we require that M itself has, at least locally, the Euclidean metric topology. Thus we restrict to the case M is a manifold.
So let M be a manifold possibly with boundary. (In this paper manifold means a Hausdorff topological manifold.) We will refer to M as the ambient space.
Lemma 4.6. Let (G, ○ G , e G ) be a group and X a set. Suppose we have a set map where PG is the power set of G, such that (I) for all x, y, z ∈ G, and for all g 1 ∈ κ(x, y) and g 2 ∈ κ(y, z), then g 1 ○ G g 2 ∈ κ(x, z), and Then there is a groupoid G κ such that • objects are the elements of X, • the set of morphisms between x, y ∈ X is the set κ(x, y); • composition of morphisms g 1 ∈ κ(x, y) and The identity morphism at any object where Notice each self homeomorphism f of M will belong to many Homeo A M (N, N ′ ). From here we will denote a self homeomorphism f ∈ Homeo A M (N, N ′ ) as a triple f A ∶ N ↷ N ′ to make it clear which set we are considering f as an element of.
We will use just Homeo M (N, N ′ ) and f ∶ N . N ′ for the case A = ∅.
Note that a subgroup of a topological group is itself a topological group with the induced topology.
Remark. There are various ways in which we could equip the subsets of M with extra structure. For example we could let N and N ′ be submanifolds of M equipped with an orientation and then consider homeomorphisms which preserve these orientations. • composition is given by the usual composition of functions. In our notation we have: Where A = ∅ we omit all superscripts and denote the groupoid by Homeo M .
Proof. This is precisely the groupoid obtained by letting κ be K A , as defined in Lemma 4.7, in 4.6.

Motion groupoid M ot A M
The core topological ideas used in this section are present in [22] (see also [21], [14]). Here Goldsmith constructs motion groups associated to a pair (M, N ) of a manifold and a subset, group elements are then equivalence classes of paths in the space of self homeomorphisms of M .
Here we construct a motion groupoid M ot M associated to a manifold M . The objects are subsets N ⊂ M and the morphisms are equivalence classes of 'motions'. Looking at the automorphism group at N gives back Goldsmith's motion group for the pair (M, N ). To make the notation more manageable we only give the full details of proofs the when working in Homeo M . In Section 5.5 we also construct a version using Homeo

Pre-motions
We denote the set of all pre-motions in M by P remot M .
Example 5.2. For any manifold M the path f t = id M for all t, is a pre-motion. We will denote this pre-motion Id M . Example 5.3. For M = S 1 (the unit circle) we may parameterise by θ ∈ R 2π in the usual way. Consider the functions τ φ ∶ S 1 → S 1 (φ ∈ R) given by θ ↦ θ + φ, and note that these are homeomorphisms. Then consider the path f t = τ tπ ('half-twist'). This is a pre-motion.
where p 0 is the projection onto the first coordinate) on the described subset. For any f ∈ Top(M ×I, M ), the image Θ(f ) is continuous as it is continuous on each projection.
where Φ is as in Theorem 2.10 and Θ is as described in Lemma 5.4. Under this map a pre-motion is sent to a homeomorphism M × I → M × I.
Let us see that this is continuous. Lemma 4.4 gives that f −1 is a premotion and it's image is the desired map. 4 Suppose we have pre-motions f and g. We would like to define a composition of f and g. In general the path g does not start at the end of the path f so we do the composition in two steps. First we construct a new path g ′ t = g t ○ f 1 which starts at f 1 . The composition of f and g is then the usual non-associative 'stack+shrink' composition of the paths f and g ′ in Top(I, X) as in (1). (2) Notation: We will write * (f, g) as g * f .
Proof. We check that for any f, g ∈ P remot M , g * f ∈ P remot M . We have Proposition 5.8. Let M be a manifold. There exists a set map Intuitivelyf is obtained from f by first changing the direction of travel along the path and then precomposing at each t with f −1 1 to force the reversed path to start at the identity.
Note that the operation f ↦f is an involution, namelyf = f .

Motions
the union over all pairs N, N ′ ⊂ M . N is a motion. We will call this the 'trivial motion' from N to N . Note that the pre-motion Id M becomes a motion from N to N for any N , but not a motion from N to N ′ unless N = N ′ .
Example 5.11. The half-twist of S 1 (see Example 5.3) becomes a motion in N to τ π (N ) for any N .
Example 5.12. Hence the blue path is a motion from N to N and the red path is a motion from N to N ′ .
We will now give two further equivalent definitions of motions. Definition 5.9 is the one we will work with for most proofs and technical constructions but the following definitions will be helpful to aide our intuitive understanding of what motions are.
where g * f is as defined in Equation  Proof. Note from Proposition 5.7 that g * f is a pre-motion; and we have (g * f ) 1 See Fig.7 for an example in our schematic representation. Fig.7(a) simply shows the flareschematics for two motions in a formal stack -note that this is not itself a flare-schematic for a motion, since the indicative paths are not matched at the join. Fig.7(b) deals with the adjustment at the join. It remains only to 'shrink' time -i.e. to pass to double speed.
Notice that if we turn a flare schematic upside down it is not a flare schematic, corresponding to the fact that f (1−t) is not the identity at t = 0; but the initial f −1 1 'fixes' this.

Motion-equivalence of motions
In this section we impose a equivalence relation on motions which will allow us to construct the groupoid M ot M associated to a manifold M in Theorem 5.33 which has equivalence classes of motions as morphisms.
To obtain a category, this equivalence must at least make the composition associative, and such that a unit exists. A minimal way to achieve this might be, for example, letting motions f ∶ N . N ′ and f ′ ∶ N . N ′ be equivalent if f and f ′ are path-equivalent as paths in Homeo M (∅, ∅). However we will impose a stronger equivalence. We are interested in how movements of some ambient space induce movements of subsets, but we are not interested in the movement of the ambient space per se. In Section 5.4, we will define  We proceed by first constructing a category with a more general equivalence to expose the conditions necessary to obtain an equivalence from a chosen subset of M t M .
In what follows we will say two motions are path-equivalent when we mean the underlying premotions path-equivalent. Also note for any motion f ∶ N . N ′ , for any path-equivalent The following Lemma will be useful in the proofs which subsequent proofs. (II) this path g ⋅ f is path-equivalent to g * f .
(III) Furthermore ⋅ defines an associative composition of paths and Before the proof, let us fix some conventions. Motions are paths I → Homeo M (∅, ∅) and then homotopies of paths are functions H∶ I × I → Homeo M (∅, ∅). We will always think of the first copy of I in a homotopy as the one parameterising the motion and will continue to use the parameter t. For the second copy of I which parameterises the homotopy we will use s.
Proof. (I) We first check that g ⋅ f is a continuous map. This can be seen by rewriting as The map into the product is continuous because it is continuous on each projection and the second map is continuous because Homeo M is a topological group by Theorem 4.5.
(III) It is clear ⋅ is associative.
The following function is suitable path homotopy to prove the path-equivalence Notice In fact, we will see in Section 6 that we could have used the convention g ⋅ f for composition of motions. 5 The convention * is used since this fits better with the intuitive notion of performing one motion followed by another.
In what follows when we say motions are path-equivalent we mean the underlying pre-motions are path-equivalent. Notice that, since path homotopies fix the endpoints, for any motion f ∶ N .
Proof. There exists a path homotopy from f to f ′ which we denote H f,f ′ . Now we will construct a path homotopy fromf ′ * f to Id M . First notice f 1 = f ′ 1 as path-equivalent paths have the same endpoints. Consider the following function: 2) to a motion from N to N in E. This is an equivalence relation.
We will use [f ∶ N . N ′ ] E to denote the equivalence class of f ∶ N . N ′ Proof. Let f ∶ N . N ′ and g∶ N . N ′ be motions. First notice that combining (2) and (3) we have We first check reflexivity. A motion f ∶ N . N ′ is path-equivalent to itself by Lemma 3.4 and hence, For symmetry, supposeḡ * f is path-equivalent to a motion e∶ N . N in E via some path homotopy Hḡ * f . Then the function: is a path homotopy fromf * g path-equivalent toē.
Finally, for transitivity, in addition to Hḡ * f , suppose there is a motion h∶ N . N ′ and a path homotopy Hh * g fromh * g to e ′ . Now first that the following composition is a well defined composition.
Proof. We have from Lemma 5.20 that E ∼ is an equivalence relation. We check the composition is well defined on equivalence classes. Suppose we have pairs of equivalent motions f ∶ N . N ′ , Hence there exists a path homotopy fromf ′ * f ∶ N . N to a motion e∶ N . N ∈ E and a path homotopy fromḡ ′ * g∶ N ′ . N ′ to a motion e ′ ∶ N ′ . N ′ ∈ E which we will denote by Hf′ * f and Hḡ′ * g respectively. Using Lemma 5.17 there are also homotopies fromf ′ ⋅ f to e andḡ ′ ⋅ g to e ′ , which we denote Hf′ ⋅f and Hḡ′ ⋅g , where e∶ N . N and e ′ ∶ N ′ . N ′ are motions in E.
Again using Lemma 5.17 we have Combining the homotopies in Equation (5) and (4) we have a path homotopy from the identity to f ⋅f so this gives

Now the function
By assumption there is a path homotopyf * (e ′ * f ) to e ′′ and precomposing with e gives e ⋅ (f * (e ′ * f )) p ∼ e ⋅ e ′′ p ∼ e * e ′′ where e * e ′′ ∶ N . N is in E by assumption. • composition of morphisms is given by where g * f ∶ N . N ′′ is as described in Lemma 5.15 (note this is well defined by Lemma 5.22).

The identity at object N is the motion-equivalence class of Id
N ′ ] E has inverse the E-equivalence class off ∶ N ′ . N (see Proposition 5.8).
We can similarly construct a path homotopy Id M * f to f . Hence, since Id M ∶ N . N is in E, by is a path homotopy (f * g) * h to f * (g * h). Hence by Lemma 5.19 associativity is satisfied.
The homotopy in Equation (5) gives that The case forf * f ∶ N . N is similar.

Set-stationary motions
In this section we will fix our choice of a subset E to obtain the equivalence on motions we will use to construct our motion groupoid. We do this with 'set-stationary' motions. The idea is that a motion f ∶ N . N in some M is stationary if for all t ∈ I, f t (N ) = N . In our groupoid these motions will become equivalent to the identity motion at N . How restrictive this is will depend on the topology of the points in N , see Examples 5.26-5.28.
N is path-equivalent to a set-stationary motion from N to N . Proof. By Lemma 5.29, the set S is a closed motion subset and so Lemma 5.20 gives that the described relation is an equivalence.
Proof. By Lemma 5.29 the set S is a closed motion subset and a normal motion subset so by Lemma 5.22 the composition is well defined.
Remark. Notice that the equivalence in Lemma 5.31 implies that any stationary motion In fact the previous remark is closely related to the fact that this equivalence can be obtained by constructing a pre-groupoid of motions up to path-equivalence and quotienting by the setstationary motions, which form a normal subgroupoid.
It may seem surprising that we define motion-equivalence entirely in terms of motions from N to itself for some subset N , as opposed to N to N ′ . Recall we are working with a groupoid and so inverses exist. In any groupoid, let X, Y be objects, then equivalence of two morphisms F, G∶ X → Y can be expressed by saying G −1 F ∶ X → X is equivalent to the identity morphism at X, or GF −1 ∶ Y → Y is equivalent to the identity morphism at Y .
Remark. Let f ∶ N N ′ and f ′ ∶ N . N ′ be motion-equivalent, so we have a homotopy H makinḡ f ′ * f path-equivalent to a set-stationary motion. Such a homotopy exists if and only if we have a path-equivalence between and a path in Homeo M (N ′ , N ). This has the advantage of appearing more symmetric than our initial formulation of motion-equivalence, the path (f ′ * f ) ○ f −1 1 starts at f −1 1 and ends at f ′−1 1 and N and N ′ appear in the statement of equivalence. However it has the disadvantage that we are no longer working with motions. • composition of morphisms is given by So there is a path homotopy f ′ * f to a path say x such that x∶ N . N is a stationary motion, using the same homotopy.

A-fixing motions
So far we have avoided working with A-fixing homeomorphisms to avoid overloading the notation and to add make the exposition clearer. Everything we have done so far could have been done by working instead with paths in Homeo A M (∅, ∅). We have the following adjusted definitions.  (N, N ′ ). We will denote this with In practice we will mostly be interested in the case A = ∂M .
Proof. This is the same as for Lemma 5.35.

Examples
Here we look at some examples. Note that by Lemma 5.42, the groupoid M ot − is invariant up to manifold homeomorphism (up to groupoid isomorphism) so it is enough to consider one manifold for each homoemorphism class. Here we will consider some examples in dimensions 1 and 3, which serve to illustrate different key aspects of the richness of our construction.
An interesting problem in each case is to give a characterisation of a skeleton. This is far from straightforward, even if we restrict to objects that are themselves manifolds. For example homeomorphism of objects is not enough to ensure isomorphism (see Example 5.49). Our construction also allows us to ask questions about 'inner' and 'outer' automorphisms. For what manifolds M can we find isomorphic M ot M (N, N ) and M ot M (N ′ , N ′ ) with N not homoeomorphic to N ′ . Think for example about N ⊂ I 3 which is a hopf link and N ′ = N ∖ {x} where x ∈ N is any point.
Observe that even in a skeleton most objects are undefinable so it is a good exercise to restrict to a full subgroupoid of particular interest.
Given a subset Q of the object class P(M ) of M ot A M we write M ot A M Q for the corresponding full subcategory. (Note that points in the boundary must remain fixed during any motion so if we allow these, we must consider more than just cardinality.) Notice that a similar process is possible without the compactness condition but we must distinguish different types of intervals as well as things of the form (a, b) and (c, d) The automorphism groups N → N in M ot I of a subset of I with a finite number of connected components is always trivial. This changes dramatically if more complicated subsets of I are considered. Q. This can be seen by observing that there is no homeomorphism θ∶ R → R sending Q to Z, since homeomorphisms R → R must map dense subsets to dense subsets. Z such that f t (x) = x + tn.

Dimension 3
Example 5.49. Let M = I 3 . Consider a subset N ⊂ I 3 in the interior such that N is homeomorphic to the 2-sphere S 2 . Now N separates I 3 ∖ N into two connected components. Let P be a point in the connected component of I 3 ∖ N containing the boundary, and P ′ a point in the other connected component. Then N ∪ P and N ∪ P ′ are homeomorphic but there does not exist a motion from N ∪ P to N ∪ P ′ in I 3 .

The setup
For n ∈ N denote by A n the set of objects in M ot ∂I 3 I 3 which are subsets of I 3 of the following form. Each subset is made up of n connected components labelled by a i , i ∈ {1, . . . , n}. Each a i is either a circle or a point. If a i is a point then it is the point i−1 2 n , 1 2, 1 2 . If a i is a loop, then it is a circle lying in the plane y = 1 2 with centre i−1 2 n , 1 2, 1 2 and radius 1 4n. See Let us consider some motions in M ot ∂I 3 I 3 An . First let N ∈ A 2 be the subset of I 3 which consists of a single point and a circle lying in an xz plane as shown on the left hand side of Figure 9, labelled t = 0. Let N ′ ∈ A 2 be the subset of I 3 with a loop on the left and a point on the right, as shown on the right hand side of Figure 9, labelled t = 1.
There is a continuous map σ∶ I 3 × I → I 3 (equivalently a motion) with the images of σ(N × {t}) as shown for various t in Figure 9. Looking at the image of N under σ, the point passes through the circle.
There is another motion with the same source and target during which the point does not pass through the circle.
We claim that the above two motions are not representatives of the same morphism. There is also a motion from N to N which leaves the point fixed for all t but which rotates the circle by π around an axis in the xz plane which passes through the centre of the circle. (If we added an orientation this would be a motion changing the orientation of the circle.) We claim this motion is not a representative of the same motion as the identity. A conjecture for a presentation for M ot ∂I 3 I 3 An . Let G be a graph and F (G) the free groupoid generated by G. Let R be a collection of sets of relations on the hom sets F (G)(a, b). To give a presentation of a groupoid C is to give some G and R such that the quotient groupoid F (G) R is isomorphic to C. (See [32] for more.) In this section we will give a category in terms of generators and relations and conjecture that this category is isomorphic to M ot ∂I 3 I 3 An . We construct an abstract category for each n ∈ N, which we denote D n , by giving objects, generating morphisms and relations. The objects of D n are tuples of length n with entries in {p, c}. Let {s 1 , . . . , s n } denote the generators of the Coxeter presentation of the symmetric group, S n . So s i corresponds to swapping the ith and i + 1th letters. If N ∈ Ob(D n ) is a tuple we use N si to denote the tuple with the entries in position i and i + 1 permuted.
Definition 5.50. For any n ∈ N, denote by D n the category with objects, generating morphisms and relations as follows.
The objects of D n are tuples of length n with entries in {p, c}. Let {s 1 , . . . , s n } denote the generators of the Coxeter presentation of the symmetric group, S n . S n acts on N ∈ Ob(D n ) by permuting the ith and i + 1th entries. We use N si to denote the image of N under this action.
(Note that in what follows we find it convenient to label morphisms only by a triple f ∶ X → Y , so there may exist a distinct morphism f ∶ W → Z with W ≠ X and Y ≠ Z, and f alone has no meaning.) For each N ∈ Ob(D n ) we have generating morphisms ρ i ∶ N → N si for each i ∈ {1, . . . , n − 1} subject to the following relations: And also generating morphisms σ i ∶ N → N si for each i ∈ {1, . . . , n − 1} such that the entry i + 1 of N is c, subject to the following relations: And also generating morphisms τ i ∶ N → N for each i ∈ {1, . . . , n} such that the ith entry of N is c subject to the following relations: Conjecture 5.51. For each n ∈ N we have an isomorphism That is to say D n is a presentation of A n .
We conjecture such an isomorphism from the combinatorially defined groupoid above to the full subgroupoid of the motion groupoid M ot ∂ Note that if we further restrict to the subset N ∈ A n where all connected components are points, this is the usual presentation of the nth braid group [8] and if all connected components are circles this is the usual presentation of the nth loop braid group [15]. 6 Relative fundamental groupoid rpM ot A M In this section we will prove in Theorem 6.9 that there is another equivalent way to define M ot M . This gives us another way to understand motions, in particular we have a new definition of the equivalence of motions (see Lemmas 6.8 and 6.7). This equivalence the same as the equivalence of relative paths in the relative fundamental set of a pair of spaces. Hence it will allow us to use the relative homotopy long exact sequence to prove the relationship between motion groupoids and mapping class groupoids in section 8. • H(t, 0) = f (t) for all t ∈ I, and • H(t, 1) = g(t) for all t ∈ I.
We call such a homotopy a relative path homotopy.
For transitivity, suppose that (g∶ N . N ′ ) rp ∼ (h∶ N . N ′ ) via a homotopy, say H g,h . Then Proof. Lemma 5.17 gives that ⋅ is an associative composition of motions. From Lemma 6.2 we have that rp ∼ is an equivalence relation. We check that the composition is well defined on equivalence classes. • composition of morphisms is given by where g ⋅ f is described in Lemma 5.17 (note this is well defined by Lemma 6.4).
Each morphism [f ∶ N . N ′ ] rp has inverse the relative path-equivalence class of Proof Remark. Note that by the following Lemma we could have defined the composition in the groupoid using f * g.
Proof. Let H be a relative path homotopy from f ∶ N . N ′ to f ′ ∶ N . N ′ . We must show that N is path-equivalent to a stationary motion from N to N .
Notice first that H(1, 1 − s) is a path f ′ 1 to f 1 which is in Homeo M (N, N ′ ) for all t, we relabel this path as γ. We defineγ asγ = γ ○ f ′−1 1 , soγ ∶ N ′ . N ′ is a stationary motion withγ 1 = f 1 ○ f ′−1 1 . First note that we can use H to construct a path homotopy from f to the path composition γf ′ . Explicitly a suitable function is: For fixed s ∈ I the path H 1 (t, s) starts at the identity, traces the whole of the path H(t, s) followed by the part of the path γ starting from γ 1−s = H(1, s) and ending at γ 1 . Note that the path composition, γf ′ is precisely the motion compositionγ * f ′ , so f p ∼γ * f ′ . Hence by gluing H 1 with the trivial homotopy along t = 1 2, we have thatf ′ * f is pathequivalent tof ′ * (γ * f ′ ). Now using the normalcy of stationary motions proved in Lemma 5.29, we have that the motion f ′ * (γ * f ′ )∶ N . N is path-equivalent to a stationary motion from N to N .
N ) by uniqueness of inverses. Let H be the path homotopy f ′ * f ∶ N ′ . N ′ to a stationary motion γ ∶ N ′ . N ′ . Consider the following function: Using Lemma 5.17 we have that γ * f Theorem 6.9. Let M be a manifold. There is an isomorphism of categories Proof. We first check Φ is a functor. Lemma 6.7 gives that the map Φ is well defined. These are in general a simpler construction than motions groupoids and there are many known results already in the literature [23,7,25]. In Section 8 we will construct a functor from the motion groupoid of a manifold M to the mapping class groupoid and obtain conditions for this to be an isomorphism.
Recall from Section 4 that for a manifold M , the notation f∶ N ↷ N ′ means a self homeomor- We call such a map an isotopy from f∶ N ↷ N ′ to g∶ N ↷ N ′ .
For transitivity suppose also that (g∶ Lemma 7.4. Let M be a manifold, isotopy is a congruence on Homeo M . This means that for any subsets N, N ′ , N ′′ ⊂ M , the following composition of motion-equivalence classes is well defined Moreover, this is precisely the composition in the quotient category. Notation: We will use [f∶ Proof. (This follows from the Alexander trick [1].) Suppose we have Notice that f 0 = id D 2 and f 1 = f and each f t is continuous. Moreover: is a continuous map. So we have constructed an isotopy from any boundary preserving self homeomorphism of D 2 to id D 2 .
Note that a lot more is true. The same argument gives that the space of maps D 2 → D 2 fixing the boundary is contractible; see [23].
Remark. Note that if K is a finite subset of D 2 ∖ ∂D 2 then the morphism group M CG ∂D 2 D 2 (K, K) is actually isomorphic to the braid group in K strands. For discussion see [8,7]. See also [15] for a thorough exposition of how loop braid groups arise as morphisms groups of the form M CG ∂ D 3 (L, L) where L consists of a set of unknotted loops contained in the interior of D 3 .

Functor from M ot
It is known that the braid groups and loop braid groups can be seen obtained as mapping class groups, as well as as motion groups [15,14,22,7]. In this section we use the homotopy long exact sequence of a space to prove in Theorem 8.11 that there is an isomorphism of categories M ot M to M CG M if Homeo M (∅, ∅) has only one path connected component and its fundamental group is trivial.
Proof. We first check F is well defined. By Theorem 6.9 two motions f ∶ N . N ′ and f ′ ∶ N . N ′ are motion-equivalent if and only if they are relative path-equivalent, i.e. if we have homotopy: satisfying the conditions in Definition 6.1. Then H(1, s) is a path f 1 to f ′ 1 such that for all s ∈ I, H(1, s) ∈ Homeo M (N, N ′ ). Hence they are isotopic self homeomorphisms.
We check F preserves composition.

Long exact sequence of relative homotopy groups
In Theorem 8.11 we will prove the functor F ∶ M ot M → M CG M is an isomorphism if and only if π 1 (Homeo M (∅, ∅)) and π 0 (Homeo M (∅, ∅)) are trivial. To do this we will use the homotopy long exact sequence. We briefly introduce this here but see Section 4.1 of [24] or Chapter 9 of [37] for a more thorough exposition. as follows. We say γ ∼ γ ′ if there exists H∶ I n × I → X such that • for all t ∈ I, H I n ×{t} is a map (I n , ∂I n , J n−1 ) → (X, A, x 0 ), • for all x ∈ I n , H(x, 0) = γ(x), and • for all x ∈ I n , H(x, 1) = γ ′ (x). This is an equivalence relation. Notation: We will call this set with the described equivalence the n th relative homotopy set and denote it π n (X, A, x 0 ).
Proof. We omit this proof but it is similar to Lemma 6.2. See also [24]. Proof. By definition.
Notation: Due to the fact the two equivalences coincide on sets we are interested in we will use [γ] rp for the equivalence class of γ in a relative homotopy set. Lemma 8.6. Let X be a topological space, A a subset and {x 0 } a point in A. For n ≥ 2, given continuous maps β∶ (I n , ∂I n , J n−1 ) → (X, A, {x 0 }) and γ∶ (I n , ∂I n , J n−1 ) → (X, A, {x 0 }), Define Then there is a composition Proof. Suppose β, β ′ ∶ (I n , ∂I n , J n−1 ) → (X, A, x 0 ) are equivalent in π(X, A, {x 0 }) via some homotopy, say H 1 . Similarly suppose γ and γ ′ are maps (I n , ∂I n , J n−1 ) → (X, A, x 0 ) which are equivalent in π(X, A, {x 0 }) via some homotopy, say H 2 . Then is a homotopy making γ + β equivalent to γ ′ + β ′ in (X, A, {x 0 }). Proof. See [24].
We also define a map which is the following restriction: Note that for n = 1, we have Let X be a space, A ⊂ X a subspace and x 0 ∈ A a basepoint. There is a long exact sequence: where exactness at the end of the sequence, where group structures are not defined, means the image of one map is equal to the set of maps sent to the homotopy class of the identity by the next.
Note the following long exact sequence generalises the sequence which appears in [22].
where all maps are group maps andF are Φ (the restrictions to the morphism group of ) the functors defined in Theorem 8.1 and Theorem 6.9 respectively.
Proof. We have from Lemma 7.  Proof. We have from Lemma 8.2 that F is full. We check F is faithful. Suppose we have motions

By Lemma 8.10 this is true if and only if
which is equivalent to saying Id M * (f ′ * f ) is path-equivalent to a stationary motion, and hence thatf * f is path-equivalent to the stationary motion (since Id M * (f ′ * f ) Proof. The proof proceeds exactly as for the previous theorem. Proof. We proved in Proposition 7.8 that M CG ∂D D 2 (∅, ∅) = π 0 (Homeo ∂D 2 D 2 (∅, ∅), id M ) is trivial. Also Homeo ∂D 2 D 2 (∅, ∅) is contractible, see e.g. Theorem 1.1.3.2 of [23]. Hence by Theorem 8.11 we have the result.

Examples using long exact sequence
Remark. As we recalled in Subsection 7.1, if K is a set with n-elements in the interior of D 2 , then the morphism group M CG ∂D 2 D 2 (K, K) is isomorphic to the braid group in n strands. The previous proposition hence implies that the group M ot ∂D 2 D 2 (K, K) is isomorphic to the braid group in n-strands. This isomorphism was (from what we know) first noticed in [14,22].
Remark. In fact Homeo ∂D m D m (∅, ∅) contractible for all m. This follows from the Alexander Trick [1]. Hence the same argument as for the n = 2 case proves that we have an isomorphism If L is an unlinked set of n loops in D 3 , then this means that the loop braid group [15,4] can either be defined as M CG ∂D 3 D 3 (L, L) or as M ot ∂D 3 D 3 (L, L). This latter isomorphism was also mentioned in [15,22].
We say a few words about what happens if we do not fix the boundary of the disk in the mapping class groupoid as we think it adds some nice intuition. Let P 2 ⊂ D 2 be a subset consisting of two points as shown in Figure 6. Consider a motion whose associated pre-motion is a πt rotation of the disk for any t ∈ I, which exchanges the two particles as shown in the figure.
If we do this motion once then its equivalence class is non trivial in M ot D 2 , and its end point also represents a non trivial element of M CG D 2 . Now consider the motion P 2 . P 2 obtained from the 2π rotation of D 2 , and hence 'remembering' all intermediate stages of the rotation, it is intuitively clear this motion is non-trivial in M ot D 2 by considering the as its image as a homeomorphism D 2 × I → D 2 × I, see Figure 11. A proof follows from the fact that the worldlines of the trajectory of the points in P 2 transcribe a non-trivial braid. However its endpoint is a 2π rotation, which clearly represents [id D 2 ∶ P 2 ↷ P 2 ] i in M CG D 2 . In fact, the map M ot D 2 → M CG D 2 is neither full nor faithful. The space Homeo D 2 is homotopy equivalent to S 1 ⊔ S 1 , where the first connected component corresponds to orientation preserving homeomorphisms and the second orientation reversing (see Section 1.1 of [23]). Hence we have that π 1 (Homeo D 2 (∅, ∅), id D 2 ) = Z where the single generating element corresponds to the 2π rotation. And π 0 (Homeo D 2 (∅, ∅), id D 2 ) = Z 2Z. So we have an exact sequence: . . . → π 1 (Homeo D 2 (N, N ), id D 2 ) Example 2: the 1-circle S 1 . The unit circle S 1 is an interesting example of very simple manifold with different motion and mapping class groupoids.
Let P ⊂ S 1 be a subset containing a single point in S 1 . Similarly to the disk, there is a nontrivial morphism in M ot S 1 (P, P ) represented by a 2π rotation of the circle, see Figure 12. We Figure 12: Example of motion of circle which is a 2π rotation carrying a point to itself.
can prove this using the long exact sequence. Note that the connected component containing id S 1 of Homeo S 1 (P, P ) is contractible, see section 1 of [23]. In particular π 1 (Homeo S 1 (P, P ), id S 1 ) is trivial. Also from [23] we have that S 1 is a strong deformation retract of Homeo S 1 (∅, ∅). Hence the sequence becomes . . . → {1} → Z → M ot S 1 (P, P ) → M CG S 1 (P, P ) → Z 2Z.
The exact sequence gives an injective map Z ≅ π 1 (Homeo S 1 (∅, ∅), id S 1 ) → M ot S 1 (P, P ). Explicitly this monomorphism sends n ∈ Z to the equivalence class of the pre-motion tracing a 2nπ rotation of the circle S 1 . The space Homeo S 1 (P, P ) only has two connected components, consisting of orientations preserving and orientation reversing homeomorphisms of S 1 fixing P , each of which is connected. In particular it follows that the projection map M ot S 1 (P, P ) → M CG S 1 (P, P ) ≅ Z 2Z is the trivial group map, for its image only contains isotopy equivalence classes of orientation preserving homeomorphisms. The exact sequence hence becomes: In particular the equivalence class of the 2π rotation of S 1 is non-trivial in M ot S 1 (P, P ), even though its image in M CG S 1 (P, P ) is trivial.
This can be seen directly by choosing the points to be antipodal, say the north and south pole. Now consider a 2π rotation with axis through north and south pole. This is a path fixing both points, hence a stationary path which is equivalent to the identity.
Looking back at the exact sequence, we have that the map M ot S 2 (P 2 , P 2 ) → M CG S 2 (P 2 , P 2 ) is injective. Combining the results of Hamstrom with the result on pg.52 of [19], we have that M CG S 2 (P 2 , P 2 ) ≅ Z 2Z×Z 2Z where the non trivial element in the first copy of Z 2Z is represented by a self homeomorphism which swaps the points by an orientation preserving self homeomorphism, and the non trivial element in the second component is represented by a self homeomorphism swapping the two points with is orientation reversing. Hence a motion which swaps the two points represents a non trivial morphism in M ot S 2 (P 2 , P 2 ).
Let M CG + S 2 be the mapping class groupoid constructed using only orientation preserving homeomorphisms. Then we have a group isomorphism M ot S 2 (P 2 , P 2 ) ≃ M CG + S 2 (P 2 , P 2 ).
Note this does not extend to a category isomorphism. Considering instead the subset consisting of three points the groups are non isomorphic. Intuitively we can see this by arguing that we cannot place three points on the sphere such that any 2π rotation is a stationary motion. But as with the previous examples a 2π rotation of the sphere represents the identity morphism in the mapping class groupoid.