Theory and Applications of Categories, Vol. 9, No. 7, 2001, pp. 121–150.

THE HURWITZ ACTION AND BRAID GROUP ORDERINGS

JONATHON FUNK

ABSTRACT. In connection with the so-called Hurwitz action of homeomorphisms in ramified covers we define a groupoid, which we call a ramification groupoid of the 2sphere, constructed as a certain path groupoid of the universal ramified cover of the 2-sphere with finitely many marked-points. Our approach to ramified covers is based on cosheaf spaces, which are closely related to Fox’s complete spreads. A feature of a ramification groupoid is that it carries a certain order structure. The Artin group of braids of n strands has an order-invariant action in the ramification groupoid of the sphere with n + 1 marked-points. Left-invariant linear orderings of the braid group such as the Dehornoy ordering may be retrieved. Our work extends naturally to the braid group on countably many generators. In particular, we show that the underlying set of a free group on countably many generators (minus the identity element) can be linearly ordered in such a way that the classical Artin representation of a braid as an automorphism of the free group is an order-preserving action.

Introduction

Patrick Dehornoy [8, 9] has discovered through a connection with self-distributive operations that an Artin braid group carries a left-invariant linear ordering. Geometric explanations of the ordering have subsequently appeared. Fenn, Greene, Rolfsen, Rourke and Wiest [10] have presented such an explanation, and Thurston has suggested another approach based on Nielsen [25, 28], which has been described and analyzed in Short and Wiest [27]. Christian Kassel has provided a description of Dehornoy’s discovery and a survey of all these achievements in a recent article [17].

The author’s previous interest [13] in ramified, or branched covers (in the sense of Fox [11]) has led to the present investigation of a connection between the so-called Hurwitz action and linear orderings of braid groups. Our explanations focus on a certain path groupoid of a ramified cover, which we shall call a ramification groupoid. The objects of this groupoid are taken to be the elements of the branch-point set of the ramified cover. One of our goals is to show how these groupoids are involved in braid group orderings. In particular, we describe an order-structure that a ramification groupoid carries.

Our approach involves cosheaf spaces [12], which we review in §2. Cosheaf spaces are defined by an adjointness with cosheaves, but they have a topological characterization that is almost equivalent to Fox’s notion of a complete spread. Fox had introduced complete spreads as a framework by which to study topologically ramified covers. It may interest the reader that a cosheaf can be equivalently regarded as a topos distribution in the sense of Lawvere [20, 21]. We shall not pursue this connection with topos distributions, but the reader can consult [6] (and references cited therein) for information and recent developments concerning topos distributions. Remark 11.4 explains another connection with topos theory through the classifying topos for the theory of dense linear orders [26].

Received by the editors 2000 December 31 and, in revised form, 2001 December 17.

Published on 2001 December 21 in the volume of articles from CT2000.

2000 Mathematics Subject Classification: 18D10, 18D30, 20F36.

\begin{matrix}c \\ ◯ \end{matrix} Jonathon Funk, 2001. Permission to copy for private use granted.

The role played by the Poincaré disk and its circle at infinity in Nielsen, and in Short and Wiest, is in this investigation played by a ramified cover. Remark 6.3 explains how the ramified cover and the Poincaré disk are related. From Nielsen [28]:

It now proves useful to close the universal covering surface by addition of its “points at infinity”; in order to make these conveniently accessible, a conformal mapping of the hyperbolic plane onto the interior of a disk is used. Just as proper elements of the fundamental group are given by finite products of the generators, the attempt to characterize individual boundary points leads us to consider improper group elements, given by infinite sequences of generators.

The points at infinity that we require are represented by the objects of the ramification groupoid. We can thus avoid words with infinitely many ‘edge’-symbols by introducing a second type of symbol, a ‘vertex’-symbol, which is used to encode the groupoid objects. But we can pass from the groupoid to the collection of infinite words by a ‘spiral’ map (Remark 13.12).

If Y [figure: ψ]X is a ramified cover, then composition with a homeomorphism X [figure: h]X produces another ramified cover h · ψ. This action of homeomorphisms in ramified covers has been called the Hurwitz action (surveyed in [4], but also see [3, 18]). If we regard two ramified covers ψ and ψ′ to be equivalent just when there is a homeomorphism Y [figure]Y′ over X, then a ramified cover ψ is fixed under this action by a homeomorphism

h precisely when h can be lifted to a homeomorphism Y [figure: h̃]Y, meaning that ψ ·h̃ = h·ψ. We are particularly interested in the question: what ψ are fixed by a given collection of homeomorphisms h? Or can we even find one such non-trivial ψ? It is desirable to be able to do this functorially in the sense that for a given collection of homeomorphisms X [figure: h]X can we find a ramified cover ψ, and a section of the monoidal functor

Σ(ψ) = {(h, h̃) | h̃ is a lifting of h over ψ} [figure]Aut(X) ; (h, h̃) ↦ h ?

The morphisms in the monoidal^{1} categories Σ(ψ) and Aut(X) may vary depending on the particular purpose one has in mind. Aut(X) acts in X (or in the fundamental groupoid π_{1}(X)), but notice that a section of the above functor yields an action of Aut(X) in the map ψ (or in the homomorphism ψ_{∗}). For example, let S denote a 2-sphere, and suppose that in Aut(S) we take for morphisms just those isotopies in S that permute n marked-points and pointwise fix an open neighbourhood of an n + 1^{st} one. The group of isomorphism classes of objects S [figure]S (homeomorphisms) of this monoidal category is isomorphic to the Artin group of braids of n strands, denoted B_{n}. In effect, a section of the above functor provides a representation of B_{n}.

1These categories are only ‘almost monoidal’ because composition of isotopies is not associative, and the constant isotopies are not identities for vertical composition. This technicality does not interest us here.

In §3 we construct a certain universal ramified cover ψ_{m} of a 2-sphere with m markedpoints as a cosheaf space. (For the reader interested in ‘exponential’ matters, this ψ_{m} is an instance of a cosheaf space that is not exponentiable in spaces over the 2-sphere - see Remark 3.6.) The ramification groupoid is derived from ψ_{m}. We then produce an action of B_{n} in the ramification groupoid of ψ_{n+1} in the manner we have tried to describe in the previous paragraph. It follows that this action respects the order structure of the ramification groupoid, so that a free element of the action provides a (left)-invariant linear ordering of the braid group (§12).

In §13 we show that these methods extend naturally to the countably generated braid group B_{∞}. We obtain the following result (Corollary 13.9): the free group on countably many generators (except the identity) can be linearly ordered in such a way that the classical Artin action [1, 15] of B_{∞} in the free group is order-preserving. This ordering of the free group is not a group ordering in the usual sense of being invariant under multiplication on the left (or right). Although this action has no free elements, for any n it has elements that are free for the subgroup B_{n}. Free elements for the whole action may be obtained by admitting infinite words. We arrive at Corollary 13.9 by considering another action of B_{∞} in the free group, which is isomorphic to Artin’s. Larue [19] has considered this other action in connection with the “Dehornoy bracket.” Remark 13.11 explains another connection with Dehornoy [9].

An appropriate setting in which to examine paths in a ramified cover is the Sierpinski fibration, as we shall call it. We shall begin in §1 by providing some details about homotopy theory in the Sierpinski fibration.

Acknowledgements: The author has benefited greatly from discussions with several people, and it is a pleasure to take this opportunity to thank them. Bert Wiest has explained to the author aspects of his work with H. Short [27]. Christian Kassel and Patrick Dehornoy have also offered the author with valuable advice and information. Many thanks to the organizers of the Como 2000 category theory conference, and also to the organizers of the Marshall Colloquiumfest where preliminary reports had been made. The author would also like to thank the participants of the topology seminar at the University of Saskatchewan. Lastly, one of the referees has offered several clarifications that have improved the paper, for which the author is most grateful.

1. Homotopy theory over Sierpinski space

Let Tsp denote the category of topological spaces and continuous functions (henceforth called maps). Let Sier (for Sierpinski) denote the space consisting of two elements {0, 1}, such that {1} is open and {0} is not. An object of Tsp//Sier is a map X [figure: U]Sier, which

we also denote as X_{U}, and a morphism X_{U} [figure]Y_{V} is a map X [figure: f]Y such that U ≤ V · f in the specialization order for maps in Tsp. Equivalently, an object X_{U} is a topological space X together with a designated open set U, and a morphism X_{U} [figure]Y_{V} is a map

X [figure: f]Y such that U ⊆ f^{−1}V. Let [X_{U}, Y_{V}] denote the set of such morphisms.

We have a functor

X : Tsp//Sier [figure]Tsp ,

which associates with an object X_{U} the space X (X_{U}) = X. This functor has left and right adjoints given by L(Z) = Z_{ϕ}, where ϕ denotes the empty space, and respectively R(Z) = Z_{Z}. These adjoints are full and faithful. Furthermore, R has a right adjoint U given by U(X_{U}) = U. Thus, we have adjoints

L ⊣ X ⊣ R ⊣ U .

The functors U and X are related by a natural transformation t : U ⇒ X, where the component morphism t_{X_{U}} is equal to the inclusion U ⊆ X. The natural transformation Rt is equal to the composite of the counit RU ⇒ id with the unit id ⇒ RX.

1.1. Definition. We shall refer to X (with all the adjoints, and the base and total

categories) as the Sierpinski fibration. A morphism X_{U} [figure: f]Y_{V} for which U = f^{−1}V is said to be cartesian.

Every map X [figure: f]X (Y_{V}) = Y has a cartesian lifting: f^{⋆}(Y_{V}) [figure: f]Y_{V}, where f^{⋆}(Y_{V})

denotes X_{f^{−1}V}. Furthermore, every morphism X_{U} [figure: f]Y_{V} factors uniquely as a vertical morphism X_{U} [figure]f^{⋆}(Y_{V}) followed by a cartesian morphism.

The functors L, X, and R preserve colimits, and X, R, and U preserve limits. We use these facts to examine limits and colimits in Tsp//Sier. For instance, the product X_{U} × Y_{V} must be isomorphic to (X × Y)_{U×V}, but an infinite product may not exist.

We next investigate exponentials in the Sierpinski fibration. For the following proposition, recall that if an exponential Y^{X} exists in Tsp, then its underlying set must be equal to the collection of maps X [figure]Y.

1.2. Proposition. If Y_{V} ^{X_{U}} exists, then Y^{X} exists, given by X (Y_{V} ^{X_{U}}), and the designated open set U(Y_{V} ^{X_{U}}) is equal to [X_{U}, Y_{V}]. Conversely, if Y^{X} exists in Tsp and if [X_{U}, Y_{V}] is an open set of Y^{X}, then Y_{V} ^{X_{U}} exists, given by Y^{X} with designated open set [X_{U}, Y_{V}].

Proof. The adjoint pair L ⊣ X satisfies

L(Z × X (Y_{V})) ≅ L(Z) × Y_{V} .  (1)

Therefore, we have

X (Y_{V} ^{X_{U}}) ≅ X (Y_{V})^{X (X_{U})} = Y^{X} .

Of course, the elements of the open set U(Y_{V} ^{X_{U}}) are in bijection with maps 1 [figure]U(Y_{V} ^{X_{U}}), which are in bijection with morphisms R(1) [figure]Y_{V} ^{X_{U}}. In turn, these correspond to morphisms X_{U} [figure]Y_{V}. The converse is just as easily established.

1.3. Remark. R ⊣ U does not satisfy condition (1), and U does not preserve exponentials.

We are going to consider homotopy theory in the Sierpinski fibration. Let I denote the real unit interval [0, 1].

1.4. Definition. A homotopy between two morphisms in the Sierpinski fibration is a morphism

H : X_{U} × R(I) [figure]Y_{V} ,

where X_{U} × R(I) = (X × I)_{U×I}. An isotopy is a homotopy H such the pairing

X_{U} × R(I) [figure]Y_{V} × R(I)

of H with the projection X_{U} ×R(I) [figure]R(I) is an isomorphism in the Sierpinski fibration.

If Y_{V} ^{X_{U}} exists, then by transposing, a homotopy may be equivalently given as a map

Ĥ : I [figure]U(Y_{V} ^{X_{U}}) .

Let (I) denote the open unit interval, and consider the object I_{(I)}. By a path p in X_{U} we shall mean a morphism

p : I_{(I)} [figure]X_{U}

in the Sierpinski fibration.

1.5. Example. Let S denote a 2-sphere, and let U denote an open subset of S. Then S_{U} is an object of the Sierpinski fibration. When S − U is a finite set we refer to S_{U} as a finitely marked 2-sphere. A path in S_{U} is a path in S that maps (I) into U.

Since the inverse operation in Aut(S) (= homeomorphisms S [figure]S) is continuous for the compact-open topology, it follows that an isotopy in S_{U} is equivalently given as a map H : S × I [figure]S such that each map H( , t) is a homeomorphism of S that permutes the marked-points.

Let Path(X_{U}) denote the set of paths in X_{U}, which is [I_{(I)}, X_{U}] in our previous notation. It is well-known that the exponential X^{I} always exists in Tsp: it is the set of paths in X, endowed with the compact-open topology. Thus we may always consider Path(X_{U}) as a topological space by regarding it as a subspace of X^{I}. By Proposition 1.2, if Path(X_{U}) is an open set of X^{I}, then X_{U}^{I_{(I)}} exists and we have Path(X_{U}) = U(X_{U}^{I_{(I)}}). Then by adjointness, the bijection expressed in the following proposition must hold. This bijection holds even when X_{U}^{I_{(I)}} does not exist. We omit the straightforward proof.

1.6. Proposition. We have a functor

Path : Tsp//Sier [figure]Tsp .

There are natural bijections

\frac{Z [figure]Path(X_{U})}{I_{(I)} × R(Z) [figure]X_{U}}

between maps in Tsp and morphisms in the Sierpinski fibration. Thus, Path has a left adjoint. In particular, Path preserves limits.

1.7. Remark. Our concept of path in the Sierpinski fibration is apparently flawed because there may not be a way to compose paths (as in S_{U}). For our purposes this apparent flaw is really a valuable feature because for a particular object X_{U} a natural composition for (homotopy classes of) paths in X_{U} may be available. The universal ramified cover of S_{U} provides an illustration of this (§3).

1.8. Definition. A homotopy of paths in X_{U} is a morphism

H : I_{(I)} × R(I) [figure]X_{U}

in the Sierpinski fibration.

By Proposition 1.6, a homotopy may be equivalently regarded as a map

Ĥ : I [figure]Path(X_{U}) .

We may speak of paths in the Sierpinski fibration with common endpoints by using the domain and codomain natural transformations: Path ⇒ X. Then we may consider homotopy equivalence classes of paths with common endpoints, in the sense of Definition 1.8.

1.9. Definition. Let us denote the collection of homotopy classes of paths in X_{U} with common endpoints by π_{1}(X_{U}). Its objects (or vertices) are the points of X, while U supplies the morphisms (or edges).

1.10. Remark. The collection π_{1}(X_{U}) is a directed graph with an involution, but as we have said it may not have a law of composition.

When we use π_{1}(X) with no subscript on X, we shall mean the ordinary fundamental groupoid.

2. Review of cosheaf spaces and complete spreads

We review the notion of cosheaf space [12] and the slightly more general notion of complete spread, due to R. Fox. Our terminology is a mixture coming from [2, 11, 12]. We first review complete spreads. Following [11], a spread is a continuous map φ : Y [figure]X, where Y is locally connected, such that the components of sets φ^{−1}(U), for U open in X, are a base for the topology on Y. We shall assume throughout that the domain space of a spread is locally connected, even though the notion can be sensibly generalized to the case of an arbitrary domain space by using quasi-components [11, 22]. In order to formulate completeness we recall that a cogerm of a map φ : Y [figure]X at x^{∈} X (with Y locally connected) is a consistent choice of components α = {α_{U} ⊆ φ^{−1}(U)}, where U ranges over all neighbourhoods of x. By consistent, we mean that U ⊆ V ⇒ α_{U} ⊆ α_{V}. Then a spread over a space X is complete if for every x^{∈} X, and every cogerm α at x, the intersection ⋂_{x ∈ U} α_{U} is non-empty.

We next turn to cosheaf spaces. For a given map φ : Y [figure]X (with Y locally connected) consider the collection

Ỹ = {(x, α) | α is a cogerm of φ at a point x^{∈} X} .

Ỹ is topologized by the basic sets

(U, β) = {(x, α) | x^{∈} U, α_{U} = β} ,

where U is an open set of X, and β is a connected component of φ^{−1}(U). We refer to the topological space Ỹ as the display space associated with the cosheaf U ↦ π_{0}(φ^{−1}(U)), where π_{0} denotes connected components. Ỹ is continuously fibered over X in the obvious way. A fiber over a given x_{0} ^{∈} X is sometimes called a costalk; it consists of the collection of pairs {(x_{0}, α)}. A costalk may also be regarded as the limit

\underleftarrow{lim}_{x_{0} ∈ U}π_{0}(φ^{−1}(U)) ,

taken over the filter of open neighbourhoods of x_{0}. Ỹ is locally connected. Every element y^{∈} Y determines a cogerm at φ(y): take α_{U} to be the unique component of φ^{−1}(U) that contains y. This defines a continuous map from Y into Ỹ (over X), denoted

η : Y [figure]Ỹ ; η(y) = (φ(y), α) .

The inverse image set η^{−1}(U, β) is equal to β. A cosheaf space is then a map φ : Y [figure]X for which Y is locally connected and η is a homeomorphism. If Y is locally connected and

Y [figure: φ]X is any map, then the canonical projection Ỹ [figure: φ̃]X is a cosheaf space. Furthermore, there are adjoint functors connecting cosheaves and cosheaf spaces, which induces an equivalence between the category of cosheaf spaces over X and a full subcategory of cosheaves on X (called the spatial cosheaves in [12]). The above map η is the unit of this adjointness. The inclusion of spatial cosheaves in cosheaves has a right adjoint.

Cosheaf spaces and complete spreads are nearly the same. The space Ỹ is precisely Fox’s construction of the completion of a spread. A map with locally connected domain over a T_{1} space is a cosheaf space if and only if it is a complete spread with T_{1} domain ([12], 5.17). The following result from [14] may help clarify the notion of cosheaf space and its connection with complete spreads. 2.1. Proposition. (Topological characterization of cosheaf spaces [14].) For any map φ : Y [figure]X with locally connected domain, the following are equivalent:

1. φ is a cosheaf space.

2. φ is a spread and η is a bijection, in which case the inverse of η is given by:

η^{−1}(x, α) = ( ⋂_{x ∈ U} α_{U}) ∩ φ^{−1}(x) .

3. φ is a spread, and for every x^{∈} X and every cogerm α of φ at x, the set

( ⋂_{x ∈ U} α_{U}) ∩ φ^{−1}(x)

is equal to a singleton.

2.2. Remark. By Proposition 2.1.3, a cosheaf space is a complete spread with locally connected domain.

The display space construction applies to any map Y [figure]X for which Y is locally connected. Thus, we may always factor such a map as

Y [figure: η]Ỹ [figure: ψ]X ,

where ψ is a cosheaf space. The map η is pure in the sense that for every non-empty, connected open set U, η^{−1}(U) is again non-empty and connected. (In particular, η is dense.) This factorization of a map with locally connected domain into its pure and cosheaf space parts (such that the middle space is also locally connected) is unique up to unique homeomorphism.

3. The universal ramified cover of a marked sphere

Consider the finitely marked 2-sphere regarded as an object S_{U} of the Sierpinski fibration (Eg. 1.5). We refer to the open set U as a punctured sphere. Let us denote the universal covering space of U by P : Y [figure]U. If m = |S − U|, let ψ_{m} : Ỹ [figure]S denote the cosheaf

space factor of the map Y [figure: P]U [figure]S (as in §2). We have

[figure: U; S; Y; U; P; Y; Ỹ; η; Ỹ; S; ψ_{m}]

where the first factor η is pure. The open inclusion U [figure]S is also pure. The cosheaf space ψ_{m} is a ramified, or branched covering space in the sense of [11]. We shall call ψ_{m} the universal ramified cover of the m-marked 2-sphere.

The following is an instance of a general fact about the pure, cosheaf space factorization of a map such as Y [figure: P]U [figure]S ([13], Lemma 4.2).

3.1. Proposition. The above square is a pullback, so that η : Y [figure]Ỹ is an open inclusion, and Y is homeomorphic to ψ_{m}^{−1}(U).

3.2. Proposition. The space Ỹ is connected, simply connected, locally path-connected, and locally simply connected.

Proof. By using Proposition 3.1 and well-known properties of Y, the stated connectedness properties of Ỹ can be argued directly from the definition of the cosheaf space topology that Ỹ carries. Another way to see that Ỹ has these properties is to use the Poincaré disk with its boundary at infinity. Let D denote this space. Topologically, D is a closed disk in the plane, but with the following additional open sets: in the following diagram, B ∪ {p} is also a basic open set in D, where B is an open disk with tangent point p.

[figure: D; B; p]

Then (for m ≥ 3) Ỹ is homeomorphic to a subspace of D in such a way that (only) the branch-points of Ỹ correspond to points on the boundary of D.

A fact that may be of independent interest is that ψ_{m} is an open map. We have the following explanation of this fact. For any given element c of a Heyting algebra (H, ≤, ⇒) one may define a ‘thinner’ partial order on the elements of H:

a ≤_{c} b if a ≤ b and a ⇒ c ≤ b ⇒ c .

Consider our situation where H is the open set lattice of S, and c is taken to be U. Since S − U is a closed set that lies discretely in S, for any two open sets V and V′ of S we have

V ⊆_{U} V′ iff V ⊆ V′ and V ∩ (S − U) = V′ ∩ (S − U) .

Let C_{ψ_{m}} denote the cosheaf associated with ψ_{m}. By definition, for any open set W ⊆ S we have

C_{ψ_{m}}(W) = π_{0}(P^{−1}(U ∩ W)) .

If W ⊆ W′, then there is an obvious transition map C_{ψ_{m}}(W) [figure]C_{ψ_{m}}(W′). The proof of Proposition 3.4 depends on the following lemma. By convention, a connected set is always non-empty.

3.3. Lemma. The cosheaf C_{ψ_{m}} has the following property: for any two connected, simply connected open subsets V, V′ of S, we have

V ⊆_{U} V′ ⇒ C_{ψ_{m}}(V) [figure]C_{ψ_{m}}(V′) is an isomorphism .

3.4. Proposition. The universal ramified cover ψ_{m} is an open surjection.

Proof. The collection of (non-empty) connected, simply connected open subsets of S is a base for S. Let (V, β) be a basic open subset of Ỹ, where V is connected and simply connected, and β is a component of P^{−1}(U ∩ V). By definition, we have

ψ_{m}(V, β) = {y^{∈} V | there is a cogerm α at y for which α_{V} = β} ⊆ V .

We shall show that ψ_{m}(V, β) is equal to V, hence open. Let y^{∈} V. If y^{∈} U, then clearly there is a cogerm α at y for which α_{V} = β because we have only to work with the cover P, and not with ψ_{m}. On the other hand, if y is a marked-point, it suffices to define a cogerm α at y on the connected, simply connected open subsets B ⊆ S that satisfy B ∩ (S − U) = {y}. For any two such B, B′ obviously we have B ⊆ B′ iff B ⊆_{U} B′. Choose one such B and any member of C_{ψ_{m}}(B), which we denote α_{B}, such that α_{B} goes to β under C_{ψ_{m}}(B) [figure]C_{ψ_{m}}(V). Then we are able to define an α at y because if B ⊆ B′, then by Lemma 3.3 the transition map C_{ψ_{m}}(B) [figure]C_{ψ_{m}}(B′) is an isomorphism.

We have the object Ỹ_{Y} = ψ_{m}^{⋆}(S_{U}) and the morphism ψ_{m} : Ỹ_{Y} [figure]S_{U} in the Sierpinski fibration, which is a cartesian morphism in the sense of fibrations (Definition 1.1).

3.5. Remark. The closed complement Ỹ − Y is discrete in Ỹ, and countable. Furthermore, for m ≥ 2, the space Ỹ is not locally compact (m = 1 is an exception because ψ_{1} is a homeomorphism). Indeed, local compactness fails at every point of Ỹ − Y. In particular, Ỹ is not a topological manifold, m ≥ 2.

3.6. Remark. The cosheaf space ψ_{m} cannot be exponentiable [23, 24, 7] in the category of topological spaces over S (in contrast with sheaf spaces, which are always exponentiable). Indeed, if ψ_{m} were exponentiable, then Ỹ would be exponentiable in topological spaces because S is exponentiable and such things compose. Hence, Ỹ would be locally compact. All assertions made here about exponentiability are explained in [23]. In this connection, Susan Niefield’s [24] recent example of a poset morphism that is a discrete opfibration, and whose induced essential geometric morphism is not exponentiable, provides another example of a cosheaf space that is not exponentiable. Indeed, a discrete opfibration of posets is a cosheaf space when regarded as a continuous map with respect to the down-closed topology (use Prop. 2.1 to see this).

4. The ramification groupoids π_{1}^{r}(S, m)

For each natural number m we now define a groupoid, which we shall call a ramification groupoid of the m-marked 2-sphere, denoted π_{1}^{r}(S, m). Here m = |S − U| denotes the number of marked-points on S. By definition, π_{1}^{r}(S, m) is the full subgraph of π_{1}(Ỹ_{Y}) whose vertices are the elements of Ỹ − Y (Definition 1.9 and §3).

We make π_{1}^{r}(S, m) into a groupoid as follows. Suppose that we have paths

u [figure: p]x [figure: q]v

in Ỹ_{Y}, where u, x, v^{∈} Ỹ − Y. It is possible to find an open set V ⊆ Ỹ that contains x, such that V − {x} ⊆ Y, and such that V − {x} is connected and simply connected. There is a t such that p(t, 1] ⊆ V and q[0, 1 − t) ⊆ V. Choose s such that t < s < 1. There is a path γ in V − {x}, which is unique up to homotopy in V − {x}, joining the points p(s) and q(1 − s). We use γ to connect p and q, producing a path u [figure]v in Ỹ_{Y}. This is a well-defined associative composition for homotopy classes of paths in Ỹ_{Y}. Furthermore, it is readily verified that this composition has identities and inverses.

4.1. Example. π_{1}^{r}(S, 1) has a single object, and is equal to the trivial group.

4.2. Remark. The groupoid π_{1}^{r}(S, m) is connected and simply connected in the sense that between any two objects there is exactly one isomorphism. It may seem that we could have saved ourselves the trouble of doing homotopy theory in the Sierpinski fibration because π_{1}^{r}(S, m) is isomorphic to the full subgroupoid of the ordinary fundamental groupoid π_{1}(Ỹ) determined by the elements of Ỹ − Y. However, we wish to emphasize that one of our aims is to describe an order structure that π_{1}^{r}(S, m) carries. We shall do this not in terms of Y or even in terms of Ỹ, but rather in terms of the object Ỹ_{Y} of the Sierpinski fibration.

5. A combinatorial presentation of π_{1}^{r}(S, n + 1)

Our later description of an order structure carried by the groupoid π_{1}^{r}(S, n+1) is in terms of the following combinatorial presentation of π_{1}^{r}(S, n+1). (Our change in notation in this section from m to n + 1 is convenient for our later purposes.) First consider the following presentation of the ordinary fundamental groupoid π_{1}(U) of the n + 1-punctured sphere U, as we have been denoting it. Let

v_{0} [figure: e_{1}]v_{1} [figure: e_{2}]v_{2} ⋯ v_{n−1} [figure: e_{n}]v_{n}

denote pairwise disjoint line segments connecting n + 1 marked-points v_{i} on the sphere, which when deleted gives U. Up to homotopy, we may encode any path in U (whose endpoints do not lie on a line segment) as a word in the names e_{i} of the line segments, and their ‘inverses’ e_{i}^{−1}. We also call e_{i} an edge-symbol, or just an edge. We build such a word from a path in U by including e_{i} just when the path crosses the line segment e_{i} going up the page, and e_{i}^{−1} when the path crosses the line segment e_{i} going down the page. Thus, we have the well-known fact that we may present any homotopy class of paths in U as an element of the free group on the symbols e_{1}, …, e_{n}. (Perhaps it should be emphasized that we are describing the fundamental groupoid of U, so it makes sense to describe paths in U using line segments without equal endpoints as generators.)

For our later purposes we need a presentation of just the isomorphisms with common domain x^{∈} Ỹ − Y. We denote this collection of isomorphisms by x/π_{1}^{r}(S, n + 1). Let us fix such an x. As above, we label the n + 1 marked-points on the sphere by v_{0}, …, v_{n}, such that

ψ_{n+1}(x) = v_{n} .  (2)

We introduce vertex-symbols v_{i}^{+} and v_{i}^{−}, which are not to be confused with the markedpoints v_{i} on the sphere. Consider all words of the form

v_{i}^{±}wv_{n} ; i^{∈} {0, …, n} ,

where w is an element of the free group on the edge-symbols e_{1}, …, e_{n}, and where v_{i}^{±} means either v_{i}^{−} or v_{i}^{+}, subject to the following ‘codomain winding relations:’

\begin{matrix}v_{i−1}^{+}e_{i}^{−1} ∼ v_{i−1}^{−} \\ v_{i}^{+}e_{i}^{−1} ∼ v_{i}^{−} \\ v_{i−1}^{−}e_{i} ∼ v_{i−1}^{+} \\ v_{i}^{−}e_{i} ∼ v_{i}^{+} \end{matrix} \} i = 1, …, n .

The meaning of these relations is clarified in the proof of Proposition 5.5. We also identify the ‘ends’ by including the relations:

\begin{matrix}v_{0}^{+} ∼ v_{0}^{−} \\ v_{n}^{+} ∼ v_{n}^{−} . \end{matrix}

5.1. Remark. We shall write v_{0} = v_{0}^{+} = v_{0}^{−}, and v_{n} = v_{n}^{+} = v_{n}^{−}, which should not lead to any confusion.

5.2. Proposition. Every such equivalence class of words v_{i}^{±}wv_{n} has a unique member of least length (up to exchanging v_{n}^{+} with v_{n}^{−}, or v_{0}^{+} with v_{0}^{−}).

Thus, there is no need to make a distinction in the notation between an equivalence class and a representing word, since we can assume it is the word of least length. Let us call these words of least length reduced words.

5.3. Example. For instance, we have v_{n}e_{n}v_{n} ∼ v_{n}v_{n}. But v_{n}e_{n−1}e_{n}v_{n} is a reduced word.

In the proof of the next proposition we shall consider leaves of the universal cover P : Y [figure]U. A leaf of P is a connected component of P^{−1}(U − ⋃ e_{i}). Thus, each leaf of P is canonically identified with U − ⋃ e_{i}. An edge of a leaf of P is the part of the closure of the leaf in Y that is mapped by P to a line segment e_{i}. The edges and branch-points of the closure in Ỹ of a leaf of P may be labelled by edge and vertex-symbols in a circular fashion illustrated by Figure 1 (§6). A leaf of a point y^{∈} Ỹ is a leaf of P whose closure in Ỹ contains y.

5.4. Remark. If a path exits a leaf across an edge e_{i}^{ν}, ν = ±1, then the adjacent leaf it enters is labelled so that the edge just crossed is now named e_{i}^{−ν}. The endpoints of e_{i}^{ν} also change sign.

5.5. Proposition. With x as in (2), every isomorphism x ≅ x′ in π_{1}^{r}(S, n + 1) may be uniquely presented by a reduced word v_{i}^{±}wv_{n}, where ψ_{n+1}(x′) = v_{i} (without ±) for some i^{∈} {0, …, n}. The identity on x is presented by the reduced word v_{n}v_{n}. Conversely, every such reduced word presents such an isomorphism.

Proof. We shall establish a bijection between reduced words and homotopy classes of paths by first arbitrarily fixing a leaf of x. Suppose we have a path p : x [figure]x′ in Ỹ_{Y}. There is a connected and simply connected open neighbourhood B ⊆ Ỹ of x, such that B − {x} ⊆ Y is open, connected and simply connected, and contained in the union of the leaves of x. By suitably winding finitely many times around x in B − {x}, the homotopy class of p can be represented by a path that exits x via the fixed leaf. The winding around x is recorded in the word v_{i}^{±}wv_{n} by the power of e_{n} occurring as the rightmost symbols of w. For instance, if w = ze_{n}^{2}, then the ‘path’ v_{i}^{±}wv_{n} exits x via the fixed leaf and cycles around x crossing e_{n} twice. But we cannot identify e_{n}^{2}v_{n} with v_{n} in v_{i}^{±}wv_{n}, for then we would identify two possibly different homotopy classes of paths in Ỹ_{Y}.

Now consider the codomain x′ of the path p. Again we may find a neighbourhood B′ as above such that B′ − {x′} is contained in the union of the leaves of x′. There is a 0 < t_{0} < 1 such that p[t_{0}, 1] lies in B′, and furthermore p crosses only finitely many leaf edges between leaving B and reaching the point p(t_{0}). The word we are building to represent the homotopy class of p may be ended with a vertex-symbol v_{i}^{±} that labels x′. This vertex-symbol is determined by the labelling of the leaf of x′ in which p(t_{0}) lies. The codomain winding relations arise because any path from any point in B′ into x′ is presented by the vertex-symbol that labels x′.

6. The order structure carried by π_{1}^{r}(S, n + 1)

From the point of view of analytic topology a circle is a continuum that is disconnected by the omission of any two of its points ([29], Chap. 3, §7). A linear order is a reflexive, transitive, and anti-symmetric relation, which is total. Consider the following property of a connected groupoid G: for every object A^{∈} G, the collection

(A/G)^{⋆} = { isomorphisms g : A ≅ B | g ≠ id_{A} }

forms a linear order. If for any object A we denote the linear ordering in (A/G)^{⋆} by ≤_{A}, then we furthermore require that the composition of the groupoid cooperates with the linear orderings as follows: for any two distinct isomorphisms g : A ≅ B and h : A ≅ C other than id_{A}, we have

g ≤_{A} h ⇒ hg^{−1} ≤_{B} g^{−1} .  (3)

The picture below may help explain this rule. In this picture regard ≤_{A} and ≤_{B} as increasing in the clockwise direction (so that g ≤_{A} h starting from A).

[figure: A; B; C; g; h; hg^{−1}]

G is thus a ‘circle.’

The groupoid π_{1}^{r}(S, m) carries an order structure such as defined above. (Since π_{1}^{r}(S, m) is connected and simply connected, x/π_{1}^{r}(S, m) is in bijection with the objects of π_{1}^{r}(S, m) (= elements of Ỹ − Y), but really it is the collection of isomorphisms with domain x, excluding id_{x}, that are linearly ordered.) In Definition 6.1 we define a linear ordering of (x/π_{1}^{r}(S, n + 1))^{⋆} in terms of the reduced words v_{i}^{±}wv_{n} (except v_{n}v_{n}, although w may be empty in which case v_{i}^{±} may not be v_{n}, for this would give v_{n}v_{n}). A word is a right subword of another word if the former appears in the latter regarded from the right.

6.1. Definition. Let v_{i}^{±}wv_{n} and v_{j}^{±}w′v_{n} be two reduced words as above. Let e_{k}^{ν}, ν = ±1, be the leftmost symbol of the greatest common right subword of w and w′ (the righthand v_{n} in the word has an inert role in this ordering criteria). Let a and b be the next symbols to the left of e_{k}^{ν} in v_{i}^{±}w, respectively v_{j}^{±}w′. Starting from the edge-symbol that immediately follows e_{k}^{−ν} (see Remark 5.4) clockwise in the circular list in Figure 1, we declare v_{i}^{±}wv_{n} < v_{j}^{±}w′v_{n} if a occurs before b (clockwise). (See Remark 5.1 about v_{0} and v_{n}.)

6.2. Remark. In Definition 6.1 it may happen that w is a right subword of w′ so that a = v_{i}^{±} (or that w′ is a right subword of w so that b = v_{j}^{±}), and hence we must compare the vertex v_{i}^{±} with an edge of w′, but the list in Figure 1 accounts for this. (If w = w′ and a = b, then of course the two words v_{i}^{±}wv_{n} and v_{j}^{±}w′v_{n} are identical.) It may happen that the greatest common right subword of w and w′ is empty, in which case by definition we start with e_{n}^{−1} when finding the positions of a and b in the circular list.

v_{0}e_{1}^{−1} v_{1}^{−} e_{2}^{−1} e_{1}v_{1}^{+} e_{2}  v_{n} e_{n} v_{n−1}^{+} e_{n−1} e_{n}^{−1} v_{n−1}^{−} e_{n−1}^{−1} · ·  · ·  ⋯  · ·  ·

Figure 1: The circular list of vertex and edge-symbols.

6.3. Remark. From a geometric point of view, the term “hyperbolic” for the above ordering seems appropriate because the space Ỹ is a subspace of the Poincaré disk and its circle at infinity (as described in the proof of Prop. 3.2) such that certain geodesics map to the ‘combinatorial geodesics’ v_{i}^{±}wv_{n}. An analysis of this connection with hyperbolic geometry is given in [27]. The ordering imposed by Definition 6.1 might also be called warped, or twisted-lexicographical (Franz-Victor Kuhlmann’s suggestion) because although the ordering has a lexicographical aspect, the order of the alphabet varies (by a rotation) depending on the last common letter encountered when comparing two words.

The following is not difficult to establish.

6.4. Proposition. The ordering of (x/π_{1}^{r}(S, n + 1))^{⋆} given by Definition 6.1 is a dense linear order (“dense” means that between any two isomorphisms p < q with domain x there is another one p < r < q).

6.5. Proposition. The order structure in π_{1}^{r}(S, n + 1) has property (3).

We omit a proof of Proposition 6.5. In any case, we do not use (3) to obtain linear orderings of a braid group.

6.6. Remark. There is a one-to-one function into x/π_{1}^{r}(S, n + 1) from the subset of the free group on e_{1}, …, e_{n} consisting of all reduced words not beginning on the left with e_{n} or with e_{n}^{−1}, which sends such a word w to v_{n}wv_{n}. The word w should not begin on the left with e_{n} or e_{n}^{−1} because of the codomain winding relations v_{n}e_{n} ∼ v_{n} and v_{n}e_{n}^{−1} ∼ v_{n}. Therefore, we may linearly order this particular subset of non-empty reduced words in e_{1}, …, e_{n} The action of a braid group restricts to this subset, and it is order-preserving. By admitting countably many generators e_{i} it is possible to avoid the codomain winding relations. In effect we may work without the vertex-symbols. The details are provided in §13.

7. Lifting a homeomorphism of S_{U} to one of Ỹ_{Y}

In this section we show how to lift a homeomorphism of S_{U} to one of Ỹ_{Y} = ψ_{m}^{⋆}(S_{U}) in the Sierpinski fibration, where m = |S − U|. (As we were saying in the Introduction, such a lifting exists just when ψ_{m} is fixed under the Hurwitz action.) Much of what we do holds in greater generality, but we shall work only with the universal ramified cover ψ_{m}. Later we shall derive, by lifting homeomorphisms, an action of the braid group on n strands in the ramification groupoid π_{1}^{r}(S, n + 1).

Recall that we denote the universal covering space of U by P : Y [figure]U. We shall use P_{∗} to denote the induced functor of fundamental groupoids. By definition, for any y^{∈} Y, the object P_{∗}(y) is equal to the point P (y).

7.1. Proposition. Let h be a homeomorphism of S_{U}. Then a point b on Y, and an isomorphism γ_{b} : P (b) ≅ h(P (b)) of the groupoid π_{1}(U) define a homeomorphism h̃_{γ_{b}} of Ỹ_{Y}, which furthermore is a lifting of h. Moreover, if γ_{c} : P (c) ≅ h(P (c)) is another isomorphism such that h_{∗}(P_{∗}(ρ)) · γ_{b} = γ_{c} · P_{∗}(ρ) in π_{1}(U), where ρ : b ≅ c is the unique isomorphism of π_{1}(Y), then h̃_{γ_{b}} = h̃_{γ_{c}} (so we may denote the lifting as h̃_{γ}).

[figure: S_{U}; S_{U}; h; Ỹ_{Y}; S_{U}; ψ_{m}; Ỹ_{Y}; Ỹ_{Y}; h̃_{γ}; Ỹ_{Y}; S_{U}; ψ_{m}]

If there is isotopy between h and h′ in S_{U} (Example 1.5) with induced natural isomorphism i : h_{∗} ≅ h′_{∗}, and if for some b^{∈} Y, we have isomorphisms γ_{b} and γ′_{b} such that γ′_{b} = i_{P (b)} ·γ_{b}, then the given isotopy lifts to one of h̃_{γ} and h̃_{γ′}.

Proof. By the well-known procedure of path-lifting, there is a homeomorphism h̄_{γ} such that the following diagram commutes.

[figure: U; U; h; Y; U; P; Y; Y; h̄_{γ}; Y; U; P]

(Here we mean the restriction of h to U, or U(h) in the notation of §1.) Indeed, given

y^{∈} Y, we define h̄_{γ}(y) by lifting the path P (b) [figure: γ_{b}]h(P (b)) [figure: h(P (!))]h(P (y)) in U to a path

b [figure]h̄_{γ}(y) in Y such that P (h̄_{γ}(y)) = h(P (y)), where b [figure: !]y denotes the unique path in Y (up to homotopy in Y). The stated uniqueness property is straightforward to establish. We may lift isotopies in U along P in essentially the same manner, but working over I.

Any homeomorphism h̄ of Y over a homeomorphism h : S_{U} [figure]S_{U} may be uniquely completed to one of Ỹ_{Y} over h, as depicted in the following diagram.

[figure: Ỹ; Ỹ; h̃; Y; Ỹ; η; Y; Y; h̄; Y; Ỹ; η; S; S; h; S; ψ_{m}; S; ψ_{m}]

The reason that there is such an h̃ is explained by the following simple fact about factorization systems, applied in our case to the pure, cosheaf space factorization. Suppose we have a commutative diagram

[figure: B; B; B; B; A; B; e; A; A; f; A; B; e; C; C; g; C; m; C; m]

where e and m are orthogonal, and where f and g are isomorphisms. Then because g · m

and e · f are orthogonal, there is a unique isomorphism B [figure: g̃]B making both squares commute.

Finally suppose we have an isotopy H̄ : Y ×I [figure]Y over an isotopy H : S_{U} ×I_{I} [figure]S_{U}. The previously mentioned fact about factorization systems also applies for completing H̄ to an isotopy H̃ : Ỹ_{Y} ×I_{I} [figure] Ỹ_{Y} over H. This fact applies because the pure, cosheaf space factorization is stable under pullback along a locally 0-acyclic map. A locally 0-acyclic map f : A [figure]B is an open map with the property that A has a base of open sets each of which meet any fiber of f in a connected or empty set. This pullback stability property has a topos theory proof: a locally 0-acyclic map induces what is usually called a locally connected geometric morphism of sheaf toposes [16]. The pullback stability of the pure, cosheaf factorization along a locally connected geometric morphism has been established in [5]. The projection map S × I [figure]S is locally 0-acyclic, so that ψ_{m} × I is a cosheaf space and η × I is pure.

Lifting a homeomorphism and lifting a braid are not exactly the same matter because a braid is really an isotopy-equivalence class of homeomorphisms. We review this in the next section.

8. Artin braids as homeomorphisms of S_{U}

Let B_{n} denote the Artin group of braids of n strands. Let D_{n} denote the closed disk with n interior marked-points. Consider the collection of homeomorphisms and isotopies of D_{n} that pointwise fix the boundary and permute the marked-points. The group of isotopyequivalence classes of this collection is called a mapping class group^{2} Mcg(D_{n}) [3]. It is well-known that B_{n} is isomorphic to Mcg(D_{n}). Generators of Mcg(D_{n}), which we denote by σ_{1}, …, σ_{n−1}, may be described as follows. The well-known Dehn twist about a circle is defined by choosing a cylinder that contains the circle so that the cylinder is parameterized by (y, θ), for −1 ≤ y ≤ 1 with the circle at y = 0. Then the Dehn twist about this circle is the function ( y, θ + π(1 − y) ) on the cylinder. To describe σ_{i}, let v_{0}, …, v_{n−1} denote the n marked-points of the disk. Consider the homeomorphism defined to be the Dehn twist about the circle on which v_{i−1} and v_{i} lie diametrically opposed (with v_{i} at (0, 0)) in a cylinder that contains no other marked-point, and defined to be the identity elsewhere on the disk. This homeomorphism exchanges v_{i−1} with v_{i} and leaves the other marked-points fixed. Let σ_{i} denote the isotopy class of this homeomorphism.

We shall want to regard a braid β^{∈} B_{n} as an isotopy-equivalence class β = [h] of homeomorphisms h : S_{U} [figure]S_{U}, where n + 1 = |S − U|. This introduces a complication because isotopies on the sphere are not the same as on the disk, but we feel that it is worth the trouble because working with ramified covers seems better suited to the sphere. We shall say that an isotopy H : S × I [figure]S fixes an open neighbourhood V ⊆ S if

[figure: S × I; S; H; V × I; S × I; V × I; V; V; S]

commutes, where the top horizontal arrow denotes the projection map. (This also makes sense for isotopies in the Sierpinski fibration.) In other words, an isotopy fixes an open set V if every homeomorphism of the isotopy fixes V. (Note: by fix we shall always mean fix pointwise.) Now consider S_{U}, where S − U = {v_{0}, …, v_{n−1}, v_{n}}. There are n ‘Dehn twist’ generators σ_{1}, …, σ_{n} of the mapping class group of S_{U}. An Artin braid may always be represented by a homeomorphism of S_{U} that fixes an open neighbourhood of v_{n}. (Such a homeomorphism necessarily preserves orientation.) Represented this way, the Artin braids are generated by the Dehn twists σ_{1}, …, σ_{n−1}, just as on the disk. Furthermore, two such homeomorphisms represent the same Artin braid if and only if they are isotopic in S_{U} under an isotopy that fixes an open neighbourhood of v_{n}.

8.1. Remark. The notions of isotopy in S_{U} and isotopy in S_{U} fixing an open neighbourhood are not equivalent. For instance, if S − U = {v_{0}, v_{1}, v_{2}}, with Dehn twist generators σ_{1}, σ_{2}, then the square of the Dehn twist that represents σ_{1} is isotopic to the identity homeomorphism of S_{U};^{3} however, σ_{1}^{2} is not equal to the identity Artin braid. I.e., the square of this Dehn twist is homotopic to the identity through orientation-preserving homeomorphisms that permute (in this case, fix) v_{0}, v_{1}, v_{2}, but this homeomorphism is not isotopic to the identity under an isotopy that fixes an open neighbourhood of v_{2}.

2A member of a mapping class group is required to preserve orientation, but in the case of D_{n}, a homeomorphism that pointwise fixes the boundary necessarily preserves orientation.

9. The action of B_{n} in π_{1}^{r}(S, n + 1)

9.1. Proposition. Depending on a chosen isomorphism γ_{b} : P (b) ≅ h(P (b)) of π_{1}(U) (as in Prop. 7.1), a homeomorphism h of S_{U} induces a groupoid automorphism of π_{1}^{r}(S, m), m = |S − U|. (How this automorphism depends on the isotopy-equivalence class of h and on the isomorphism γ_{b} is explained in Prop. 7.1.)

Proof. From Proposition 7.1, two isotopic homeomorphisms of S_{U} have liftings to isotopic homeomorphisms of Ỹ_{Y}. Since the branch-point set Ỹ −Y (= objects of π_{1}^{r}(S, m)) is a discrete subspace of Ỹ, it follows that two isotopic homeomorphisms of Ỹ_{Y} must induce the same automorphism of the branch-point set. Further, two homotopic paths (in the Sierpinski fibration) are homotopic under isotopic images. We omit further details.

By Proposition 9.1, a braid of n strands induces an automorphism of π_{1}^{r}(S, n + 1), but when working with braid homeomorphisms and the restricted notion of ‘fixed open set’ isotopy in S_{U} we can do better: for any given object x of π_{1}^{r}(S, n + 1) we may define a covariant (or left) action of B_{n} in π_{1}^{r}(S, n + 1) by groupoid automorphisms, such that every braid fixes x. We now explain how to do this. Let x be any object of π_{1}^{r}(S, n + 1). We label S − U = {v_{0}, …, v_{n−1}, v_{n}} such that

ψ_{n+1}(x) = v_{n} .

Once (pairwise disjoint) line segments e_{i} joining the v_{i} have been chosen, we may define leaves of the cover Y [figure: P]U. (A leaf of P is a connected component of P^{−1}(U − ⋃ e_{i}).) Let A ⊆ Y be a leaf of x, i.e., a leaf of P whose closure in Ỹ contains x. Let β = [h] denote an arbitrary member of B_{n}, where h is a homeomorphism of S_{U} that fixes an open neighbourhood V ⊆ S of v_{n}. Let b^{∈} A be any point such that P (b)^{∈} V, so that h fixes P (b). Then, as in Prop. 9.1, we may define an induced automorphism of π_{1}^{r}(S, n + 1) taking for γ_{b} the identity on P (b). The definition of this automorphism does not depend on the point b^{∈} A chosen such that P (b)^{∈} V. If h′ is another homeomorphism that is isotopic to h under an isotopy that fixes an open neighbourhood of v_{n}, then the same automorphism of π_{1}^{r}(S, n + 1) is induced. Thus, we may denote this automorphism as β_{∗}. It is clear that β_{∗}(x) = x. We denote the action by

β · g = β_{∗}(g) ,

3This can be seen using the system of generators and relations for the mapping class group of the marked sphere given in [3]. It can also be seen directly: rotate one revolution, and in opposite directions, the two closed disks that are disjoint from the open cylinder in which the Dehn twist for σ_{1} (squared) is defined, whilst keeping v_{0} and v_{1} fixed.

where g is an isomorphism of the groupoid π_{1}^{r}(S, n + 1). It may be verified that the automorphism (ββ′)_{∗} is equal to the composite automorphism β_{∗} · β′_{∗}. The definition of this action depends (only) on x and on the leaf of x chosen.

9.2. Remark. Since x is fixed under this action, we may pass to an action of B_{n} in x/π_{1}^{r}(S, n + 1) that fixes the identity isomorphism id_{x} : x ≅ x.

10. A presentation of the action of B_{n} in π_{1}^{r}(S, n + 1)

The order structure of π_{1}^{r}(S, n) has been defined combinatorially, so in order to show (in §11) that the action of B_{n} in π_{1}^{r}(S, n + 1) respects this order structure we shall provide a combinatorial presentation of the action (which is (4) below). This action is isomorphic to the Artin action (Proposition 13.8). Larue [19] has considered (4) in his study of the ‘Dehornoy bracket.’

In §5 we have described a geometric representation of a free group(oid) using line segments and edge-symbols. In order to prescribe the action of B_{n} in π_{1}(U) precisely, let σ_{i}, 1 ≤ i ≤ n − 1, denote the braid group generators. We have previously explained in §8 that σ_{i} may be represented by a Dehn twist (on the sphere), which permutes v_{i−1} and v_{i}, where S − U = {v_{0}, …, v_{n}}. From this geometric representation of σ_{i} we are led to define the action of the σ_{i} on the generators e_{j} of π_{1}(U) as follows:

\begin{matrix}σ_{i} · e_{i} = e_{i−1}e_{i}^{−1}e_{i+1} \\ σ_{i} · e_{j} = e_{j} ; j ≠ i \end{matrix} \} i = 1, …, n − 1 .  (4)

The first identity in (4) arises by observing what the Dehn twist σ_{i} does to a simple path that crosses the line segment e_{i+1} going up the page (which we regard as the positive direction). When i = 1, the first identity does not make sense because there is no edge e_{0}. In this case, we define

σ_{1} · e_{1} = e_{1}^{−1}e_{2} .

The braid relations are easily seen to hold for the above definitions, so that we have a well-defined action of B_{n} in π_{1}(U). This action is faithful in the sense that for any braid β, if for every member w of the free group on the e_{i}, we have β · w = w, then β = id.

We complete the presentation of the action of B_{n} in π_{1}^{r}(S, n + 1) by defining the effect on the vertex-symbols v_{0}, v_{1}^{±}, …, v_{n−1}^{±}, v_{n}.

\begin{matrix}σ_{i} · v_{i−1}^{+} = v_{i}^{+} \\ σ_{i} · v_{i}^{+} = v_{i−1}^{−}e_{i+1} \\ σ_{i} · v_{i−1}^{−} = v_{i}^{+}e_{i−1}^{−1} \\ σ_{i} · v_{i}^{−} = v_{i−1}^{−} \end{matrix} \} i = 1, …, n − 1  (5)

The third line above does not make sense when i = 1 again because there is no edge e_{0}. We define

σ_{1} · v_{0}^{−} = v_{1}^{+} .

This is consistent with the other definitions. For instance, since v_{0}^{+} ∼ v_{0}^{−}, we have

σ_{1} · v_{0}^{−} = σ_{1} · v_{0}^{+} = v_{1}^{+} ,

where the second equality is by the first identity in (5). In all other cases, we set σ_{i} · v_{k}^{±} = v_{k}^{±}. The definition extends to arbitrary words v_{i}^{±}wv_{n}, and braids β in the obvious way. For instance, we have σ_{i} · v_{n} = v_{n} for i = 1, …, n − 1, so that every braid fixes the word v_{n}v_{n}. Also, the action of σ_{i}^{−1} on the vertex-symbols may be inferred from the above definitions and the codomain winding relations. For instance, we have

σ_{i}^{−1} · v_{i}^{−} = v_{i−1}^{+}e_{i+1}^{−1} .

This completes the combinatorial presentation of the action of B_{n} in π_{1}^{r}(S, n + 1).

11. The action of B_{n} in π_{1}^{r}(S, n + 1) respects order

We shall prove the following.

11.1. Theorem. The action of B_{n} in π_{1}^{r}(S, n + 1) respects the order structure carried by π_{1}^{r}(S, n + 1): for every object x of π_{1}^{r}(S, n + 1), isomorphisms g, h^{∈} (x/π_{1}^{r}(S, n + 1))^{⋆}, and braid β^{∈} B_{n}, we have

g ≤_{x} h ⇒ β · g ≤_{x} β · h ,

where the action is defined in §9 (depending on x and a choice of leaf of x), and where ≤_{x} denotes the linear ordering in (x/π_{1}^{r}(S, n + 1))^{⋆} (given by Definition 6.1).

Our argument for Theorem 11.1 is combinatorial in nature since that is how we have defined the order structure of π_{1}^{r}(S, n + 1). For this argument only, we shall write the words v_{i}^{±}wv_{n} just as v_{i}^{±}w since the right-hand v_{n} is inert anyway. We begin with some terminology.

11.2. Definition. Let 1 ≤ k ≤ n.

1. We shall say that a reduced word b in the e_{i} is k-positive definite if only positive exponents of e_{k} appear in b, and if the first and last symbol of b is e_{k}. Similarly, we define k-negative definite.

2. Let sgn_{k}(w) denote the sum of the exponents of e_{k} in w. Let |w| denote the positive word obtained from w by replacing every exponent by its absolute value.

3. Obviously any reduced word w in the e_{i} can be written uniquely as

w_{0}b_{1}w_{1} ⋯ b_{m}w_{m} ,

where each b_{j} is either k-positive definite or k-negative definite, sgn_{k}(b_{j}) alternates between positive and negative, and sgn_{k}(|w_{j}|) = 0. We shall call this the k-definite form of w.

An examination of the definitions (4) and (5) reveals the following. As always, σ_{i} denotes a braid group generator.

11.3. Lemma. Fix h^{∈} {1, …, n − 1}, and let v_{k}^{±}z denote σ_{h} · v_{i}^{±}w.

1. If sgn_{h}(|w|) = 0, then either z = e_{h−1}^{−1}w, z = e_{h+1}w, or z is a (possibly empty) right subword of w.

2. Assume that sgn_{h}(|w|) ≠ 0, and write w in its h-definite form w_{0}b_{1}w_{1} ⋯ b_{m}w_{m} (each b_{j} is either h-positive definite or h-negative definite, sgn_{h}(b_{j}) alternates between positive and negative, and sgn_{h}(|w_{j}|) = 0). Then v_{k}^{±}z has a similar form v_{k}^{±}z_{0}c_{1}z_{1} ⋯ c_{m}z_{m}, where sgn_{h}(|z_{j}|) = 0, and sgn_{h}(c_{j}) = −sgn_{h}(b_{j}). Furthermore, if w_{m} is empty, then z_{m} = e_{h+1}^{ν}, where ν is equal to the sign of sgn_{h}(b_{m}).

We proceed with the proof of Theorem 11.1. Assume that v_{k}^{±}z < v_{i}^{±}w. We may assume that the rightmost symbols of these two words are different, so that the start-symbol in the circular list in Figure 1 is e_{n}^{−1}. There must be vertex-symbols v_{j}^{±}, v_{j′}^{±} such that

v_{k}^{±}z ≤ v_{j}^{±} ≤ v_{j′}^{±} ≤ v_{i}^{±}w ,

where in the circular list v_{j}^{±} is clockwise adjacent to the rightmost symbol of v_{k}^{±}z, and v_{j′}^{±} is clockwise adjacent to the rightmost symbol of v_{i}^{±}w. We thus have three cases to argue: when w is empty and z is not, when both w and z are empty, and when z is empty and w is not. We shall argue only the case when z is empty and w is not: this is the case v_{k}^{±} < v_{i}^{±}w, where v_{k}^{±} is clockwise adjacent to the rightmost symbol of v_{k}^{±}w. This symbol is an edge-symbol since we are assuming that w is non-empty. Furthermore, we shall assume that v_{k}^{±} = v_{n−1}^{−} (the general case is similarly argued), so that the rightmost symbol of v_{k}^{±}w is therefore e_{n−1}^{−1}. To summarize: we are assuming that v_{n−1}^{−} < v_{i}^{±}w (starting from e_{n}^{−1}), and that the rightmost symbol of w is e_{n−1}^{−1}. We must show that σ_{h} · v_{n−1}^{−} < σ_{h} · v_{i}^{±}w and σ_{h}^{−1} · v_{n−1}^{−} < σ_{h}^{−1} · v_{i}^{±}w, for h = 1, …, n − 1.

First we consider the case h ≠ n − 1, so that σ_{h}^{±1} · v_{n−1}^{−} = v_{n−1}^{−}. The word v_{i}^{±}w may be uniquely expressed as

v_{i}^{±}w_{0}e_{n}^{p_{1}}w_{1} … e_{n}^{p_{m}}w_{m} ; m ≥ 0 ,  (6)

where every subword w_{j} does not contain e_{n}^{−1}, and the p_{j}’s are negative integers. For j ≥ 1, w_{j} is non-empty, but w_{0} may be empty, in which case we must have m ≥ 1. For h ≠ n − 1, the action of σ_{h}^{±1} (on vertex or edge-symbols) does not introduce or remove e_{n}^{±1}, and we have σ_{h}^{±1} · e_{n}^{±1} = e_{n}^{±1}. Hence, for j ≥ 1, σ_{h}^{±1} · w_{j} is non-empty and contains no e_{n}^{−1}, so that no change in m or in the p_{j} in (6) can occur under the action of σ_{h}^{±1}. In particular, the rightmost symbol of σ_{h}^{±1} · v_{i}^{±}w (= rightmost symbol of σ_{h}^{±1} · w_{m}) cannot be e_{n}^{−1}. Hence, v_{n−1}^{−} < σ_{h}^{±1} · v_{i}^{±}w (starting from e_{n}^{−1}).

Next assume that h = n − 1. We shall show that σ_{n−1}^{−1} · v_{n−1}^{−} < σ_{n−1}^{−1} · v_{i}^{±}w, leaving σ_{n−1} · v_{n−1}^{−} < σ_{n−1} · v_{i}^{±}w for the reader. We have σ_{n−1}^{−1} · v_{n−1}^{−} = v_{n−2}^{+}e_{n}^{−1}. Let v_{k}^{±}z denote σ_{n−1}^{−1} · v_{i}^{±}w. If the rightmost symbol of v_{k}^{±}z is not e_{n}^{−1}, then the largest common right subword of v_{n−2}^{+}e_{n}^{−1} and v_{k}^{±}z is empty, which means that we begin from e_{n}^{−1} in Figure 1. But then v_{n−2}^{+}e_{n}^{−1} < v_{k}^{±}z. Hence, we can assume that the rightmost symbol of v_{k}^{±}z is e_{n}^{−1}. Then the greatest common right (edge) subword of v_{k}^{±}z and σ_{n−1}^{−1} · v_{n−1}^{−} = v_{n−2}^{+}e_{n}^{−1} is e_{n}^{−1}. Thus, we determine the order of v_{n−2}^{+}e_{n}^{−1} and v_{k}^{±}z by comparing v_{n−2}^{+} with the right-penultimate symbol of v_{k}^{±}z, starting from the edge-symbol that immediately follows e_{n} clockwise in the circular list, which is again e_{n}^{−1}. By assumption, the rightmost symbol of v_{i}^{±}w is e_{n−1}^{−1}. Hence sgn_{n−1}(|w|) ≠ 0, so that when we consider v_{i}^{±}w written in its n − 1-definite form, we have m ≥ 1. Moreover, in this case w_{m} is empty. So according to Lemma 11.3.2 (for σ_{n−1}^{−1}), we have z_{m} = e_{n}^{−1}, and that the right-penultimate symbol of v_{k}^{±}z is e_{n−1} (= rightmost symbol of c_{m}). This concludes the argument because v_{n−2}^{+} comes before e_{n−1} starting from e_{n}^{−1}.

11.4. Remark. Kassel [17] notes that Nielsen [25, 28] gives an action of B_{n} in the unit circle that preserves circular order and has a fixed point. Furthermore, this action has free elements. We have the following topos theory version of Nielsen’s result. Let B_{n} denote the topos of (left) B_{n}-sets. Let A denote the topological group of order preserving bijections of the rationals in (0, 1). It is well-known that the topos A of continuous (left) A-sets classifies dense linear orders without endpoints [26]. Thus, corresponding to the dense linear order that we have constructed in B_{n} there is a geometric morphism B_{n} [figure]A.

12. Linear orderings of B_{n}

A preorder is a reflexive and transitive relation. A partial order is a preorder that is also anti-symmetric. If a preorder (A, ≤) is equipped with an order-preserving (left) action by a group G, then an element a^{∈} A induces a left-invariant preorder ≤_{a} in G by:

g ≤_{a} h if g · a ≤ h · a ,

for any g, h^{∈} G. Moreover, if (A, ≤) is a partial order, then ≤_{a} is anti-symmetric if and only if a is free (meaning that the stabilizer of a is trivial). We shall also use the symbol <, which means ≤ and not equal. For our purposes, we are only interested in total or linear orders. Everything we have said about preorders and partial orders applies equally well to linear preorders and orders.

Thus, we may produce left-invariant linear orderings of B_{n} by producing free elements of the left B_{n}-set (x/π_{1}^{r}(S, n + 1))^{⋆}. We have the following.

12.1. Proposition. For any n, the isomorphism x ≅ x′ of π_{1}^{r}(S, n + 1) presented by the word

d_{n} = v_{n}e_{n−1}^{−1}e_{n} ⋯ e_{2}^{−1}e_{3}e_{1}^{−1}e_{2}e_{1}v_{n}

is free under the action of B_{n}. (Note: we have ψ_{n+1}(x) = ψ_{n+1}(x′) = v_{n}.)

Proof. We sketch a topological argument. Suppose that β^{∈} B_{n} fixes d_{n}. Then we may find a path p in Ỹ_{Y} that represents d_{n} (with ψ_{n+1}(p(0)) = ψ_{n+1}(p(1)) = v_{n}), and a homeomorphism h of S_{U} representing β (that fixes an open neighbourhood of v_{n}), such that h pointwise fixes the path ψ_{n+1} · p in S_{U}. Now consider the components of the open set S − ψ_{n+1} · p, of which there are just finitely many. Each marked-point except v_{n} is an interior point of some component, while some components may contain no marked point. The components must be permuted under h; however, further straightforward argumentation shows that this permutation must be the identity. In particular, h must fix the points v_{i} (hence, the permutation of β is the identity). To each component we may now apply a classical result due to Alexander (called the Alexander trick - proof provided in [15]): a homeomorphism of a closed disk that pointwise fixes the boundary is isotopic to the identity under an isotopy that also pointwise fixes the boundary. Furthermore, if the homeomorphism fixes an interior point, then the isotopy may be chosen to fix that point. Note: we can find a closed disk C that contains v_{n} as an interior point, that is fixed by h, and which meets only those components of S − ψ_{n+1} · p whose closure contains v_{n} (there is at least one such component). We now just delete the interior of C, and apply the Alexander trick (without interior point) to the components of S − ψ_{n+1} · p that meet C, using the boundary of C wherever necessary. The result is that we are able to build an isotopy of h with the identity that fixes the interior of C. After all this, we finally conclude that the h with which we began is isotopic to the identity, under an isotopy that fixes an open neighbourhood of v_{n}. Thus, β = id.

Let ≤_{d_{n}} denote the linear ordering induced in B_{n} by the free element d_{n}.

12.2. Remark. By the results of Short and Wiest [27], it may be deduced through the connection with their work explained in Remark 6.3 that (B_{n}, ≤_{d_{n}}) is the Dehornoy ordering. A more direct link with the Dehornoy ordering is explained in Remark 13.11.

12.3. Example. We calculate the order of the braid generators σ_{1}, σ_{2}, σ_{3} in (B_{4}, ≤_{d_{4}}), where d_{4} = v_{4}e_{3}^{−1}e_{4}e_{2}^{−1}e_{3}e_{1}^{−1}e_{2}e_{1}v_{4}. Direct calculation yields

σ_{1} · d_{4} = v_{4}e_{3}^{−1}e_{4}e_{2}^{−1}e_{3}e_{2}^{−1}e_{1}e_{2}e_{1}^{−1}e_{2}v_{4}
σ_{2} · d_{4} = v_{4}e_{3}^{−1}e_{4}e_{3}^{−1}e_{2}e_{1}^{−1}e_{3}e_{2}^{−1}e_{3}e_{1}v_{4}
σ_{3} · d_{4} = v_{4}e_{3}e_{2}^{−1}e_{4}e_{3}^{−1}e_{4}e_{1}^{−1}e_{2}e_{1}v_{4} ,

so that id ≤_{d_{4}} σ_{3} ≤_{d_{4}} σ_{2} ≤_{d_{4}} σ_{1}.

13. Ordering B_{∞}

Our methods extend naturally to the braid group B_{∞}. This group is the group colimit of the B_{n} organized by the natural inclusions B_{n} [figure]B_{m}, n < m. B_{∞} is also isomorphic to the quotient of the free group in countably many generators, given by the same relations as in the finite case (but now there are countably many). A member of B_{∞} may be thought of as a braid of countably many strands for which all but finitely many strands are unbraided.

Consider a collection of countably many marked points v_{i} on the sphere S that has just one limit point v_{ω}. We suppose that these marked points lie on a closed line segment with endpoints v_{0} and v_{ω}.

v_{0} [figure: e_{1}]v_{1} [figure: e_{2}]v_{2} ⋯ v_{n−1} [figure: e_{n}]v_{n} ⋯ v_{ω}

Let U denote the open set S −{v_{0}, …, v_{n}, …, v_{ω}}. U is connected, locally path-connected and locally simply connected with a universal covering space P : Y [figure]U. Consider the

pure, cosheaf space factorization of Y [figure: P]U [figure]S.

[figure: U; S; Y; U; P; Y; Ỹ; η; Ỹ; S; ψ_{ω+1}]

The open subset U [figure]S is pure, the square is a pullback, and ψ_{ω+1} is a ramified covering space. As in the case of finitely many marked-points, Ỹ is connected, locally pathconnected, semi-locally simply connected, and simply connected.

An interesting feature of Ỹ is that no branching occurs at the points in the fiber of v_{ω}, even though these points are members of the branch-point set Ỹ −Y. A precise expression of this fact is as follows.

13.1. Proposition. If x^{∈} Ỹ lies over v_{ω}, then x lies in the closure of exactly one leaf of P. (Whereas a point over v_{i}, i < ω, lies in the closures of countably many leaves of P.)

The objects of the groupoid π_{1}^{r}(S, ω + 1) are the elements of Ỹ − Y as in the finite case. Morphisms and composition in π_{1}^{r}(S, ω + 1) are also defined as in the finite case. Composition at points of the fiber ψ_{ω+1}^{−1}(v_{ω}) is simpler than over a v_{i}, i < ω, because there can be no cycling around a point over v_{ω}. On S_{U}, we compose two paths (that can be composed) at v_{ω} by moving their end-points appropriately to the ‘right’ of v_{ω}.

Let F (X) denote the free group on a set X. Let E denote a set of countably many ‘edge’-symbols e_{1}, e_{2}, . . .. Fix a point x^{∈} Ỹ − Y lying over v_{ω}. The paths in Ỹ_{Y} with domain x and codomain in Ỹ − Y may be encoded as words

v_{i}^{±}wv_{ω} ; w^{∈} F (E) , i^{∈} {0, 1, 2, …, ω} = ω + 1 ,  (7)

where we identify v_{0}^{+} ∼ v_{0}^{−} and v_{ω}^{+} ∼ v_{ω}^{−}. There are codomain winding relations for the v_{i}^{±}, i < ω, so if w^{∈} F (E) is a reduced word, then v_{i}^{±}wv_{ω} may not be reduced. But there are no codomain winding relations for v_{ω}, so a word v_{ω}wv_{ω} is reduced if w^{∈} F (E) is. We have the following.

13.2. Proposition. The following collections are in bijective correspondence (in an obvious way).

1. The fiber ψ_{ω+1}^{−1}(v_{ω}) = {y^{∈} Ỹ | ψ_{ω+1}(y) = v_{ω}}.

2. For any x such that ψ_{ω+1}(x) = v_{ω}, the subset of x/π_{1}^{r}(S, ω + 1) consisting of those isomorphisms whose codomain also lies over v_{ω}.

3. The collection of reduced words v_{ω}wv_{ω}.

4. F (E).

If G is a group, let G^{⋆} denote G−{id}. For instance, F (E)^{⋆} consists of all non-empty reduced words in E. The collection of reduced words (7) (except v_{ω}v_{ω}) may be linearly ordered just as in the finite case. Hence, the subset of the collection (7) consisting of the reduced words v_{ω}wv_{ω} (except v_{ω}v_{ω}) can also be linearly ordered. This subset corresponds to F (E)^{⋆}. We shall use the following terminology.

13.3. Definition. We shall refer to the ordering of F (E)^{⋆} given by Definition 6.1 (ignoring the vertex-symbols and admitting countably many edge-symbols) as the hyperbolic ordering of F (E)^{⋆}.

13.4. Remark. We can work without the vertex-symbols v_{i}^{±}, but we cannot ignore the vertex-symbol v_{ω} in Definition 13.3. For instance, if we wish to compare e_{2}e_{3} and e_{3}, we compare e_{2} with v_{ω} starting from the edge that is clockwise adjacent to e_{3}^{−1}, which is e_{2}^{−1}. Then e_{2}e_{3} ≤ e_{3} since e_{2} occurs before v_{ω} clockwise from e_{2}^{−1}. If the largest common right subword of two words is empty, then we start from v_{ω}. (In the finite case we start from the ‘least’ edge e_{n}^{−1}.) How to compare the empty word with another one is not defined because in this case we must compare an e_{i}^{±1} with v_{ω} starting from v_{ω}; however, v_{ω} is neither the top nor the bottom of the order.

13.5. Proposition. The hyperbolic ordering of F (E)^{⋆} is a dense linear ordering (without endpoints).

13.6. Remark. The hyperbolic ordering of F (E)^{⋆} is neither invariant under left multiplication nor under right multiplication. For instance, we have e_{1} ≤ e_{3}, but e_{3}e_{2}^{−1} ≤ e_{1}e_{2}^{−1}. If we multiply e_{1} ≤ e_{3} on the left by e_{1}e_{3}^{−1}, then we have e_{1} ≤ e_{1}e_{3}^{−1}e_{1}. Thus, the hyperbolic ordering is not a group ordering in the usual sense. (In anycase, the identity is left out of the ordering.) But the hyperbolic ordering does satisfy property (3), which in the present context reads as follows:

∀ w, z^{∈} F (E)^{⋆}, w < z ⇒ zw^{−1} < w^{−1} .  (8)

For instance, we have e_{3}e_{2}^{−1} < e_{4}^{−1}e_{2}^{−1}. Then e_{4}^{−1}e_{3}^{−1} < e_{2}e_{3}^{−1} as predicted by (8).

A member of B_{∞} may be regarded as a homeomorphism class β = [h] of S_{U}, where U = S−{v_{0}, …, v_{n}, …, v_{ω}}, such that the homeomorphism h fixes an open neighbourhood of v_{ω}. If x lies over v_{ω}, then we may lift such a homemorphism to a homeomorphism of Ỹ_{Y} that fixes x. In this way we obtain an action of B_{∞} in π_{1}^{r}(S, ω + 1) such that every braid fixes the chosen x. Then we may pass to an action of B_{∞} in (x/π_{1}^{r}(S, ω + 1))^{⋆}. (Unlike the finite case, by Proposition 13.1 we do not need to choose a leaf of x to define the action because x has just one leaf.) This action is presented by (4) and (5) just as in the finite case.

The action of B_{∞} in (x/π_{1}^{r}(S, ω + 1))^{⋆} is order-preserving. Moreover we may restrict the action to those isomorphisms whose codomain also lies over v_{ω}, which we have encoded as precisely the members of F (E)^{⋆}. We have the following.

13.7. Proposition. The following action of B_{∞} in F (E) (as in (4)) preserves the hyperbolic ordering of F (E)^{⋆}.

\begin{matrix}σ_{i} · e_{i} = e_{i−1}e_{i}^{−1}e_{i+1} \\ (σ_{1} · e_{1} = e_{1}^{−1}e_{2}) \\ σ_{i} · e_{j} = e_{j} ; j ≠ i \end{matrix} \} i = 1, 2, ….  (9)

For any n, the only member of the subgroup B_{n} [figure]B_{∞} that stabilizes the word

d_{n} = e_{n−1}^{−1}e_{n} ⋯ e_{2}^{−1}e_{3}e_{1}^{−1}e_{2}e_{1}

is the identity braid. (We shall say that d_{n} is n-free.) Thus, the linear preorder ≤_{d_{n}} that d_{n} induces in B_{∞} is a linear order in B_{n} (meaning anti-symmetric). The natural inclusions B_{n} [figure]B_{m} are compatible with the linear orderings induced by d_{n} and d_{m}, so that all the d_{n} taken together induce a linear order ≤_{d} of B_{∞}: define id ≤_{d} β if for some N for which β^{∈} B_{N}, we have id ≤_{d_{N}} β in B_{N}.

Let V denote a set of countably many ‘vertex’-symbols v_{0}, v_{1}, . . .. The classical Artin action of B_{∞} in F (V) is given by:

\begin{matrix}σ_{i} · v_{i−1} = v_{i} \\ σ_{i} · v_{i} = v_{i}^{−1}v_{i−1}v_{i} \end{matrix} \} i = 1, 2, ….  (10)

Of course, the map v_{i} ↦ e_{i+1} provides a group isomorphism of F (V) with F (E), but we wish to know how F (V) and F (E) are related as B_{∞}-sets. Proposition 13.8 has been observed by Larue [19].

13.8. Proposition. (Larue) The group homomorphism a : F (V) [figure]F (E) given by a(v_{0}) = e_{1} and a(v_{i}) = e_{i}^{−1}e_{i+1}, i = 1, 2, …, is an isomorphism of B_{∞}-sets. (The inverse of a is given by a^{−1}(e_{i}) = v_{0}v_{1} ⋯ v_{i−1}.)

By Proposition 13.8, we may rephrase Proposition 13.7 as follows.

13.9. Corollary. F (V)^{⋆} may be given the structure of a dense linear order such that the Artin representation (10) is an order-preserving action. In this ordering (which is not a group ordering) we have v_{i}^{−1} < v_{i} < v_{i+1}^{−1}. For any n, the word v_{n−1} ⋯ v_{1}v_{0} is n-free, so that the linear preorder it induces in B_{∞} is a linear ordering of the subgroup B_{n}. All the words v_{n−1} ⋯ v_{1}v_{0} taken together induce a linear order in B_{∞} as explained in Proposition 13.7.

Proof. The ordering of F (V)^{⋆} is given by w ≤ w′ just when a(w) ≤ a(w′) in the hyperbolic ordering of F (E)^{⋆}. For instance, we have v_{i} < v_{i+1}^{−1} because e_{i}^{−1}e_{i+1} < e_{i+2}^{−1}e_{i+1}. To see this we must first determine the starting symbol: it is the symbol clockwise adjacent to e_{i+1}^{−1}, which is e_{i}^{−1}. But then e_{i}^{−1} occurs before e_{i+2}^{−1} (clockwise) starting from e_{i}^{−1}. The word v_{n−1} ⋯ v_{1}v_{0} = a^{−1}(d_{n}) is n-free by Proposition 12.1.

13.10. Remark. It ought to be possible to explicitly describe the ordering in F (V)^{⋆} given by Corollary 13.9. We leave this as an exercise. 13.11. Remark. Dehornoy [9] has provided an analysis of his ordering in terms of the Artin action of a braid group in a free group. Using Corollary 13.9 as a link to Dehornoy’s analysis should provide another way to show the the ordering ≤_{d_{n}} in B_{n} coincides with Dehornoy’s. We shall not carry out the details of this.

The action (9) of B_{∞} in F (E) has no free elements. We can obtain free elements by admitting (reduced) infinite words

w = ⋯ e_{i_{3}}^{ν_{3}}e_{i_{2}}^{ν_{2}}e_{i_{1}}^{ν_{1}} ; ν_{j} ^{∈} {−1, 1} , i_{j} ^{∈} {1, 2, …} .  (11)

Let F̂ (E) denote the collection of infinite reduced words. By convention, F̂ (E) includes all finite words as well. Geometrically, such an infinite word w can be thought of as an ‘open-ended’ path

p : [0, 1)_{(0,1)} [figure] Ỹ_{Y} ,

where ψ_{ω+1}(p(0)) = v_{ω}. 13.12. Remark. If ψ_{ω+1}(x) = v_{ω}, then there are two ‘spiral’ maps (clockwise and counter-clockwise):

x/π_{1}^{r}(S, ω + 1) [figure]F̂ (E) .

For instance, the counter-clockwise spiral map is given by

\begin{matrix}v_{i}^{+}wv_{ω} ↦ ⋯ e_{i+1}e_{i}^{−1}e_{i+1}w \\ v_{i}^{−}wv_{ω} ↦ ⋯ e_{i}^{−1}e_{i+1}e_{i}^{−1}w \end{matrix} \} i < ω ,

v_{ω}wv_{ω} ↦ w (these are the finite words).

On the sphere, all three paths above begin at v_{ω}, but whereas the first two appear to spiral into v_{i}, the third one makes finitely many loops around the marked-points and then returns to v_{ω}.

Consider the infinite word

d = ⋯ e_{2}^{−1}e_{3}e_{1}^{−1}e_{2}e_{1} = a(⋯ v_{2}v_{1}v_{0}) .

On S_{U} we think of d as a presentation of a ‘wild’ path p_{d} that begins and ends at v_{ω}, and loops once around every v_{i} in a loop whose diameter converges to 0. Such a path p_{d} : [0, 1] [figure]S_{U} may be lifted (uniquely) to an open-ended one

\overline{p_{d}} : [0, 1)_{(0,1)} [figure] Ỹ_{Y} ,

provided that \overline{p_{d}}(0) is given lying over v_{ω}. However, it is not possible to continuously define \overline{p_{d}}(1) such that ψ_{ω+1}(\overline{p_{d}}(1)) = v_{ω}.

The hyperbolic ordering of F (E)^{⋆} may be extended to the collection F̂ (E)^{⋆} of nonempty infinite words (11). Likewise, the action (9) of B_{∞} in F (E)^{⋆} may be extended to F̂ (E)^{⋆}. Although we cannot necessarily calculate this action effectively, we can effectively compare d and β · d because d has the property that for any braid β^{∈} B_{∞}, β · d differs from d only by an effectively computable finite right subword. We have the following.

13.13. Proposition. The infinite word d is free for the action of B_{∞} in F̂ (E)^{⋆}. The linear order it induces in B_{∞} is precisely the one ≤_{d} obtained in Proposition 13.7. It is the Dehornoy ordering of B_{∞}.

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Dept. of Mathematics and Statistics University of Saskatchewan Saskatoon, SK., S7N 5E6 CANADA

Email: funk@snoopy.usask.ca

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