Homological stability for Artin monoids

We prove that certain sequences of Artin monoids containing the braid monoid as a submonoid satisfy homological stability. When the K(π,1) conjecture holds for the associated family of Artin groups, this establishes homological stability for these groups. In particular, this recovers and extends Arnol'd's proof of stability for the Artin groups of type A , B and D .


Introduction
A sequence of groups or monoids with maps between them is said to satisfy homological stability if the induced maps on homology are isomorphisms for n sufficiently large compared to i.
This paper concerns homological stability for sequences of Artin monoids and groups, and in this paper the associated maps will always be inclusions.In particular, we consider sequences of Artin groups that have the braid group as a subgroup, and the corresponding sequences of monoids.
We recall the definition of Artin groups.Given a finite set Σ, to every unordered pair {σ s , σ t } ∈ Σ × Σ associate either a natural number greater than 2 or the symbol ∞, and denote this by m(s, t).An Artin group A with generating set Σ has the following presentation A = Σ | π(σ s , σ t ; m(s, t)) = π(σ t , σ s ; m(s, t)) , where π(σ s , σ t ; m(s, t)) is the alternating product of σ s and σ t starting with σ s and of length m(s, t).The braid group with its standard presentation is the archetypal example of an Artin group, with presentation Every Artin group has an associated Coxeter group (discussed in Section 2), and in fact Artin groups were first introduced by Brieskorn [2] as the fundamental groups of hyperplane complements built from Coxeter groups.The information of the presentation can be packaged into a Coxeter diagram.This diagram has vertex set Σ and edges corresponding to m(s, t) for each pair of vertices: no edge when m(s, t) = 2, an unlabelled edge when m(s, t) = 3 and an edge labelled with m(s, t) otherwise.For example, the braid group B n has diagram which is known as the Coxeter diagram of type A and has corresponding Coxeter group the symmetric group S n .The sequences of Artin groups studied in this paper correspond to the following sequence of diagrams: where the grey box indicates that the sequence begins with an arbitrary diagram: an arbitrary Artin group with finite generating set.The type A subdiagram corresponds to a subgroup of A n being the braid group B n+1 , with an increasing number of generators as n increases.This gives rise to a sequence of groups and inclusions and the goal of this paper is to discuss stability for sequences of Artin groups of this form.This was motivated by work of Hepworth [13], who proved homological stability for the associated sequence of Coxeter groups.
While the argument used for the proof of homological stability is similar to that used by several authors, the novel part of this paper comes from dealing with Artin groups and monoids.Very little is known for Artin groups in general, for instance, the centre of a generic Artin group is unknown and it is not known whether all Artin groups are torsion free.In particular, there are no tools to date for working with Artin cosets (for example, there is no canonical way to choose a coset representative), something that is usually desirable when proving homological stability for a family of groups.Therefore, the results of this paper are stated and proved for the corresponding Artin monoids, for which a technical 'coset' theory is developed in Section 4 (the notion of coset of a submonoid is not defined in general).Key properties of Artin monoids, such as the existence of a well-defined length function and the existence of lowest common multiples under certain conditions, allow us to define a canonical choice of 'coset representative' for these monoids.From the monoid result, we then deduce homological stability for Artin groups that satisfy the K(π, 1) conjecture (discussed in more detail below).
We denote the Artin monoid corresponding to A n by A + n .The inclusion map between the monoids is denoted s and called the stabilisation map.The main result of this paper is the following, which to my knowledge is the first instance where homological stability is proved for monoids.

Theorem A. The sequence of Artin monoids
More precisely, the induced map on homology 2 and a surjection when * = n 2 .Here, homology is taken with arbitrary constant coefficients, that is, coefficients in an abelian group.
The classifying space of an Artin monoid BA + is homotopy equivalent to some interesting spaces that arise naturally in mathematics.One manifestation of this is that in the study of Artin groups, there is a well-known conjecture by Arnol'd, Brieskorn, Pham and Thom called the K(π, 1) conjecture (discussed in Section 3).For this introduction, it suffices to know the following fact due to Dobrinskaya.1460244x, 2020, 3, Downloaded from https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/plms.12335 by University Of Glasgow, Wiley Online Library on [02/02/2023].See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions)on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License Theorem [10,Theorem 6.3].Given an Artin group A and its associated monoid A + , the K(π, 1) conjecture holds if and only if the induced map between their classifying spaces BA + → BA is a homotopy equivalence.Thus, if the K(π, 1) conjecture holds for a family of Artin groups, Theorem A establishes homological stability as below.
Corollary B. When the K(π, 1) conjecture holds for all A n , the sequence of Artin groups satisfies homological stability.More precisely, the induced map on homology is an isomorphism when * < n 2 and a surjection when * = n 2 .Here, homology is taken with arbitrary constant coefficients, that is, coefficients in an abelian group.
the categorical set up of [21], however, the classifying spaces of the sequence can be shown to assemble into an E 1 -module over an E 2 algebra, as in [14].Combining the results of [14] with the high connectivity results established in this paper, our homological stability result with constant coefficients (Theorem A) can most likely be enhanced to one with abelian coefficients and coefficient systems of finite degree in the sense of [14,Section 4].

Outline of proof
The proof follows the outline of a standard homological stability argument, which we describe below for the benefit of the reader.We indicate where the new ingredients are used.
The proof of Theorem A requires the introduction of a semi-simplicial space A n • for each monoid in the sequence A + n such that: (1) there exist homotopy equivalences A n p BA + n−p−1 for every p 0; and (2) there is a highly connected map from the geometric realisation of A n • to the classifying space BA + n , which we denote φ an isomorphism on a large range of homotopy groups.

The skeletal filtration of A n
• gives rise to a spectral sequence E 1 p,q = H q (A n p ) ⇒ H p+q ( A n • ).From Point (1), it follows E 1  p,q = H q (A n p ) = H q (BA + n−p−1 ).We prove that on the E 1 page under the above equality the differentials are given by either the zero map or the stabilisation map s * : H q (BA + n−p−1 ) → H q (BA + n−p ).Following this, applying the inductive hypotheses that previous monoids in the sequence satisfy stability gives that in a range (when q is small compared to n) the spectral sequence converges to H q (BA + n−1 ): E 1  p,q = H q (BA + n−p−1 ) ⇒ H p+q ( A n • ) = H q (BA + n−1 ) in a range.The highly connected map of Point (2) above now gives that in a range the spectral sequence also converges to the homology of BA + n , which completes the proof.For sequences of discrete groups, a usual candidate for A n • would be built out of cosets of previous groups in the sequence.However, the fact that no tools exist for manipulating Artin cosets means that this approach cannot be taken.The coset theory developed in this paper for the corresponding sequence of Artin monoids is used to build A n • .The main obstacle in the proof is the high connectivity argument for Point 2 which follows a 'union of chambers' argument inspired by, but more involved than, work of Paris [19] and Davis [8].

Organisation of the paper
Sections 2 and 3 provide background on Coxeter groups and Artin groups, and the K(π, 1) conjecture, respectively.Section 4 then introduces Artin monoids and develops a novel theory of 'cosets' and corresponding technical results.Following this, Section 5 details the required semisimplicial background and particular monoid constructions used in the proof, some of which are new.Section 6 applies the theory of Section 4, and introduces notation used throughout the proof.Section 7 introduces the semi-simplicial space A n • and the map φ • described above.High connectivity of φ • is then the topic of Section 8, in which the general method of proof for the high connectivity argument is introduced, before the proof is split into several cases, due to the complexity of using Artin monoids.Finally, the spectral sequence argument and homological stability result are given in Section 9.
Here, m(s, t) = ∞ means there is no relation between s and t.We call (W, S) a Coxeter system.We adopt the convention that (W, ∅) is the trivial group.
Remark 2.3.Note that the condition m(s, s) = 1 on the Coxeter matrix implies that the generators of the group are involutions, that is, s 2 = e for all s in S. Remark 2.6.The relations (st) m(s,t) = e can be rewritten as π(s, t; m(s, t)) = π(t, s; m(s, t)) when m(s, t) = ∞.Therefore, the presentation of a Coxeter group W can also be given as Definition 2.7.Given a Coxeter matrix corresponding to a Coxeter system (W, S), there is an associated graph called the Coxeter diagram, denoted D W .It is the graph with vertices indexed by the elements of the generating set S. Edges are drawn between the vertices corresponding to s and t in S when m(s, t) 3 and labelled with m(s, t) when m(s, t) 4, as shown below: The following is a complete list of the diagrams corresponding to finite irreducible Coxeter groups.
Definition 2.9.We say that a finite irreducible Coxeter group W is of type D if its corresponding diagram is given by D, and we denote this Coxeter group W (D).
Remark 2.10.The Coxeter group W (A n ) is isomorphic to S n+1 , the symmetric group, which is the reflection group of the regular (n + 1)-simplex.Definition 2.11.Let (W, S) be a Coxeter system.For each T ⊆ S, T generates a subgroup W T such that (W T , T ) is a Coxeter system in its own right.We call subgroups that arise in this way parabolic subgroups.If the subgroup is finite, we call it a spherical subgroup.

Artin groups
This section follows Charney [6, Section 1] and notes by Paris [19].Given a Coxeter system (W, S), the corresponding Artin group is given by forgetting the involution relations, that is, setting m(s, s) = ∞.Definition 2.12.For every Coxeter system (W, S), there is a corresponding Artin system (A W , Σ) comprising an Artin group A W with generating set Σ := {σ s for s ∈ S} We note that the Coxeter diagram D W contains all the information about the Artin group presentation.
Example 2.13.The Artin group A W corresponding to the Coxeter group W ∼ = S n is the braid group B n .The corresponding diagram D W is where we relabel σ si to σ i for ease of notation.The presentation is therefore given by the standard presentation for the braid group on n strands.
Example 2.16.When all of the edges in the Coxeter diagram are labelled with ∞, but not necessarily all possible edges are present (some m(s, t) may be equal to 2), then the corresponding Artin group is called a right-angled Artin group, or RAAG.Definition 2.17.When the Coxeter group W is finite, that is, when its diagram D W is a disjoint union of diagrams from Proposition 2.8, then the corresponding Artin group A W is called a finite-type Artin group, or a spherical Artin group.
Much of the known theory of Artin groups is concentrated around RAAGs and finite-type Artin groups, though we do not restrict ourselves to either of these families in our results.In general, little is known about Artin groups.For instance, the following properties hold for finite-type Artin groups [6].
• There exists a finite model for the classifying space K(A W , 1).
• A W has solvable word and conjugacy problem.
To date these properties are not known for general Artin groups.In the next section, we consider the first point in detail.

The K(π, 1) conjecture
This section introduces the K(π, 1) conjecture, following [19].In general, one can associate a hyperplane arrangement A and associated complement M (A) to each Coxeter group W , such that there is a free action of W on M (A).When we consider this hyperplane complement modulo this W action, the corresponding quotient M (A)/W has as its fundamental group the Artin group A W .In some cases, this quotient space is known to be a K(A W , 1), in particular we recall Deligne's theorem for finite-type Artin groups.
Theorem 3.1 (Deligne's theorem [9]).For W a finite Coxeter group and A W the associated Artin group, M (A)/W is aspherical with fundamental group A W , that is, M (A)/W is a K(A W , 1).
For arbitrary Artin groups, the K(π, 1) conjecture was formulated by Arnol'd, Brieskorn, Pham and Thom, and states than an analogue of Deligne's theorem holds for all Artin groups.The analogue of the hyperplane complement was formulated by Vinberg.For a more detailed description, see Davis [8], notes by Paris [19] and the introduction to a paper on RAAGs by Charney [6].
Remark 3.2.It is worth noting here a reformulation of the conjecture in terms of a finitedimensional CW-complex called the Salvetti complex, denoted by Sal(A) and introduced by Salvetti in [22], for a hyperplane arrangement A in a finite-dimensional real vector space V .The Salvetti complex is defined in terms of cosets of finite subgroups of the Coxeter group [23].Paris extended this definition to any infinite hyperplane arrangement in a non-empty convex cone I [19] and proved that Sal(A) and M (A) have the same homotopy type.The K(π, 1) conjecture can therefore be restated as a conjecture about the Salvetti complex.
The K(π, 1) conjecture has been proven for large classes of Artin groups [19].However, the conjecture has not been proven for general Artin groups.We will apply a reformulation of the K(π, 1) conjecture to our results, involving the Artin monoid A + and discussed in Section 4.

Artin monoids
The start of this section follows Jean Michel's A note on words in braid monoids [16] and Brieskorn and Saito's Artin-Gruppen und Coxeter-Gruppen [4].Much of the material in Section 4.3 is new.

Definition and examples
Definition 4.1.The Artin monoid system (A + W , Σ) associated to a Coxeter system (W, S) is given by the generating set Σ for the corresponding Artin system (A W , Σ), and the monoid with the same presentation as the Artin group A W : W are therefore strings of letters for which the alphabet consists of σ s in Σ.
Remark 4.2.The group completion of A + W is A W .
Example 4.3.The braid monoid B + n is the monoid associated to the Coxeter group S n , the symmetric group, with group completion the braid group B n .Given the standard generating set for the symmetric group, the braid monoid consists of words in the braid group made from the positive generators σ i .In terms of the braid diagrams these can be viewed as braids consisting of only positive twists.Definition 4.4.We call a submonoid M + of an Artin monoid A + a parabolic submonoid if the monoid M + is generated by the set M + ∩ Σ.We call this generating set for the monoid Σ M , giving a system (M + , Σ M ).
In this paper, by convention every submonoid of an Artin monoid considered will be a parabolic submonoid.

Divisors in Artin monoids
Definition 4.5.Define the length function on an Artin monoid A + with system (A + , Σ) : A + → N to be the function which maps an element α in A + to the unique word length required to express α in terms of the generators in Σ.
Remark 4.6.Note here that since there are no inverses in Artin monoids, multiplication corresponds to addition of lengths, that is, (ab) = (a) + (b) ( is a monoid homomorphism).Definition 4.7.For elements α and β in an Artin monoid A + with system (A + , Σ), we say that α R β if for some γ in A + we have β = γα, that is a word representing α appears on the right of some word representing β, in terms of the generating set Σ. We say that β is right-divisible by α, or alternatively that α right-divides β.We now consider work by Brieskorn and Saito in their 1972 paper Artin-Gruppen und Coxeter-Gruppen [4].They consider notions of least common multiples and greatest common divisors of sets of elements in the Artin monoid.We are interested in the notion of least common multiple.Definition 4.9.Given a set of elements {g j } j∈J in an Artin monoid A + with system (A W , Σ), a common multiple β is an element in A + which is right-divisible by all g j .That is g j R β for all j in J.A least common multiple of {g j } is a common multiple that right-divides all other common multiples.Let E be a set of elements in the Artin monoid A + .Denote the least common multiple (if it exists) of E by Δ(E).For α and β two elements in A + denote the least common multiple of α and β (if it exists) by Δ(α, β).Remark 4.10.Should a least common multiple exist, it will be unique.Proposition 4.11 (Brieskorn and Saito [4,4.1]).A finite set of elements in an Artin monoid either has a least common multiple or no common multiple at all.Remark 4.12.Since the relations in an Artin monoid have the same letters appearing on each side, the set of letters present in any word representing an element of an Artin monoid is fixed.Therefore, the notion of 'letters appearing in an element' is well defined.(Brieskorn and Saito [4]).For a fixed generating set, the letters arising in a least common multiple of a set of elements in an Artin monoid are only those letters which appear in the elements themselves.Definition 4.14.Consider a submonoid M + of an Artin monoid A + , with system (M + , Σ M ).Given an element α in A + we define two end sets Remark 4.15.EndGen M (α) is exactly the letters σ s in Σ M that a word representing α can end with, and EndMon M (α) is exactly the elements in M + that a word representing α can end with.Note that EndGen M (α) is a subset of EndMon M (α), consisting of words of length 1 and EndMon M (α) = ∅ if and only if α has no right-divisors in M + .

Required theory
Much of the proof of Theorem A is concerned with algebraic manipulation of words in the Artin monoid.Here we introduce some technical definitions and lemmas used in the proof.We build up a theory of cosets in the case of Artin monoids, which is new unless cited.Lemma 4.16.Given α in A + , and M + a submonoid of A + , the set EndMon M (α) has a least common multiple Δ(EndMon M (α)) = β which lies in the submonoid M + .That is, there exists β in M + and γ in A + such that α = γβ for some words representing α, β, γ, and if β in A + and γ in A + satisfy α = γ β , it follows that β R β .
Proof.From Proposition 4.11 if a common multiple exists, then Δ(EndMon M (α)) exists.We have that α itself is a common multiple of all elements in EndMon M (α), by definition of EndMon M (α).Furthermore, Lemma 4.13 notes that only letters appearing in EndMon M (α) will appear in Δ(EndMon M (α)).By definition, these are letters in M + and so Δ(EndMon M (α)) lies in M + .Remark 4.17.For n element α in A + let Δ(EndMon M (α)) = β.We write α with respect to M + for the element α in A + such that α = αβ.It will always be clear in the text with respect to which submonoid M + we are taking the reduction.Definition 4.18.For A + an Artin monoid and M + a submonoid, let A + (M ) be the following set That is, A + (M ) is the set of elements in A + which are not right-divisible by any element of M .Lemma 4.19.For all α in A + and all β in M + , α = αβ where the reduction is taken with respect to M + .Proof.Let α = γ, so α = γη for some η in M + , and EndMon M (γ) = ∅, that is, γ has no right-divisors in M + .Then αβ = γηβ and since η and β are both in M + , it follows that ηβ ∈ EndMon M (αβ).If ηβ is the least common multiple of EndMon M (αβ), then αβ = γ = α so we are done.Suppose for a contradiction that ηβ is not the least common multiple of EndMon M (αβ), and note that ηβ is a right-divisor of the least common multiple.Then there exists some ζ in M + of length at least 1 such that ζηβ where M + is a submonoid of A + .When we use this relation it will be made clear which submonoid M + is being considered.The relation ∼ is symmetric and reflexive.Let ≈ be the transitive closure of ∼.That is, α 1 ≈ α 2 if there is a chain of elements in A + : for some k.Denote the equivalence class of α in A + under the relation ≈ with respect to the submonoid M + as [α] M .Definition 4.21.Let q : A + (M ) → A + / ≈ be the quotient map, taken with respect to the equivalence relation ≈.

3]). Given an Artin group A W and its associated monoid A +
W , the K(π, 1) conjecture holds if and only if the natural map between their classifying spaces, BA + W → BA W is a homotopy equivalence.
This theorem has been reproved using a different Morse-theoretic approach by Ozornova [17] and her result has in turn been strengthened by Paolini [18].

Semi-simplicial constructions with monoids
This section is split into three subsections.The first introduces background semi-simplicial theory before the second introduces theory for generic monoids and submonoids, including some new results.The third subsection focuses on Artin monoids and contains results required later in the proof.

Semi-simplicial objects
This subsection consists of the required background and follows Ebert and Randal-Williams [12]. .Let Δ denote the category which has as objects the non-empty finite ordered sets [n] = {0, 1, . . ., n}, and as morphisms monotone increasing functions.These functions are generated by the basic functions which act on the ordered sets as follows: The opposite category Δ op is known as the simplicial category.We denote the opposite of the maps D i by ∂ i and the opposite of the maps S i by s i .We call these the face maps and the degeneracy maps, respectively.
Let Δ inj ⊂ Δ be the subcategory of Δ which has the same objects but only the injective monotone maps as morphisms, generated by the D i .The opposite category Δ op inj is known as the semi-simplicial category and its morphisms are therefore generated by the face maps ∂ i .
for all p 1 and 0 i p.
Example 5.5 [12, 1.2].The standard n-simplex has two equivalent manifestations: as a simplicial object in Set and as an object in T op.When viewed as a simplicial set the standard nsimplex is denoted Δ n • and is defined via the functor ) for all [m] in Δ op .When viewed as an object in T op the standard n-simplex is denoted |Δ n | and defined to be One can associate to a morphism φ : is the first face map to be applied, followed by ∂ p−2 i2 , etc.For ease of notation we dispense with superscripts, writing the tuple as (∂ i1 , ∂ i2 , . . ., ∂ i k ) and assuming the domain and targets are such that the composite map is defined.
Lemma 5.6.With the above notation, the tuple of face maps can be written such that i j+1 i j for all j.
This procedure reduces the sum k j=1 i j by one, and therefore upon iteration must terminate.Applying this process enough times gives i j+1 i j for all j.Definition 5.7 [12, 1.2].The geometric realisation of a semi-simplicial set or space is the topological space denoted by X • and defined to be The geometric realisation is an example of a coequaliser or colimit (see Dugger [11]).Definition 5.8.Given a semi-simplicial map f ).We write the image of the standard face maps in each simplicial direction (∂ i × id) and (id ×∂ j ), as ∂ i,• and ∂ •,j .We note that and we denote this map ∂ i,j .When C is equal to Top the bi-semi-simplicial object is called a bi-semi-simplicial space.
Remark 5.10.A bi-semi-simplicial space can be viewed as a semi-simplicial object in ssT op in two ways: Definition 5.11 [12, 1.2].Given a bi-semi-simplicial space X •,• we define its geometric realisation to be where ∼ is generated by (x, t 1 , t 2 ) ∼ (y, u 1 , u 2 ) whenever (∂ i,j )(x) = y, D i (u 1 ) = t 1 and D j (u 2 ) = t 2 .This is equivalent to taking the geometric realisation of the semi-simplicial space first in the p direction, followed by the q direction, or in the q direction followed by the p direction.This is due to the following homeomorphisms [12, 1.9 and 1.10] :

Semi-simplicial constructions using monoids and submonoids
The following description of the geometric bar construction and related definitions loosely follows Chapter 7 of May's Classifying spaces and fibrations [15].In this section, we view monoids and groups as discrete spaces.
Definition 5.12.Let M be a monoid and let X and Y be spaces with a left and right action of M , respectively.Then the bar construction denoted B(Y, M, X) is the geometric realisation of the semi-simplicial space B • (Y, M, X) given by Elements in B n (Y, M, X) are written as y[g 1 , . . ., g n ]x for y ∈ Y , g i ∈ M for 1 i n and x ∈ X. Face maps are given by Elements are written as [m 1 , . . ., m j ]y for m i in M for 1 i j and y in Y .Face maps are given by In This is the geometric realisation of the semi-simplicial space B • ( * , N, M) given by Elements are written as [n 1 , . . ., n j ]m for n i in N for 1 i j and m in M .Face maps are given by We can build a similar homotopy quotient for a submonoid N acting on Proof.Writing down the simplices and face maps for the homotopy quotients G \ \ * and G / / * gives exactly the simplices and face maps for the standard resolution or bar resolution of G, which is a model for BG (see, for example, [5]).This holds similarly for the monoid M (see, for example, [15, p. 31]).
Proof.This is a consequence of [12, Lemma 1.12] using the augmentation to a point.Lemma 5.17.Let N be a monoid and S be a space with right N action.Suppose S can be decomposed as S ∼ = X × Y and, under this decomposition, the action of N restricts to a right action on the Y component and trivial action on the X component.Then the map given by the geometric realisation of the levelwise map on the bar construction for x ∈ X, y ∈ Y and n i ∈ N for all i is a homotopy equivalence.That is, the homotopy quotient satisfies where the homotopy equivalence is given by the geometric realisation of the levelwise map on the bar construction Proof.The homotopy quotient S / / N is the geometric realisation of the simplicial space B • (S, N, * ) with j-simplices given by B j (S, N, * ) = S × N j and face maps given by Definition 5.13, the first face map ∂ 1 encoding the right action of N on S.Under the decomposition S ∼ = X × Y the j-simplices are given by where the second isomorphism highlights that the action of N on S can be restricted to a right action on Y , since the action is trivial on the X component.Note that the second factor is precisely the j-simplices in B j (Y, N, * ), and since the face maps act trivially on the X factor, the face maps in B j (S, N, * ) induce face maps in B j (Y, N, * ) under the decomposition.The proof is concluded by taking the geometric realisation of B • (S, N, * ) and the geometric realisation of

Semi-simplicial constructions for Artin monoids
Given an Artin monoid A + and a parabolic submonoid M + , recall from Section 4 that A + (M ) is the set of elements in A + which do not end in elements in M + and there is a decomposition as sets (Proposition 4.23), where α = αβ (as defined in Remark 4.17) and the right action of M + on A + descends to a trivial action on A + (M ) and a right action on M + .In this section, we view monoids and sets as discrete spaces.
Proposition 5.18.The map defined levelwise on the bar construction B • (A + , M + , * ) by Proof.From Proposition 4.23, A + ∼ = A + (M ) × M + as sets, hence as discrete spaces, and this decomposition respects the right action of M + on A + .Then where the first homotopy equivalence uses Lemma 5.17 and the second homotopy equivalence uses Lemma 5.16.The levelwise map given by the composition of the maps in these two lemmas is precisely the map in the statement.
Proposition 5.19.Let A + be a monoid and M + be a submonoid.Consider two maps f and g : A + \ \ A + → A + \ \ A + which are both equivariant with respect to the action of M + on the right of A + \ \ A + .Then there exists an M + equivariant homotopy between the two maps.Let f k be the restriction of the map f to the k-cells of A + \ \ A + and similarly for g k .We first define an equivariant homotopy between f 0 and g 0 .Under the above decomposition, (A + \ \ A + ) 0 ∼ = (A + (M ) × M + ).Consider f 0 (α) and g 0 (α) in A + \ \ A + for α in A + (M ).Then since A + \ \ A + * by Lemma 5.16 it follows that there exists a path between f 0 (α) and g 0 (α): call this h 0 (α, t) for t ∈ [0, 1].Extend this homotopy to all 0-cells by setting h 0 (αm, t) = h 0 (α, t) • m for all m in M + .Then, since f 0 and g 0 are M + equivariant,

Proof. Denote the set of k-cells in A
Now assume that we have built an equivariant homotopy h k−1 (x, t) on the (k − 1)-skeleton and we show how to extend it to the k-cells.The homotopy Then its boundary consists of (k − 1)-cells and it follows that h k−1 defines a homotopy (∂(D(p 1 , . . ., p k )) • α) × I → A + \ \ A + and the maps f k and g k also define maps The union of these three maps defines a map from ∂((D(p 1 , . . ., p k ) • α) × I) to A + \ \ A + , but this boundary is a (k − 1)-sphere and so, since A + \ \ A + is contractible the (k − 1)-sphere bounds a (k)-disk.We can compatibly extend the map over this disk to create the required homotopy which agrees on the boundary with the three maps above.Now define h k on any k-cell D(p 1 , . . ., p k ) • αm by the following: for x in D(p 1 , . . ., p k ) • α we set Then by construction h k is M + equivariant and, since both f k and Definition 5.20.Given a monoid M and two submonoids N 1 and N 2 we can define the double homotopy quotient N 1 \ \ M / / N 2 to be the geometric realisation of the bi-semi-simplicial space (recall Definition 5.9) defined by taking the two simplicial directions arising from the bar constructions B • ( * , N 1 , M) and B • (M, N 2 , * ).The p, q level of the associated bi-semi-simplicial space X •• has simplices and face maps inherited from B • ( * , N 1 , M) in the p direction (∂ p,• ) and B • (M, N 2 , * ) in the q direction (∂ in N 1 and n j in N 2 for 1 i p and 1 j q.We note that the face maps on the left and right commute, since the only maps which act on the same coordinates are ∂ p,• in the p direction and ∂ •,0 in the q direction and these commute: )).

Preliminaries concerning the sequence of Artin monoids
This section introduces notation used throughout the remainder of the proof.
We consider the sequence of Artin monoids and inclusions with Artin monoid systems (A + n , Σ n ) given by the following diagrams.Here, the Artin Monoid A i corresponds to the Coxeter group W i , as defined in [13], and so we denote the corresponding Coxeter diagrams D Wi .Definition 6.1.Let (A 0 , Σ 0 ) be the Artin system corresponding to the Coxeter diagram D W1 , but with the vertex σ 1 and all edges which have vertex σ 1 at one end removed.We depict the diagram as above.Note that A 0 → A 1 .Remark 6.2.With the generating sets corresponding to the above sequence of diagrams, for all p every generator and hence every word in the monoid A + p commutes with σ j for j p + 2.
We now apply the theory developed in Section 4.3 to the specific case of a monoid A + n in the sequence of monoids and inclusions (1) and the submonoid of A + n , given by a previous monoid in the sequence A + p , where p < n.We adopt the following notation for the remainder of this paper.The generating set of A + n will always be given by Σ n , the generating set specified by the diagram D Wn .
• Let EndMon p (α) = EndMon Ap (α) and EndGen p (α) = EndGen Ap (α) for α in A + n , as in Definition 4.14.Then that if β is the least common multiple of EndMon p (α), then we define α in A + n to be the element such that α = αβ.Then A + (n; p) is the set of all such α and for all α 1 and α 2 in A + n : We also have from Proposition 4.23 the decomposition 7. The semi-simplicial space A n

•
We now build the semi-simplicial space A n • as promised in Section 1.1.
Definition 7.1.Define a semi-simplicial space C n • by, for 0 p (n − 1), setting levels C n p to be the discrete space of equivalence classes A + n / ≈ where the equivalence relation is taken with respect to the submonoid A + n−p−1 , that is, ≈ is the transitive closure of the relation ∼ on A + n given by Face maps are given by

Proof.
We want that if and so the face maps are well defined.
Lemma 7.4.Recall the notation A + (n; n − p − 1), as defined in Section 6.Then the realisation of the map defined levelwise on the bar construction by Proof.This is a direct application of Propositions 4.23 and 5.18 which gives the decomposition and ∂ p k acts as the face map ∂ p k from Definition 7.1 on the C n p factor of each simplex in the homotopy quotient, and as the identity on the other factors.
Diagrammatically, A n • can be drawn as: Lemma 7.6.The factorwise definition of the face maps ∂ p k in Definition 7.5 gives well-defined maps on the homotopy quotients at each level of A n • .
Proof.The set of j-simplices in A + n \ \ C n p is identified with (A + n ) j × C n p and a generic element in this set is given by [a 1 , . . ., a j ][α] n−p−1 , where the a i and α are in A + n .Then the map ∂ p k acts on this simplex as Lemma 7.7.The face maps ∂ p k on A n • defined in Definition 7.5 satisfy the simplicial identities, that is for 0 i < j p: Proof.This follows directly from the fact that the simplicial identities are satisfied for C n • (Lemma 7.3), since the face maps for A n • are defined via the maps for C n • .
We now show that there exist homotopy equivalences A n p BA + n−p−1 for every p 0, as promised in Section 1.1.
and the projection where α and

Then these maps are homotopy equivalences
Proof.From Lemma 7.4,

and this induces
with the homotopy equivalence given by the required map.We then have the following The central equality is due to the fact that the double homotopy quotient is the geometric realisation of a bi-simplicial-set and therefore we can take the realisation in either direction first.The second map in the previous equation is a homotopy equivalence by Lemma 5.16.

We now define the map from the geometric realisation of A n
• to the classifying space BA + n promised in Section 1.1: In Section 8, we will show that φ • is highly connected.
Lemma 7.9.The geometric realisation Proof.The face maps in the bar construction for all p 0. The map φ • is defined levelwise as the projection -connected it follows that φ p is k-connected and in particular it is (k − p)-connected for all p 0. Thus, the geometric realisation φ • is k-connected.

High connectivity
This section is concerned with the proof of the following theorem.
Combining this theorem with Proposition 7.11, it follows that the map φ • is (n − 1)connected as promised in Section1.1.For the remainder of this paper, we will refer to the geometric realisation of the semi-simplicial space as a complex (the geometric realisation is, by definition, a cell complex: note that it is not necessarily a simplicial complex).

Union of chambers argument
There is a specific argument, called a union of chambers argument that is often used to prove high connectivity of a complex.It is closely related to the notion of shellability.
In [8], Davis used a union of chambers argument to prove that the Davis complex Σ W associated to a Coxeter group is contractible.He did this by showing that the Davis complex is an example of a so-called basic construction.Hepworth's high-connectivity results relating to homological stability for Coxeter groups [13] also used such an argument.In [19], Paris used a union of chambers argument to show that the universal cover of an analogue of the Salvetti complex for certain Artin monoids is contractible, proving the K(π, 1) conjecture for finitetype Artin groups.In this chapter, we use a similar union of chambers argument to prove high connectivity.Loosely, the argument consists of breaking the complex up into high-dimensional chambers and considering how connectivity changes as they are glued together to create the complex.While applying the argument in the case of Artin monoids and the complex we have constructed, numerous technical challenges arise, leading to the proof being split into many separate cases.
To prove high connectivity in our setup, we use a union of chambers argument applied to the complex C n • .Recall that C n • has dimension n − 1.We filter the top dimensional simplices by the natural numbers as follows.
Consider EndGen 1 (α).If this is non-empty, then there exists η with length at least 1 in and as before (α ) k.This shows that the facet To complete the proof of point (A) we must show that if a lower dimensional face of We first describe a general form for faces of [[α]] 0 .), and we let a j := σ ij −1+j . . .σ j .That is, the (j − 1)st map in the tuple corresponds to right multiplication by a j .We note here that a j has length i j and ends with the generator σ j , unless i j = 0 in which case a j = e.

Proof of point (A): low-dimensional faces of [[α]]
From now on we assume that the first map in a tuple maps from C n n−1 to C n n−2 , the second map from C n n−2 to C n n−3 and so on.We therefore dispense of the superscripts in the ∂ notation for the face maps when we write these tuples.
Proof.This is a direct consequence of Lemma 5.6.
In general these facets are given by the face map Proof.It is enough to show that ∂ ij +(j−2) can act as the first face map in the tuple (∂ i2 , ∂ i3 , . . ., ∂ ip+1 ) for all j.Recall from Lemma 8.8 that in the tuple i j+1 i j for all j.Using the simplicial identities, the tuple is equivalent to the tuple For the remainder of this section, let α in A + n with (α) = k + 1.The aim of this section is to show that if the (n , then it follows that one of the facets of [[α]] 0 from Lemma 8.9 is also in C n (k).The proof of point (A) will follow.
Proof.Suppose that β and b j are chosen such that p+1 k=2 l k is minimal, and furthermore suppose that (β) < (α) − 1 and p+1 k=2 l k > 0. Then some l k = 0: set j to be minimal such that l j = 0. Then b j = σ lj −1+j . . .σ j = e and

But this is the tuple of face maps (∂
. However, the tuple for βσ lj −1+j has the sum of its corresponding l j less than the original tuple for β.This is a contradiction, as β was chosen to have minimal where either b = e, or (β) = (α) − 1.We recall that this is equivalent to αa = βb in A + (n; p).Let γ := αa = βb, and define u and v in A + p such that αa = γu and βb = γv.
] 0 , and we wish to prove this is contained in a facet of [[α]] 0 (from Lemma 8.9) which is contained in C n (k).We complete the proof of this by splitting into three cases: and since multiplication in the Artin monoid corresponds to adding lengths the conditions of these cases correspond to analogous conditions on the lengths of u and v. Remark 8.13.Note that if p+1 k=2 l k = 0, then b = e, and since (β) < (α) it follows we are therefore in case (i): (βb) < (αa).
We prove the three cases one by one in the following subsections.This involves some technical lemmas, and in particular computation of least common multiples of strings of words.We therefore include these technical lemmas in a separate section and refer to them as required.

Proof of point (A): preliminary lemmas
Recall from Definition 8.7 that a face of [[α]] 0 is obtained by applying a series of face maps to [[α]] 0 .We denote the series of face maps by a tuple ( ), and we let a j = σ ij −1+j . . .σ j and when i j = 0 let a j = e.That is, the (j − 1)st map in the tuple corresponds to right multiplication by a j .We let a = a 2 . . .a p+1 .Recall also that if Recall from Definition 4.9 that for α and β two words in A + , we denote the least common multiple of α and β (if it exists) by Δ(α, β).Lemma 8.14.For all k > j, the generators σ i satisfy Proof.We proceed by induction on k − j.For the base case let k − j = 1, that is, k = j + 1.Then the left-hand side of the above equation evaluates to σ j+1 σ j σ j+1 and the right-hand side evaluates to σ j σ j+1 σ j .These are equal by the Artin relations.For the inductive hypothesis, we assume the Lemma is true for k − j < r, and we prove for k − j = r, that is, k = j + r.We manipulate the left-hand side of the equation, and show equality to the right-hand side: where the final equality applies the inductive hypothesis.Proof.The proof is the same for both the a j and b j case, so we prove it for the a j case.We must show: n is a common multiple of a j+1 and σ j , then a j+1 σ j a j+1 R x. Recall a j+1 := σ ij+1+j . . .σ j+1 .Without loss of generality, we relabel j = 1 and i j+1 + j = k.Then a j+1 = σ k . . .σ 2 and σ j = σ 1 .
To prove (a) note that a j+1 R a j+1 σ j a j+1 by observation, and also by Lemma 8.14, so σ 1 = σ j R a j+1 σ j a j+1 .
When j = 1, the expression satisfies that is, the same equality holds, setting a 1 := σ 1 .The analogous statements hold for the b j . Proof.
For the inductive hypothesis we assume true for (a j+1 ) r − 1 and prove for (a j+1 ) = r − 1, that is, k = r.Recall from Lemma 8.14 that where the final equality applies the inductive hypothesis.Then σ r commutes with (σ r−2 . . .σ l+1 )a j since (a j ) (a j+1 ).This gives the following: Since i j i j−1 it follows that l − 1 is the maximal index of a generator appearing in a j−1 and hence in the string a 2 . . .a j−1 .Therefore, âj letterwise commutes with a 2 . . .a j−1 since the indices of the generators in each word pairwise differ by at least two.
Since the b j have the same form as the a j , with difference only in word length, the analogous statements hold for the b j .
Recall the definition of a j and b j for 2 j p + 1, from Definitions 8.7 and 8.10, respectively.1460244x, 2020, 3, Downloaded from https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/plms.12335 by University Of Glasgow, Wiley Online Library on [02/02/2023].See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions)on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License Definition 8.17.For 2 j p + 1 define c j as follows Proof.We prove that: n is a common multiple of a and b, then c R x.To prove (a), we show that c = a a: the proof that c = b b is symmetric.It follows from the definitions that c j = a j a j .The smallest generator index in a j is (i j + j) and the largest generator index in a 2 . . .a j−1 is (i j−1 + (j − 1) − 1).The elements a j and a 2 . . .a j−1 letterwise commute, since i j i j−1 so We prove in the next three Lemmas that in the case EndGen p (αa) = ∅ we are done.
Proof.Consider τ in EndGen 0 (αa).Then since the generators S 0 of A + 0 commute with σ 2 , . . ., σ n it follows that τ letterwise commutes (Definition 4.26) with a, because a = a 2 . . .a p+1 only contains generators in the set of {σ 2 , . . .σ n }.Since τ and a are both in EndMon n (αa) and they letterwise commute, it follows from Lemma 4.28 that τ is in EndMon n (α), that is, some Here, the final equality is due to α a 2 τ = α a 2 where the reduction is taken with respect to A + 1 (from Lemma 4.19).The penultimate equality is due to the fact τ and a 2 letterwise commute.
and this completes the proof.
Lemma 8.20.Suppose a j = e, then the words a j and a j+1 as in Definition 8.7 satisfy a j+1 σ j = āj a j , for some āj in A + n with (ā j ) 1. Furthermore, āj letterwise commutes with a 2 . . .a j−1 .Regardless of whether or not a j = e, a j+1 σ j corresponds to the face map ∂ n−j+1 ij+1+1 .The analogous results hold for the b j .

Proof of point (B)
Recall point B: Suppose that u = e.Then by Proposition 8.22, it follows that a facet of [

Proof of Theorem A
This section proves the required results on the differentials of the spectral sequence introduced in Subsection 1.1, before putting together the results of the previous sections and running the spectral sequence argument to complete the proof of Theorem A.

Results on face and stabilisation maps
Recall the definition of the face maps of A n • from Definition 7.5: is given by right multiplication of the central term in the double homotopy quotient by ( The second map ι p is the map  We note that the dotted map is precisely the map which defines the natural inclusion BA + n−p−1 → BA + n−p under the identification of * / / A + r with BA + r for all r.The natural inclusion is in turn induced by the stabilisation map A + r s → A + r+1 and so we denote it s * .This gives that the left-hand square commutes up to homotopy.

Spectral sequence argument
In this section, we run a first quadrant spectral sequence for filtration of A n • , see, for example, Randal-Williams [20,2 (sSS)].Recall the points we proved regarding A n • : (1) there exist homotopy equivalences A n p BA + n−p−1 for p 0; (2) there is an (n − 1) connected map φ • from the geometric realisation of A n • to the classifying space BA + n : A n
The first quadrant spectral sequence of the simplicial filtration of A n • satisfies By point (1), the left-hand side is given by . The first page of the spectral sequence is depicted in Figure 1.By point (2), φ • induces an isomorphism Therefore, the alternating sum of face maps in the differential d 1 will cancel out to give the zero map when there are an even number of terms, and will give the stabilisation map when there are an odd number of terms, that is,   even,l → E 1 odd,l odd number of terms, so equals the stabilisation map s * , d 1 : E 1 odd,l → E 1 even,l even number of terms, so equals the zero map 0. This gives the E 1 page shown in Figure 2.
We proceed by induction on n, for the sequence of monoids A + n , and assume that homological stability holds for previous groups in the sequence.Inductive hypothesis: The map induced on homology by the stabilisation map −→ H i (BA + k ) is an isomorphism for k > 2i and is a surjection for k = 2i whenever k < n.
Here, we note that Theorem A holds for the base case n = 1, since we have to check H 0 (BA + 0 ) → H 0 (BA + 1 ) is a surjection, which is true since BA + n is connected for all n.
Lemma 9.6.Under the inductive hypothesis, the E 0,l terms stabilise on the E 1 page for 2l < n, that is, E 1 0,l = E ∞ 0,l when 2l < n.In particular, the d 1 differential does not alter these groups, and all possible sources of differentials mapping to E 0,l for 2l < n are trivial from the E 2 page.
Proof.The d 1 differentials are given by either the zero map or the stabilisation map as shown in Figure 2. The d 1 differentials are given by the zero map, and the E 1 −1,l terms are zero, since this is a first quadrant spectral sequence.Therefore, the E 2 0,l terms are equal to the E 1 0,l terms.To show that the sources of all other differentials to E 0,l for 2l < n are zero, we invoke the inductive hypothesis.This implies that on the E 1 page in the interior of the triangle of height n 2 and base n, the stabilisation maps, or d 1 differentials satisfy the inductive hypothesis.The resulting maps are shown in Figure 3, for the cases n odd and n even.Since the d 1 differentials going from the odd to the even columns are zero it follows that many groups in the interior of the triangle are zero on the E 2 page.This is shown in detail in Figure 4 for the   cases n odd and n even.These groups include all the sources of differentials to E 0,l for 2l < n, hence E 2 0,l = E ∞ 0,l for 2l < n.
We are now in a position to prove Theorem A.

Definition 2 . 4 .
Define the length function on a Coxeter system (W, S) : W → N to be the function which maps w in W to the minimum word length required to express w in terms of the generators.Definition 2.5.Define π(a, b; k) to be the word of length k, given by the alternating product of a and b, that is, π(a, b; k) = length k abab . . . .

zτ 1 =
vβτ 1 and so by cancellation of τ 1 , it follows that z = vβ.Reinserting this in the previous equation gives x = zγ = vβγ and so βγ = γβ is in EndMon M (x) as required.Lemma 4.28.If words α, a and b in A + are such that b R αa and a and b letterwise commute, then it follows that b R α. Proof.An equivalent way of writing m R n for m, n in A + is m ∈ EndMon A (n) where the end set is taken with respect to the full monoid A + .Since a and b are both in EndMon A (αa) it follows that Δ(a, b) is in EndMon A (αa), from Lemma 4.25.Since a and b letterwise commute, Δ(a, b) = ab = ba from Lemma 4.27.Therefore, ba is in EndMon A (αa) and, by cancellation of a, b is in EndMon A (α) as required.

4. 4 .
Relation to the K(π, 1) conjecture In 2002, Dobrinskaya published a paper relating the classifying space of the Artin monoid BA + W to the K(π, 1) conjecture.This was later translated into English as Configuration Spaces of Labelled Particles and Finite Eilenberg -MacLane Complexes [10].The main result of the paper was the following.

1460244x, 2020, 3 ,
Downloaded from https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/plms.12335 by University Of Glasgow, Wiley Online Library on [02/02/2023].See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions)on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License Definition 5.1 [12, 1.1] a natural transformation of functors, and in particular has components f n : X n → Y n .Simplicial objects in C form a category denoted sC, and semi-simplicial objects a category denoted ssC.When C equals Set a (semi-)simplicial object is called a (semi-)simplicial set and when C equals Top it is called a (semi-)simplicial space.Remark 5.3.A semi-simplicial object in a category C is equivalent to the following data.(a) An object X p in C, for p 0. (b) Morphisms in C ∂ p i : X p → X p−1 for 0 i p and all p 0 called face maps, which satisfy the following simplicial identities

1460244x, 2020, 3 ,
Downloaded from https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/plms.12335 by University Of Glasgow, Wiley Online Library on [02/02/2023].See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions)on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License That is, morphisms send the jth vertex of the simplex |Δ n | to the φ(j)th vertex of |Δ m | and extend linearly.Under this viewpoint the map D i * sends |Δ n | to the ith face of |Δ n+1 | and the map S i * collapses together the ith and (i + 1)st vertices of |Δ n+1 | to give a map to |Δ n |.A tuple (∂ p−1 i1 , ∂ p−2 i2 , . . ., ∂ p−k i k ) denotes the application of several face maps in a row, where ∂ p−1 i1

Definition 5 . 13 .
Consider the bar construction B( * , M, Y ) for Y a space with an action of the monoid M on the left and * a point on which M acts trivially.Define this to be the homotopy quotient of Y over M (or M under Y ) and denote it B( * , M, Y ) =: M \ \ Y .This is the geometric realisation of the semi-simplicial space B • ( * , M, Y ) given by the situation of a monoid M acting on a space Y on the right we define the homotopy quotient to be B(Y, M, * ) =: Y / / M. Example 5.14.Consider the bar construction B( * , N, M), for N a submonoid of M acting on M on the left, by left multiplication, and * a point, on which N necessarily acts trivially.Then the homotopy quotient of M over N is M on the right by right multiplication.Then the associated homotopy quotient is the geometric realisation B(M, N, * ) = M / / N. Lemma 5.15.The homotopy quotient of a group G or monoid M under a point * is a model for the classifying space of the group or monoid, that is, BG G \ \ * * / / G and BM M \ \ * * / / M.
as in Definition 5.13) be denoted by the tuple (p 1 , . . ., p k , a), with p i and a in A + .There is a right action of A + on the k-cells given by (p 1 , . . ., p k , a) • μ = (p 1 , . . ., p k , aμ).Define the set of elementary k-cells to be those with tuple (p 1 , . . ., p k , e), where e is the identity element in the monoid, and denote this cell D(p 1 , . . ., p k ).Then 1460244x, 2020, 3, Downloaded from https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/plms.12335 by University Of Glasgow, Wiley Online Library on [02/02/2023].See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions)on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License every k-cell is uniquely determined by an elementary k-cell and an element a in A + , since (p 1 , . . ., p k , a) = D(p 1 , . . ., p k ) • a.The isomorphism of Proposition 4.23 shows that A + = A + (M ) × M + and we let a = ām under this decomposition.Then we get the following description for k-cells:

For example, ∂ p 0
acts on the equivalence class representative by right multiplication by e, and ∂ p p acts by right multiplication by σ n . . .σ n−p−1 .The motivation for this choice of face maps follows Hepworth, as discussed in [13, Example 35].Lemma 7.2.The face maps of Definition 7.1 are well defined.
where the bar is taken with respect to A + n−p−1 .Set α = γ (recall the definition of α from Remark 4.17).It follows that there exist a and b in A + n−p−1 such that α = γa and η = γb.Then since a and b only contain letters in A + n−p−1 and all of these letters commute with (σ n−p+k σ n−p+k−1 . . .σ n−p+1 ) it follows that a and b letterwise commute with the face map.Taking equivalence classes with respect to A p and since the multiplication by (σ n−p+k σ n−p+k−1 . . .σ n−p+1 ) is on the right it follows that ∂ p k commutes with all face maps of the bar construction B • ( * , A + n , C n p ) for each k.Therefore, the definition of ∂ p k on the simplicial level induces a map on the homotopy quotient A + n \ \ C n p .

Define a = a 2 Lemma 8 . 18 .
. . .a p+1 and b = b 2 . . .b p+1 .With c, a and b as in Definition 8.17 and a and b as defined in Definition 8.12, we have c = Δ(a, b) and in particular c = a a = b b.
we are left with the case that u = v = e, giving αa = γ = βb and since (α) = (β) it follows that (a) = (b).Since α = β it follows a = b.Recall the definition of c, a and b from Definition 8.17.From Lemma 8.18, c = Δ(a, b) and c = a a = b b.Since (a) = (b), then (a ) = (b ).Suppose a = e, then (a ) = (b ) gives b = e and hence c = a = b.But a = b so it follows that a = e and in particular (a ) 1. From Lemma 4.27, since a and b are in EndMon n (αa) it follows that Δ(a, b) = c is in EndMon n (αa), so αa = α c = α (a a) for some α in A + n .By cancellation of a we have α = α a and (α ) < (α).Then [[αa]] p = [[(α a )a]] p = [[α c]] p and [[α c]] p is in C n (k) since c represents a series of face maps originating at [[α ]] 0 , with each face map given by the map corresponding to right multiplication by c j , which is either the face map corresponding to a j or b j .1460244x, 2020, 3, Downloaded from https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/plms.12335 by University Of Glasgow, Wiley Online Library on [02/02/2023].See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions)on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License This completes the proof of point (B), and hence by Proposition 8.4, it follows that C n • is (n − 2) connected.

1 A
where ∂ p k is induced by the face maps of C n• , which we recall are a composite of right multiplication of the representative for the equivalence class inC n p = A + (n; n − p − 1) by (σ n−p+k σ n−p+k−1 . . .σ n−p+1), before the inclusion to the equivalence class in C n p−1 .Recall from Lemma 7.4 that for each 0 p n − 1 there is a homotopy equivalenceA + n / / A + n−p−+ (n; n − p − 1) = C n p ,given by the map defined levelwise on the bar construction byB k (A + n , A + n−p−1 , * ) → A + (n; n − p − 1)α[m 1 , . . ., m k ] → α, where α ∈ A + n , m i ∈ A + n−p−1 for all i and α = αβ for α ∈ A + (n; n − p − 1) and β ∈ A + n−p−1 .Definition 9.1.Define the map

Figure 1 .
Figure 1.The E 1 page of the spectral sequence, with the groups identified.

H
k+l ( A n • ) ∼ = H k+l (BA + n ) when (k + l) < n − 1 and a surjection H k+l ( A n • ) H k+l (BA + n ) when (k + l) = n − 1.The differentiald 1 is given by an alternating sum of face maps in A n • .By Corollary 9.4, the face maps are all homotopic to each other and by Lemma 9.5 they are all homotopic to the stabilisation map s * , via A n p BA + n−p−1 .

Figure 2 .
Figure 2. The E 1 page of the spectral sequence, with groups and d 1 differentials identified.

Figure 3 .
Figure 3.The E 1 page of the spectral sequence, with possible non-zero groups represented as circles and the inductive hypothesis applied to the d 1 differentials.

Figure 4 (
Figure4(colour online).The E 2 page of the spectral sequence, under the inductive hypothesis.To the left of the red line, all groups are zero except at positions E 2 0,l for 2l < n -these are highlighted in blue.

2 .
Coxeter groups and Artin groups 2.1.Coxeter groups This section follows The Geometry and Topology of Coxeter Groups by Davis [8].Definition 2.1.A Coxeter matrix on a finite set of generators S is a symmetric matrix M indexed by elements of S, that is, with entries m(s, t) in N ∪ ∞ for {s, t} in S × S.This matrix must satisfy: • m(s, s) = 1 for all s in S; • m(s, t) = m(t, s) must be either greater than 1, or ∞, when s = t.Definition 2.2.A Coxeter matrix M with generating set S has an associated Coxeter group W , with presentation 1460244x, 2020, 3, Downloaded from https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/plms.12335 by University Of Glasgow, Wiley Online Library on [02/02/2023].See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions)on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License When the diagram D W is connected, W is called an irreducible Coxeter group.The disjoint union of two diagrams gives the product of their corresponding Coxeter groups.Theorem 2.8 (Classification of finite Coxeter groups [7]).A Coxeter group is finite if and only if it is a (direct) product of finitely many finite irreducible Coxeter groups.
1460244x, 2020, 3, Downloaded from https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/plms.12335 by University Of Glasgow, Wiley Online Library on [02/02/2023].See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions)on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License Example 2.14.When all possible edges in the Coxeter diagram D W are present and labelled with ∞ the corresponding Artin group is the free group on |S| generators.The group has presentation A Example 2.15.When there are no edges in the Coxeter diagram D W , the corresponding Artin group is the free abelian group on |S| generators.The group has presentation W = σ s for s ∈ S .
2020, 3, Downloaded from https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/plms.12335 by University Of Glasgow, Wiley Online Library on [02/02/2023].See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions)on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License a generic element is given by [a 1 , . . ., a j ]a[a 1 , . . ., a k ], where a i and a are in A + n and a i are in A + n−p−1 .The map dp k acts on this simplex as dp k , 2020, 3, Downloaded from https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/plms.12335 by University Of Glasgow, Wiley Online Library on [02/02/2023].See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions)onWileyOnlineLibraryforrules of use; OA articles are governed by the applicable Creative Commons License Lemma 9.2.The map dp k in Definition 9.1 gives a well-defined map on the double homotopy quotient A + n \ \ A + n / / A + n−p−1 .Proof.Since (σ n−p+k σ n−p+k−1 ...σ n−p+1 ) letterwise commutes with every word in A + n−p−1 , it follows that dp k commutes with all face maps of the bi-semi-simplicial space A + n\ \ A + n / / A + n−p−1 .Therefore, the map on the central term of each simplex gives a map on the whole bi-semi-simplicial space, and hence its geometric realisation: the double homotopy quotientA + n \ \ A + n / / A + n−p−1 .Proof.Recall from Lemma 7.8 that the horizontal homotopy equivalence is given by the levelwise maps on (j, k)-simplices of A + + n−p−1 .Diagram chasing using the definition of d p k in Definition 9.1 gives that levelwise these maps commute, and so taking homotopy quotients and the corresponding maps induced by these levelwise maps yields the required result.Lemma 9.4.The face maps ∂ p k of A n • are all homotopic to the zeroth face map ∂ p 0 .Proof.The map dpk restricted to A + n \ \ A + n is A + n−p−1 -equivariant, and the same holds for the identity map id A + n \ \A + n .Applying Proposition 5.19 to these two maps therefore gives an A + n−p−1 -equivariant homotopy between them.It follows that they induce homotopic maps dp .Applying the inclusion ι p to both maps and the homotopy between them yields a homotopy h k from d p k to ι p .However, ι p is precisely the map d p 0 , and thus h k is a homotopy from d p k to d p 0 for all k.Then the image of h k under the homotopy equivalence in Lemma 9.3 yields a homotopy from ∂ p k to the zeroth face map ∂ k 0 , as required.Lemma 9.5.The following diagram commutes up to homotopy.ishomotopyequivalenttothe map s * : BA + n−p−1 → BA + n−p induced by the stabilisation map s: A + n−p−1 → A + n−p .Proof.From the proof of Lemma 9.4, the right-hand square commutes up to homotopy.From Lemma 7.8, the map from the centre to the left is given on the (j, k)-simplices of the geometric realisation byf p (j,k) : (A + n \ \ A + n / / A + n−p−1 ) (j,k) → ( * / / A + n−p−1 ) k [a 1 , . ..,a j ]a[a 1 , . . ., a k ] → * [a 1 , . . ., a k ], where a and a i are in A + n and a i is in A + n−p−1 .The map d p 0 is the map d p + n and the a i are in A + n−p−1 , hence a i is in A + n−p .Applying this map before the homotopy equivalence to the classifying space gives and on a (j, k) simplex this map is given by 1460244x, 2020, 3, Downloaded from https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/plms.12335 by University Of Glasgow, Wiley Online Library on [02/02/2023].See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions)on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License Note that dp 0 is the identity map, and therefore d p 0 = ι p .1460244xn \ \ A + n / / A + n−p−1 : (A + n \ \ A + n / / A + n−p−1 ) (j,k) → (A + n \ \ C n p ) j [a 1 , ..., a j ]α[a 1 , ..., a k ] → [a 1 , ..., a j ]α,where α and a i are in A + n , the a i are in A + n−p−1 , and α = αβ for α in A + (n; n − p − 1) andβ in A 0 : A + n \ \ A + n / / A + n−p−1 → A + n \ \ A + n / / A + n−p induced by the inclusion A + n−p−1 → A + n−p .Restricting this map to (j, k)-simplices of the double homotopy quotient gives(d p 0 ) (j,k) : (A + n \ \ A + n / / A + n−p−1 ) (j,k) → (A + n \ \ A + n / / A + n−p ) (j,k)[a 1 , . .., a j ]a[a 1 , . .., a k ] → [a 1 , . .., a j ]a[a 1 , . .., a k ],where a and a i are in A Theorem 9.7.The sequence of monoids A + n satisfies homological stability, that is H , 2020, 3, Downloaded from https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/plms.12335 by University Of Glasgow, Wiley Online Library on [02/02/2023].See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions)on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License i (BA + n−1 ) ∼ = H i (BA + n )when 2i < n, and the mapH i (BA + n−1 ) → H i (BA + n ) is surjective when 2i = n.1460244x