Weighted sheaves and homology of Artin groups

TL;DR

This paper develops a theory of weighted sheaves over posets to analyze the homology of Artin groups, providing explicit formulas and computations for various cases, including new affine types.

Contribution

It introduces a novel approach using weighted sheaves and discrete Morse theory to compute Artin group homology, extending previous methods to affine cases.

Findings

Established relations between braid group homology and graph independence complexes.

Derived explicit formulas for Morse complex homology using incidence matrices.

Performed new homology computations for affine Artin groups  C_n, complementing existing results.

Abstract

In this paper we expand the theory of weighted sheaves over posets, and use it to study the local homology of Artin groups. First, we use such theory to relate the homology of classical braid groups with the homology of certain independence complexes of graphs. Then, in the context of discrete Morse theory on weighted sheaves, we introduce a particular class of acyclic matchings. Explicit formulas for the homology of the corresponding Morse complexes are given, in terms of the ranks of the associated incidence matrices. We use such method to perform explicit computations for the new affine case $\tilde C_n$, as well as for the cases $A_n$, $B_n$ and $\tilde{A}_n$ (which were already done before by different methods).

Tables

Table 1. Table 1. Matching in the case A n subscript 𝐴 𝑛 A_{n} with n = 5 𝑛 5 n=5 , d = 3 𝑑 3 d=3 , f = 1 𝑓 1 f=1 . The last columns indicates the case where simplices occur. When case (e) is reached, the arrow “ ↝ ↝ \rightsquigarrow ” indicates how the recursion continues (after K 5 , 1 subscript 𝐾 5 1 K_{5,1} , the recursion involves K 3 , 0 subscript 𝐾 3 0 K_{3,0} , K 2 , 1 subscript 𝐾 2 1 K_{2,1} and K 0 , 0 subscript 𝐾 0 0 K_{0,0} ).

Simplices			$v_{\phi }\left(\sigma \right)$	Step
	$⟶$		2	(a)
	$⟶$		1	(a)
	$⟶$		1	(a)
	$⟶$		1	(a)
	$⟶$		1	(e) $\rightsquigarrow$ (a)
	$⟶$		0	(e) $\rightsquigarrow$ (c)/(d)
	$⟶$		0	(e) $\rightsquigarrow$ (c)/(d)
		(critical)	1	(e) $\rightsquigarrow$ (e) $\rightsquigarrow$ (a)
		(critical)	0	(e) $\rightsquigarrow$ (e) $\rightsquigarrow$ (e) $\rightsquigarrow$ (b)

Table 2. Table 2. Description of the critical simplices for A n subscript 𝐴 𝑛 A_{n} .

Case		Simplices	$\left|\sigma \right|$	$v_{\phi }\left(\sigma \right)$
	$n=kd+f$ (Fig. 4)	${\left(1^f{01}^{d-2-f}0\right)}^{k-1}{1}^f{01}^{d-1}$	$n-2k+1$	1
$n>f$		${\left(1^f{01}^{d-2-f}0\right)}^k{1}^f$	$n-2k$	0
$f\le d-2$	$n=kd-1$ (Fig. 5)	${\left(1^f{01}^{d-2-f}0\right)}^{k-1}{1}^{d-1}$	$n-2k+2$	1
		${\left(1^f{01}^{d-2-f}0\right)}^{k-1}{1}^f{01}^{d-2-f}$	$n-2k+1$	0
$n=f$		$1^f$	$n$	1 or 0

Table 3. Table 3. Matching in the case B n subscript 𝐵 𝑛 B_{n} with n = 4 𝑛 4 n=4 and d = 4 𝑑 4 d=4 .

Simplices			$v_{\phi }\left(\sigma \right)$	$q$	Step
	$⟶$		1	1	(a)/(b)
	$⟶$		0	0	(a)/(b)
	$⟶$		0	0	(a)/(b)
	$⟶$		0	0	(a)/(b)
	$⟶$		0	0	(a)/(b)
		(critical)	2	2	(b)
		(critical)	1	1	(c)
		(critical)	1	0	(c)
	$⟶$		0	0	(c)
		(critical)	0	0	(c)

Table 4. Table 4. Description of the critical simplices for B n subscript 𝐵 𝑛 B_{n} , where d 𝑑 d is even and n = k ​ d 2 𝑛 𝑘 𝑑 2 n=k\frac{d}{2} .

Simplices	$\left|\sigma \right|$	$v_{\phi }\left(\sigma \right)$
${\sigma }_q=1^{q\frac{d}{2}}{\left({01}^{\frac{d}{2}-1}\right)}^{k-q-2}{01}^{d-1}$	$n-k+q+1$	$q+1$	$0\le q\le k-2$
${\sigma }_q^{\prime }=1^{q\frac{d}{2}}{\left({01}^{\frac{d}{2}-1}\right)}^{k-q}$	$n-k+q$	$q$	$0\le q\le k$

Table 5. Table 5. Critical simplices for B n subscript 𝐵 𝑛 B_{n} by dimension, where d 𝑑 d is even and n = k ​ d 2 𝑛 𝑘 𝑑 2 n=k\frac{d}{2} .

$\left|\sigma \right|$	$v_{\phi }\left(\sigma \right)$	Simplices
$n$	$k$	${\sigma }_k^{\prime }$
$n-1$	$k-1$	${\sigma }_{k-2}$, ${\sigma }_{k-1}^{\prime }$
$n-2$	$k-2$	${\sigma }_{k-3}$, ${\sigma }_{k-2}^{\prime }$
⋮	⋮	⋮
$n-k+1$	$1$	${\sigma }_0$, ${\sigma }_1^{\prime }$
$n-k$	$0$	${\sigma }_0^{\prime }$

Table 6. Table 6. The only alternating path from σ q ′ superscript subscript 𝜎 𝑞 ′ \sigma_{q}^{\prime} to σ q − 2 subscript 𝜎 𝑞 2 \sigma_{q-2} for n = 10 𝑛 10 n=10 , d = 4 𝑑 4 d=4 , k = 5 𝑘 5 k=5 and q = 2 𝑞 2 q=2 .

Simplices			$v_{\phi }\left(\sigma \right)$
${\sigma }_2^{\prime }$	$=$		2
${\tau }_1$	$=$		1
${\rho }_1$	$=$		1
${\tau }_2$	$=$		1
${\rho }_2$	$=$		1
${\tau }_3$	$=$		1
${\rho }_3$	$=$		1
${\sigma }_0$	$=$		1

Table 7. Table 7. Description of the critical simplices for A ~ n subscript ~ 𝐴 𝑛 \tilde{A}_{n} . Below, the binary string notation is used.

Case	Simplices	$\left|\sigma \right|$	$v_{\phi }\left(\sigma \right)$
$n=kd+r$	${\tau }_q$	$n-2\left(k-q\right)+1$	$q+1$	$0\le q\le k-1$
$\not\equiv -1\left(\text{mod}d\right)$	${\tau }_q^{\prime }$	$n-2\left(k-q\right)$	$q$	$0\le q\le k$
$n=kd-1$	${\sigma }_{q,r}$	$n-2\left(k-q\right)+2$	$q+1$	$0\le q\le k-2$, $0\le r\le d-2$
	${\sigma }_{q,r}^{\prime }$	$n-2\left(k-q\right)+1$	$q$	$0\le q\le k-1$, $0\le r\le d-2$
	$\overline{\sigma }=1^{n}0$	$n$	$k$

Table 8. Table 8. Description of the critical simplices for C ~ n subscript ~ 𝐶 𝑛 \tilde{C}_{n} . When d 𝑑 d is even, set n = k ​ d 2 + r 𝑛 𝑘 𝑑 2 𝑟 n=k\frac{d}{2}+r .

Case	Simplices	$\left|\sigma \right|$	$v_{\phi }\left(\sigma \right)$
$d$ odd	$\overline{\sigma }=1^{n}0$	$n$	$1$ or $0$
$d$ even	${\sigma }_{q_1,q_2}$	$n-k+q_1+q_2+1$	$q_1+q_2+1$	$0\le q_1+q_2\le k-2$
	${\sigma }_{q_1,q_2}^{\prime }$	$n-k+q_1+q_2$	$q_1+q_2$	$0\le q_1+q_2\le k$

Table 9. Table 9. Critical simplices for C ~ n subscript ~ 𝐶 𝑛 \tilde{C}_{n} in dimension m = n − k + l 𝑚 𝑛 𝑘 𝑙 m=n-k+l ( 0 ≤ l ≤ k 0 𝑙 𝑘 0\leq l\leq k ), where d 𝑑 d is even and n = k ​ d 2 + r 𝑛 𝑘 𝑑 2 𝑟 n=k\frac{d}{2}+r . For l = k 𝑙 𝑘 l=k only the second column occurs.

Simplices
${\sigma }_{l-1,0}$	${\sigma }_{l,0}^{\prime }$
${\sigma }_{l-2,1}$	${\sigma }_{l-1,1}^{\prime }$
⋮	⋮
${\sigma }_{1,l-2}$	${\sigma }_{2,l-2}^{\prime }$
${\sigma }_{0,l-1}$	${\sigma }_{1,l-1}^{\prime }$
	${\sigma }_{0,l}^{\prime }$

Equations

$$H_{\ast }({\mathrm{Br}}_{n+1};R{)}_{{\phi }_d}\cong {\tilde{H}}_{\ast -d+1}\left({\mathrm{Ind}}_{d-2}(A_{n-d});\frac{R}{({\phi }_d)}\right).$$

$$H_m({\mathbf{G}}_{\mathbf{W}};R{)}_{{\phi }_d}\cong \{\begin{array}{ll}{\left(R/({\phi }_d)\right)}^{\oplus m+k-n+1}& \text{if n−k≤m≤n−1},\\ 0& \text{otherwise},\end{array}$$

$$\{A_x\mid x\in P\}$$

$$\{{\rho }_{x,y}:A_y\to A_x\mid x⪯y\}$$

$${\rho }_{x,x}={\mathrm{id}}_{A_x};$$

$$x⪯y⪯z\Rightarrow {\rho }_{x,z}={\rho }_{x,y}{\rho }_{y,z}.$$

$$p_1\mid p_2\iff (p_1)\supseteq (p_2)$$

$$\{R/(w(x))\mid x\in P\}$$

$$\{i_{x,y}:R/(w(y))\to R/(w(x))\mid x⪯y\}$$

$$\sigma ⪯\tau \iff \sigma \subseteq \tau .$$

$$\sigma ⪯\tau \Rightarrow w(\sigma )\mid w(\tau ).$$

$$C_k^0(K;R)=\underset{\begin{array}{c}\sigma \in K\\ |\sigma |=k\end{array}}{⨁}R{e}_{\sigma }^0$$

$${\partial }^0(e_{\sigma }^0)=\sum _{|\tau |=k-1}[\sigma :\tau ]e_{\tau }^0,$$

$$L_k=\underset{|\sigma |=k}{⨁}\frac{R}{(w(\sigma ))}{\bar{e}}_{\sigma },$$

$$\partial (a_{\sigma }{\bar{e}}_{\sigma })=\sum _{\tau \prec \sigma }[\sigma :\tau ]i_{\tau ,\sigma }(a_{\sigma }){\bar{e}}_{\tau }.$$

$$w_{\phi }(\sigma )={\phi }^{v_{\phi }(\sigma )},$$

$$v_{\phi }(\sigma )=\text{maximal r such that φr divides w(σ)}.$$

$$(L_k{)}_{\phi }=\underset{|\sigma |=k}{⨁}\frac{R}{({\phi }^{v_{\phi }(\sigma )})}{\bar{e}}_{\sigma }.$$

$$F^s(L_{\ast }{)}_{\phi }=\underset{v_{\phi }(\sigma )\le s}{⨁}\frac{R}{({\phi }^{v_{\phi }(\sigma )})}{\bar{e}}_{\sigma }.$$

$$K_{\phi ,s}=\{\sigma \in K\mid v_{\phi }(\sigma )\le s\}.$$

$$(K_{\phi ,s},R,w_{\phi }{|}_{K_{\phi ,s}}).$$

$$E_{p,q}^r\Rightarrow H_{\ast }((L_{\ast }{)}_{\phi })$$

$$E_{p,q}^1=H_{p+q}(F^p/F^{p-1})\cong H_{p+q}(K_{\phi ,p},K_{\phi ,p-1};R/({\phi }^p))$$

$$y_0⊳x_1⊲y_1⊳x_2⊲y_2⊳\cdots ⊳x_m⊲y_m(⊳x_{m+1})$$

$$(x,y)\in \mathcal{M}\Rightarrow (w(x))=(w(y))$$

$$L_{\ast }^{\mathcal{M}}=\underset{\sigma \text{ critical}}{⨁}\frac{R}{(w(\sigma ))}{\bar{e}}_{\sigma }$$

$${\partial }^{\mathcal{M}}({\bar{e}}_{\sigma })=\sum _{\begin{array}{c}\tau \text{ critical}\\ |\tau |=|\sigma |-1\end{array}}[\sigma :\tau {]}^{\mathcal{M}}{\bar{e}}_{\tau },$$

$$\sigma ⊳{\tau }_1⊲{\sigma }_1⊳{\tau }_2⊲{\sigma }_2⊳\cdots ⊳{\tau }_m⊲{\sigma }_m⊳\tau$$

$$(-1{)}^m[\sigma :{\tau }_1][{\sigma }_1:{\tau }_1][{\sigma }_1:{\tau }_2][{\sigma }_2:{\tau }_2]\cdots [{\sigma }_m:{\tau }_m][{\sigma }_m:\tau ].$$

$$H_{\ast }(L_{\ast },\partial )\cong H_{\ast }(L_{\ast }^{\mathcal{M}},{\partial }^{\mathcal{M}})$$

Full text

The final publication in Algebraic & Geometric Topology is available at http://doi.org/10.2140/agt.2018.18.3943.

1. 1 Introduction

2. 2 Preliminaries

3. 2.1 Weighted sheaves over posets

4. 2.2 Decomposition and filtration

5. 2.3 Discrete Morse theory for weighted complexes

6. 2.4 Weighted sheaves for Artin groups

7. 3 Homology of braid groups and independence complexes

8. 4 Precise matchings

9. 5 Precise matchings for Artin groups

10. 5.1 Case $A_{n}$

11. 5.2 Case $B_{n}$

12. 5.3 Case $\tilde{A}_{n}$

13. 5.4 Case $\tilde{C}_{n}$

14. Concluding remarks

1. Introduction

The topological theory of Artin groups started in the 60’s with classical braid groups [FN62, Arn69, Arn70], and was broadened to general Artin groups in the 70’s [Fuc70, BS72, Del72, Bri73, Coh76, Wei78, Gor78]. It has received much attention in connection with problems in singularity theory, homotopy theory, hyperplane arrangements, and combinatorics (see [Par14] for some references), up to recent applications of some special cases to conjectures about hyperbolic groups [AGM13].

Thanks to a good topological model for the $K(\pi,1)$ space of Artin groups [Sal87, Sal94], a combinatorial free resolution for these groups was found [DCS96], and several (co)homology computations were carried out [DCPSS99, DCPS01, CS04, Cal05, Cal06, CMS08a, CMS08b, CMS10] (equivalent resolutions also appeared in [Squ94] and [Ozo17, Pao17a]). The computational methods used in these papers are essentially based on filtering the algebraic complex and using the associated spectral sequence.

In [SV13] a more combinatorial method of calculation was introduced, based on the application of discrete Morse theory to a particular class of sheaves over posets (called weighted sheaves). It is interesting to notice that similar ideas were considered later in [CGN16], even if there the authors were mainly interested in computational aspects.

In this paper we expand the theory of weighted sheaves. First, we observe a surprising relationship between some twisted (co)homology of the classical braid groups and the (co)homology of certain independence complexes, which in their simplest form have already been studied in a combinatorial context (see for example [BK07, Koz07, Eng09]). This relationship is obtained in an indirect way, after localization. The exact formula we are going to prove is the following (it was announced in [Sal15] without proof).

Theorem 3.4.

[TABLE]

Here on the left we have the homology with local coefficients of the classical braid group and on the right we have the homology of some independence complex with trivial coefficients (we indicate by $\varphi_{d}$ the $d$-th cyclotomic polynomial).

As said before, our method is based on the notions of weighted sheaves over posets and of weighted matchings, which are interesting objects by themselves. In our situation, the $d$-weight $v_{d}(\sigma)$ of a simplex $\sigma$ is the maximal power of the $d$-th cyclotomic polynomial which divides the Poincaré polynomial of the (finite) parabolic subgroup generated by the vertices of the simplex. We allow matchings between simplices of the same weight. Applying discrete Morse theory, one obtains a Morse complex which still computes the local homology.

Then, by applying these methods to other Coxeter graphs (in particular to finite and affine type cases), we come up with the notion of precise matching. In all the cases we consider, it is possible to produce acyclic weighted matchings such that the Morse complex turns out to have a very nice property: the boundary of a critical $p$-simplex $\sigma$ has a non-zero coefficient along a critical $(p-1)$-simplex $\tau$ if and only if $v_{d}(\sigma)=v_{d}(\tau)+1.$ Then the homology of the Morse complex reduces to the computation of the ranks of the incidence matrices among critical simplices (Theorems 4.2 and 4.4). We show that the existence of precise matchings can also be interpreted in terms of second-page collapsing of a spectral sequence which is naturally associated to the weighted sheaf (Proposition 4.3).

As we discuss at the beginning of Section 5, the existence of precise matchings for a certain Artin group has strong implications for its homology. In particular we want to highlight the following non-trivial result.

Theorem 5.2.

Let ${\bf G_{W}}$ be an Artin group (corresponding to some finitely generated Coxeter group ${\bf W}$) that admits a $\varphi$-precise matching for all cyclotomic polynomials $\varphi=\varphi_{d}$ (with $d\geq 2$). Then the homology $H_{*}(\mathbf{X}_{\mathbf{W}};R)$ does not have $\varphi^{k}$-torsion for $k\geq 2$.

Here $\mathbf{X}_{\mathbf{W}}$ (introduced in Section 2.4) is a finite CW-complex which is a deformation retract of the orbit space associated to ${\bf W}$, and $\pi_{1}(\mathbf{X}_{\mathbf{W}})={\bf G_{W}}$. When ${\bf W}$ is finite and in a few other cases $\mathbf{X}_{\mathbf{W}}$ it is known to be a $K({\bf G_{W}},1)$, whereas this property is only conjectured in general [Del72, Bri73, Oko79, VdL83, Hen85, Sal87, Sal94, CD95, CMS10, Par14].

Theorem 5.2 can be seen as a result about the homology of the infinite cyclic covering of $\mathbf{X}_{\mathbf{W}}$ and its monodromy. It is known that the conclusion of this theorem holds for all Artin groups of finite type, due to geometric reasons (see Remark 5.3). Our construction provides a new combinatorial condition that applies to a wider class of Artin groups (including, among others, the affine groups of type $\smash{\tilde{A}_{n}}$ and $\smash{\tilde{C}_{n}}$).

The methods we develop allow explicit and complete computations, in general in a more direct way compared to other known methods. We use them to compute the twisted homology in the case of affine Artin groups of type $\smash{\tilde{C}_{n}}$, obtaining the following result.

Theorem 5.20.

Let ${\bf G_{W}}$ be an Artin group of type $\tilde{C}_{n}$. Then the $\varphi_{d}$-primary component of $H_{*}({\bf G_{W}};R)$ is trivial for $d$ odd, and for $d$ even is as follows:

[TABLE]

where $n=k\frac{d}{2}+r$.

We also carry out complete computations in the cases $A_{n}$, $B_{n}$ and $\smash{\tilde{A}_{n}}$: these cases were already done before by different methods [Fre88, DCPSS99, DCPS01, CMS08a], but we need them to perform computations of more complicated cases.

In a recent preprint [Pao17b], the first author succeeded in constructing precise matchings for all remaining Artin groups of finite and affine type. Therefore precise matchings seem to be a suitable and effective tool for studying the homology of Artin groups.

The paper is structured as follows. In Section 2 we recall the most important definitions and results of [SV13]. In particular we introduce the general framework of weighted sheaves over posets, and then explain how it can be used to compute the homology of Artin groups. In Section 3 we study the case of braid groups (i.e. Artin groups of type $A_{n}$), and relate their homology with that of certain independence complexes. In Section 4 we introduce precise matchings and derive a general formula for the homology of the Morse complex. In Section 5 we apply the theory developed in Section 4 to the case of Artin groups. We construct precise matchings for Artin groups of type $A_{n}$, $B_{n}$, $\smash{\tilde{A}_{n}}$ and $\smash{\tilde{C}_{n}}$, and use such matchings to explicitly compute the homology.

2. Preliminaries

In this section we are going to recall some definitions and constructions from [SV13] (see also [MSV12]).

2.1. Weighted sheaves over posets

Let $(P,\preceq)$ be a finite poset.

Definition 2.1**.**

Define a sheaf of rings over $P$ (or a diagram of rings over $P$) as a collection

[TABLE]

of commutative rings together with a collection of ring homomorphisms

[TABLE]

satisfying

[TABLE]

[TABLE]

In other words, a sheaf of rings over $P$ is a functor from $P^{\text{op}}$ to the category of commutative rings.

Fix a PID $R$; usually we take $R=\mathbb{Q}[q^{\pm 1}]$. The divisibility relation

[TABLE]

gives $R$ the structure of a small category, and to $R/R_{*}$ the structure of a poset (where $R_{*}$ is the group of units in $R$) with minimum element the class of the units and maximum element [math]. Any functor $\mathit{w}\colon(P,\preceq)\to(R,\mid)$, which maps every $x\in P$ to some $\mathit{w}(x)\in R\setminus\{0\}$, defines a sheaf over $P$ by the collection

[TABLE]

and

[TABLE]

where $i_{x,y}$ is induced by the identity of $R$.

Definition 2.2**.**

Given a poset $P$, a PID $R$ and a morphism $\mathit{w}\colon(P,\preceq)\to(R,\mid)$ as above, we call the triple $(P,R,\mathit{w})$ a weighted sheaf over $P$ and the coefficients $\mathit{w}(x)$ the weights of the sheaf.

Remark 2.3**.**

Consider the poset topology over $P$, where a basis for the open sets is given by the upper intervals $\mathcal{B}=\{P_{>p},\ p\in P\}$. Let $(P,R,\mathit{w})$ be a weighted sheaf. Then one can see $\mathit{w}$ as a functor from $(\mathcal{B},\supseteq)$ to $(R,\mid)$ and thus a weighted sheaf defines a sheaf in the usual sense.

From now on our poset will be a simplicial complex $K$ defined over a finite set $S$, with the partial ordering

[TABLE]

We adopt the convention that $K$ contains the empty simplex $\varnothing$. A weighted sheaf over $K$ is given by assigning to each simplex $\sigma\in K$ a weight $\mathit{w}(\sigma)\in R$, with

[TABLE]

Let $C^{0}_{*}(K;R)$ be the $1$-shifted standard algebraic complex computing the simplicial homology of $K$, i.e.

[TABLE]

where $e_{\sigma}^{0}$ is a generator associated to a given orientation of $\sigma$ ($C^{0}_{0}(K;R)=R\,e_{\varnothing}^{0}$). The boundary is given by

[TABLE]

where $[\sigma:\tau]$ is the incidence number (which is equal to $\pm 1$ if $\tau\prec\sigma$, and otherwise is equal to [math]).

Definition 2.4**.**

The weighted complex associated to the weighted sheaf $(K,R,\mathit{w})$ is the algebraic complex $L_{*}=L_{*}(K)$ defined by

[TABLE]

with boundary $\partial\colon L_{k}\to L_{k-1}$ induced by $\partial^{0}$:

[TABLE]

There is a natural projection $\pi\colon C^{0}_{*}(K;R)\to L_{*}$ which maps a generator $e_{\sigma}$ in $C^{0}_{*}$ to the generator $\bar{e}_{\sigma}$ in $L_{*}$.

Remark 2.5**.**

The diagram $\{R/(w(\sigma))\mid\sigma\in K\}$ also defines a sheaf over the poset $K$ in the sense of Remark 2.3. The sheaf cohomology associated to the open covering given by the upper segments coincides with the homology of the weighted sheaf.

2.2. Decomposition and filtration

Let $\mathcal{S}=(K,R,\mathit{w})$ be a weighted sheaf. For any irreducible $\varphi\in R,$ we define the $\varphi$-primary component $\mathcal{S}_{\varphi}=(K,R,\mathit{w}_{\varphi})$ of the weighted sheaf $\mathcal{S}$ by setting

[TABLE]

where

[TABLE]

Since $R$ is a PID and $w(\sigma)\neq 0$, such a maximal value exists. Notice that $\mathcal{S}_{\varphi}$ is a weighted sheaf. The weighted complex $(L_{*})_{\varphi}$ associated to $\mathcal{S}_{\varphi}$ is called the $\varphi$-primary component of the weighted complex $L_{*}$. In degree $k$ one has:

[TABLE]

The complex $(L_{*})_{\varphi}$ has a natural increasing filtration by the following subcomplexes:

[TABLE]

This filtration is associated to an increasing filtration of the simplicial complex $K$:

[TABLE]

Then $F^{s}(L_{*})_{\varphi}$ is the weighted complex associated to the weighted sheaf

[TABLE]

Theorem 2.6** ([SV13]).**

Let $(K,R,\mathit{w})$ be a weighted sheaf, with associated weighted complex $L_{*}$. For any irreducible $\varphi\in R$, there exists a spectral sequence

[TABLE]

that abuts to the homology of the $\varphi$-primary component of the associated algebraic complex $L_{*}$. Moreover the $E^{1}$-term

[TABLE]

is isomorphic to the relative homology with trivial coefficients of the simplicial pair $(K_{\varphi,p},K_{\varphi,p-1})$.

2.3. Discrete Morse theory for weighted complexes

Given $x$, $y$ in a poset $(P,\prec)$, the notation $x\lhd y$ means that $y$ covers $x$, i.e. $x\prec y$ and there is no other $z\in P$ such that $x\prec z\prec y$.

Recall the basic facts of discrete Morse theory (see for example [For98, Koz07]). A matching in a poset $P$ is a set $\mathcal{M}\subseteq P\times P$ such that $(x,y)\in M\Rightarrow x\lhd y$ and each $x\in P$ belongs to at most one pair of $\mathcal{M}$. An alternating path is a sequence

[TABLE]

such that each pair $x_{i}\lhd y_{i}$ belongs to $\mathcal{M}$ and no pair $x_{i}\lhd y_{i-1}$ belongs to $\mathcal{M}$. A cycle is a closed alternating path with $y_{0}=y_{m}$. An acyclic matching over $P$ is a matching with no cycles. We are going to describe a variant of discrete Morse theory which is suitable for our situation.

Definition 2.7**.**

A weighted acyclic matching on a weighted sheaf $(P,R,\mathit{w})$ over $P$ is an acyclic matching $\mathcal{M}$ on $P$ such that

[TABLE]

(in [SV13] we required $\mathit{w}(x)=\mathit{w}(y)$, but the above generalization does not change the results).

The standard theory generalizes as follows. Let $\mathcal{S}=(K,R,\mathit{w})$ be a weighted sheaf over a finite simplicial complex $K$, and let $\mathcal{M}$ an acyclic weighted matching. A critical simplex of $K$ is a simplex $\sigma$ which does not belong to any pair of $\mathcal{M}$. The following definition is equivalent to [SV13, Definition 3.3].

Definition 2.8**.**

The Morse complex of $\mathcal{S}$ with respect to $\mathcal{M}$ is defined as the torsion complex

[TABLE]

with boundary

[TABLE]

where $[\sigma:\tau]^{\mathcal{M}}\in\mathbb{Z}$ is given by the sum over all alternating paths

[TABLE]

from $\sigma$ to $\tau$ of the quantity

[TABLE]

The boundary map $\partial^{\mathcal{M}}$ is extended by $R$-linearity, where some care should be taken since each $\bar{e}_{\tau}$ lives in a component with a possibly different torsion ($\partial^{\mathcal{M}}$ is the same as the boundary map of [SV13]).

Theorem 2.9** ([SV13]).**

Let $\mathcal{S}=(K,R,\mathit{w})$ be a weighted sheaf over a simplicial complex $K$. Let $\mathcal{M}$ be an acyclic matching for $\mathcal{S}$. Then there is an isomorphism

[TABLE]

between the homology of the weighted complex $L_{*}$ associated to $\mathcal{S}$ and the homology of the Morse complex $L^{\mathcal{M}}_{*}$ of $\mathcal{S}$.

Remark 2.10**.**

If we forget about the weights, the matching $\mathcal{M}$ is in particular an acyclic matching for the simplicial complex $K$. Therefore the algebraic complex $C^{0}_{*}(K;A)$, which computes the (shifted and reduced) simplicial homology of $K$ with coefficients in some ring $A$, admits a “classical” algebraic Morse complex

[TABLE]

with boundary map

[TABLE]

Remark 2.11**.**

Set $C^{0}_{*}=C^{0}_{*}(K;R)$ and let $\pi\colon C^{0}_{*}\to L_{*}$ be the natural projection. The composition with the projection $L_{*}\to(L_{*})_{\varphi}$, for some fixed irreducible element $\varphi\in R$, yields a natural projection $\pi_{\varphi}\colon C^{0}_{*}\to(L_{*})_{\varphi}$. Similarly at the level of Morse complexes we have the projection $\bar{\pi}_{\varphi}\colon(C^{0}_{*})^{\mathcal{M}}\to(L_{*})_{\varphi}^{\mathcal{M}}$ which sends a generator $\bar{e}_{\sigma}$ in $(C^{0}_{*})^{\mathcal{M}}$ to the generator $\bar{e}_{\sigma}$ in $\smash{(L_{*})_{\varphi}^{\mathcal{M}}}$. Then, by construction, the maps induced in homology by the projections $\pi_{\varphi}$ and $\bar{\pi}_{\varphi}$ commute with the Morse isomorphisms of [Koz07] and [SV13]:

{H_{*}(C^{0}_{*})}$${H_{*}((L_{*})_{\varphi})}$${H_{*}((C^{0}_{*})^{\mathcal{M}})}$${H_{*}((L_{*})^{\mathcal{M}}_{\varphi}).}$$(\pi_{\varphi})_{*}$$\cong$$\cong$$(\bar{\pi}_{\varphi})_{*}

Let $\varphi\in R$ be an irreducible element, and let $\mathcal{M}$ be a weighted acyclic matching on $\mathcal{S}_{\varphi}$. Then the filtration on the weighted complex induces a filtration on the Morse complex:

[TABLE]

Consider also the following quotient complex:

[TABLE]

Theorem 2.12** ([SV13]).**

Let $\mathcal{S}$ and $\varphi$ be as above and let $\mathcal{M}$ be an acyclic matching for $\mathcal{S}_{\varphi}$. Then the $E^{1}$-page of the spectral sequence of Theorem 2.6 is identified with

[TABLE]

where $(L_{*})_{\varphi}^{\mathcal{M}}$ is the Morse complex of $\mathcal{S}_{\varphi}.$ The differential

[TABLE]

is induced by the boundary of the Morse complex, and thus it is also computed by using alternating paths.

2.4. Weighted sheaves for Artin groups

We use the constructions given above in the context of Artin groups. Let $({\bf W},S)$ be a Coxeter system, with $|S|=n$ finite, and let $\Gamma$ be the corresponding Coxeter graph (with $S$ as vertex set). Recall that ${\bf W}$ can always be realized as a group of reflections in some $\mathbb{R}^{n}$ (for example through the Tits representation, see [Bou68, Vin71], see also [Par14]), so that it has the following naturally associated objects:

1. (i)

a hyperplane arrangement in $\mathbb{R}^{n}$

[TABLE] 2. (ii)

configuration spaces

[TABLE]

where $U={\bf W}\cdot\overline{C_{0}}$ is the Tits cone (here $C_{0}$ is a fixed chamber of the arrangement, and $\overline{C_{0}}$ is its topological closure); 3. (iii)

a simplicial complex (defined over the finite set $S$)

[TABLE]

One can define the Artin group ${\bf G_{W}}$ of type ${\bf W}$ as the fundamental group $\pi_{1}({\bf Y_{W}})$. It has a presentation

[TABLE]

(both the products have $m(s,s^{\prime})$ factors, where $m(s,s^{\prime})$ is the order of $ss^{\prime}$ in ${\bf W}$). Recall also the following results [Sal94, Theorems 1.4 and 1.8].

1. (i)

The orbit space ${\bf Y_{W}}$ deformation retracts onto a finite $CW$-complex $\mathbf{X}_{{\bf W}}$ given by a union

[TABLE]

of convex polyhedra with explicit identifications of their faces. 2. (ii)

Consider the action of the Artin group ${\bf G_{W}}$ on the ring $R=\mathbb{Q}[q^{\pm 1}]$ given by

[TABLE]

Then the homology $H_{*}(\mathbf{X}_{{\bf W}};R)$ is computed by the algebraic complex

[TABLE]

with boundary

[TABLE]

where $\displaystyle{\bf W}_{\sigma}(q)=\sum_{w\in{\bf W}_{\sigma}}q^{l(w)}$ is the Poincaré polynomial of ${\bf W}_{\sigma}$.

Remark 2.13**.**

The $R$-module structure on $H_{*}(\mathbf{X}_{\mathbf{W}};R)$ is given by the transformation $\mu_{q}$ induced by $q$-multiplication. If the order of $\mu_{q}$ is $n,$ then the homology groups decompose into cyclic factors which are either free, of the form $R^{k},$ or torsion, of the form $R/(\varphi_{d})$, where $\varphi_{d}$ is the $d$-th cyclotomic polynomial with $d\mid n$. Therefore we are interested in localizing to cyclotomic polynomials.

Theorem 2.14** ([SV13]).**

To a Coxeter system $({\bf W},S)$ we can associate a weighted sheaf $(K,R,\mathit{w})$ over the simplicial complex $K=K_{{\bf W}}$, by setting $\mathit{w}(\sigma)={\bf W}_{\sigma}(q)$. Also, for any cyclotomic polynomial $\varphi=\varphi_{d}$, the map

[TABLE]

defines a weighted sheaf $(K,R,\mathit{w}_{\varphi})$ over $K=K_{{\bf W}}$.

The homology of the associated weighted complex is strictly related to the homology of $\mathbf{X}_{\mathbf{W}}$. Specifically, set $C_{*}^{0}=C_{*}^{0}(K;R)$ and consider the diagonal map

[TABLE]

By the formula for the boundary map it follows that $\Delta$ is an injective chain-complex homomorphism, so there is an exact sequence of complexes:

[TABLE]

where

[TABLE]

is the quotient complex. Passing to the associated long exact sequence we get:

[TABLE]

Then the homology of $L_{*}$ can be used to compute the homology of $C_{*}$.

The orbit space ${\bf Y_{W}}$ (and thus the CW-complex $\mathbf{X}_{\mathbf{W}}\simeq{\bf Y_{W}}$) is conjectured to be a classifying space for the Artin group ${\bf G_{W}}$ [Del72, Bri73, VdL83, Par14]. This conjecture was proved for Artin groups of finite type [Del72], for affine Artin groups of type $\tilde{A}_{n}$, $\tilde{B}_{n}$ and $\tilde{C}_{n}$ [Oko79, CMS10], and for some other families of Artin groups [Hen85, CD95]. Whenever the conjecture holds (in particular this is true for all the cases we consider in this paper), the homology $H_{*}(\mathbf{X}_{\mathbf{W}};R)$ coincides with the twisted homology $H_{*}({\bf G_{W}};R)$ of the Artin group ${\bf G_{W}}$.

3. Homology of braid groups and independence complexes

In this section we are going to show how the twisted homology of braid groups (i.e. Artin groups of type $A_{n}$) is related to the homology of suitable independence complexes. Recall that a Coxeter graph of type $A_{n}$ is a linear graph with $n$ vertices, usually labeled $1$, $2$, …, $n$ (see Figure 1), and that the corresponding Artin group is the braid group on $n+1$ strands (which we denote by $\operatorname{Br}_{n+1}$). With a slight abuse of notation, by $A_{n}$ we will sometimes indicate the graph itself.

The homology of $\operatorname{Br}_{n+1}$, with coefficients in the representation $R=\mathbb{Q}[q^{\pm 1}]$ described in the previous section, has been computed in [Fre88, DCPS01]. See also [Cal06] for coefficients in $\mathbb{Z}[q^{\pm 1}]$ (but we will not address this case).

Theorem 3.1** ([Fre88, DCPS01]).**

The $\varphi_{d}$-primary component of the twisted homology of a braid group is given by

[TABLE]

where the non vanishing term is in degree $(d-2)k$ if $n=dk$ or $n=dk-1$.

Recall that, if $G$ is a graph with vertex set $VG$, an independent set of $G$ is a subset of $VG$ consisting of pairwise non-adjacent vertices. Also, the independence complex $\operatorname{Ind}(G)$ of $G$ is the abstract simplicial complex with $VG$ as set of vertices and whose simplices are all the non-empty independent sets of $G$. Thus $\operatorname{Ind}(G)$ is the clique complex of the complement graph of $G$. In contrast with the simplicial complex $K$ introduced in Section 2, the simplicial complex $\operatorname{Ind}(G)$ does not contain the empty simplex, and the dimension of a simplex $\sigma\in\operatorname{Ind}(G)$ is given by $|\sigma|-1$. The homotopy type of $\operatorname{Ind}(A_{n})$ has been computed in [Koz07] by means of discrete Morse theory, and the result is the following.

Proposition 3.2** ([Koz07, Proposition 11.16]).**

[TABLE]

By comparing this with Theorem 3.1 we obtain the following relation between the homology of the independence complex of $A_{n}$ and the $\varphi_{3}$-primary component of the twisted homology of $\operatorname{Br}_{n+1}$.

Corollary 3.3**.**

[TABLE]

where on the left we have local coefficients and on the right we have trivial coefficients.

∎

In general, following [Sal15], define the $r$-independence complex of a graph $G$ as

[TABLE]

So $\operatorname{Ind}_{r}(G)$ is an abstract simplicial complex on the set of vertices $VG$ of the graph $G$, which coincides with $\operatorname{Ind}(G)$ for $r=1$. The case $r=0$ also makes sense: $\operatorname{Ind}_{0}(G)=\varnothing$ for any graph $G$. We are going to prove the following generalization of Corollary 3.3.

Theorem 3.4**.**

[TABLE]

where on the left we have local coefficients and on the right we have trivial coefficients.

From the expression of the $\varphi_{d}$-primary component of the local homology of the braid group (Theorem 3.1) we obtain the following consequence.

Corollary 3.5**.**

[TABLE]

Therefore the knowledge of the twisted homology of the braid group gives the homology (with trivial coefficients) of $\operatorname{Ind}_{d}(A_{n})$. Conversely the knowledge of the homology of $\operatorname{Ind}_{d}(A_{n})$ gives the twisted homology of the braid group.

The Poincaré polynomial of a Coxeter group of type $A_{k}$ is given by

[TABLE]

(see for example [BB06]). Consider now a simplex $\sigma\subseteq S\cong\{1,\dots,n\}$. Denote by $\Gamma(\sigma)$ the subgraph of $A_{n}$ induced by $\sigma$. Denote by $\Gamma_{1}(\sigma)$, $\Gamma_{2}(\sigma)$, …, $\Gamma_{m}(\sigma)$ the connected components of $\Gamma(\sigma)$, and by $n_{1}$, $n_{2}$, …$n_{m}$ their cardinalities (see Figure 2). The $i$-th connected component is a Coxeter graph of type $A_{n_{i}}$, so the entire Coxeter graph induced by $\sigma$ has Poincaré polynomial

[TABLE]

Then we are interested in the homology of the weighted complex $(L_{*})_{\varphi}$ associated to the weighted sheaf $(K,R,\mathit{w}_{\varphi})$, where $\varphi=\varphi_{d}$ is a cyclotomic polynomial (with $d\geq 2$) and

[TABLE]

Notice that only the connected components with at least $d-1$ vertices contribute to the $\varphi$-weight. Therefore, the weighted complex $(L_{*})_{\varphi}$ is generated by the subgraphs having at least one component with $\geq d-1$ vertices.

Theorem 3.6**.**

There is a weighted acyclic matching $\mathcal{M}$ on $K$ such that the set of critical simplices is given by

[TABLE]

where $A_{d-1}$ is the linear graph on the vertices $n-d+2$, …, $n$.

Proof.

First notice that removing the $d$-th vertex from an $A_{k}$ component leaves the $\varphi$-weight unchanged:

[TABLE]

(see Figure 3).

Let $K_{0}=\operatorname{Ind}_{d-2}(A_{n})\subseteq K$ (this is the set of the simplices $\sigma\in K$ such that all the connected components of the induced subgraph $\Gamma(\sigma)$ have cardinality $<d-1$). Let us define on $K^{\prime}=K\setminus K_{0}$ the following matching $\mathcal{M}$. For $\sigma\in K^{\prime}$, with $\Gamma(\sigma)=\Gamma_{1}\sqcup\dots\sqcup\Gamma_{m}$, set

[TABLE]

Then match $\sigma$ with the simplex $\tau$ obtained by adding or removing from $\sigma$ the $d$-th vertex of the component $\Gamma_{i(\sigma)}$. By the remark at the beginning of the proof, $\mathcal{M}$ is a weighted matching. We prove that it is acyclic. In fact, suppose that an alternating path contains some subpath

[TABLE]

with $\tau\neq\tau^{\prime}$. Then either $i(\tau)<i(\tau^{\prime})$ or (set $j=i(\tau)=i(\tau^{\prime})$)

[TABLE]

for some $a\leq n-d+1.$ In this case the first vertex of $\Gamma_{j}(\tau^{\prime})$ is greater than the first vertex of $\Gamma_{j}(\tau)$. Therefore an alternating path in $K^{\prime}$ cannot be closed.

The set of critical elements of $\mathcal{M}$ in $K^{\prime}$ is given by

[TABLE]

where $A_{d-1}$ is as in the statement of the theorem. ∎

Proof of Theorem 3.4.

Consider the matching $\mathcal{M}$ of Theorem 3.6. Since $K_{0}=\operatorname{Ind}_{d-2}(A_{n})$ does not contribute to the weighted complex $(L_{*})_{\varphi}$, we concentrate on the poset $K^{\prime}=K\setminus K_{0}$. Notice that there are no non-trivial alternating paths between critical elements of $K^{\prime}$. Therefore the boundaries between simplices in

[TABLE]

are the same as the boundaries of $\operatorname{Ind}_{d-2}(A_{n-d})$. In the long exact sequence (1) the algebraic complex $C^{0}_{*}$ has vanishing homology in all degrees (because $K$ is the full simplicial complex on $n$ vertices), so we have an isomorphism $H_{*+1}(L_{*})\cong H_{*}(C_{*})$. Therefore

[TABLE]

In the last isomorphism there is a $(d-1)$-shift in degree due to the loss of $d-1$ vertices when passing from $\text{Cr}^{\prime}(\mathcal{M})$ to $\operatorname{Ind}_{d-2}(A_{n-d})$; there is then a further $1$-shift in degree due to the fact that in $L_{*}$ a simplex $\sigma$ has dimension $|\sigma|$, and in $\operatorname{Ind}_{d-2}(A_{n-d})$ it has dimension $|\sigma|-1$; finally, it is necessary to pass to reduced homology because the empty simplex is missing in $\operatorname{Ind}_{d-2}(A_{n-d})$. ∎

For the sake of completeness we also determine the homotopy type of the $r$-independence complex $\operatorname{Ind}_{d-2}(A_{n})$ (obtaining again Corollary 3.5 as a consequence). This is a straightforward generalization of the case $d=3$ proved in [Koz07, Proposition 11.16].

Proposition 3.7**.**

[TABLE]

Proof.

We denote here for brevity $K_{0}=\operatorname{Ind}_{d-2}(A_{n})$. Let $n=qd+r$ be the euclidean division of $n$ by $d$.

Let $P=\{c_{d}>c_{2d}>c_{3d}>\dots>c_{qd}>c_{*}\}$ be a linearly ordered set, where $c_{*}$ is the minimum element. Define a map $f\colon K_{0}\to P$ as follows. If $\sigma\in K_{0}$ does not contain any multiples of $d$, then set $f(\sigma)=c_{*}$. Otherwise, set $f(\sigma)=c_{jd}$ if $jd\in\sigma$ but $j^{\prime}d\not\in\sigma$ for $j^{\prime}<j$. Clearly $f$ is a poset map (if we remove a vertex from $\sigma$, $j$ increases or remains the same).

In $f^{-1}(c_{d})$ consider the matching $\mathcal{M}_{1}=\{(\sigma\setminus\{1\},\sigma\cup\{1\})\mid\sigma\in f^{-1}(c_{d})\}$, which is justified by:

[TABLE]

Similarly, in $f^{-1}(c_{jd})$ for $j\leq q$, consider the matching

[TABLE]

justified by:

[TABLE]

Each $\mathcal{M}_{j}$ is acyclic in $f^{-1}(c_{jd})$, thus

[TABLE]

is an acyclic matching on $K_{0}$ by the Patchwork Theorem (see for example [Jon08, Lemma 4.2], [Koz07, Theorem 11.10]). Since each $\mathcal{M}_{j}$ is a perfect matching on $f^{-1}(c_{jd})$ for $j=1,\dots,q$, and $K_{1}=f^{-1}(c_{*})$ is a subcomplex of $K_{0}$, it follows that $K_{0}$ deformation retracts onto $K_{1}$.

Notice now that $K_{1}$ is the join of $q$ copies of $\operatorname{Ind}_{d-2}(A_{d-1})\cong S^{d-3}$ and one copy of $\operatorname{Ind}_{d-2}(A_{r})$. For $r=0$ we get

[TABLE]

and for $r=d-1$ we get

[TABLE]

which give the first case in the corollary. If $r$ is not [math] or $d-1$ then $\operatorname{Ind}_{d-2}(A_{r})$ is a full simplex $\Delta^{r-1}$ on $r$ vertices, and therefore $K_{0}$ is contractible. ∎

4. Precise matchings

In this section we are going to introduce the notion of precise matching on a weighted sheaf. The motivation comes from the study of the twisted homology of some families of Artin groups, as we will show in the subsequent sections. For now we go back to the general framework of weighted sheaves over simplicial complexes (later we will specialize in the case of Artin groups).

Assume from now on that the PID $R$ contains some field $\mathbb{K}$. Our main case of interest is $\mathbb{K}=\mathbb{Q}$ and $R=\mathbb{Q}[q^{\pm 1}]$. Let $\mathcal{S}=(K,R,\mathit{w})$ be a weighted sheaf over the finite simplicial complex $K$, with associated weighted complex $L_{*}$. Given a fixed irreducible element $\varphi$ of $R$, let $\mathcal{S}_{\varphi}$ be the $\varphi$-primary component of $\mathcal{S}$ and let $(L_{*})_{\varphi}$ be its associated weighted complex. Let $\mathcal{M}$ be a weighted acyclic matching for $\mathcal{S}_{\varphi}$.

Let $G^{\mathcal{M}}$ be the incidence graph of the corresponding Morse complex: the vertices of $G^{\mathcal{M}}$ are the critical simplices of $K$, and there is an (oriented) edge $\sigma\to\tau$ whenever $[\sigma:\tau]^{\mathcal{M}}$ is not [math] in $R$ (or equivalently in $\mathbb{K}$), where $[\sigma:\tau]^{\mathcal{M}}\in\mathbb{Z}$ is the incidence number between $\sigma$ and $\tau$ in the Morse complex of $K$. In other words, there is an edge $\sigma\to\tau$ if $[\sigma:\tau]^{\mathcal{M}}$ is not multiple of $\operatorname{char}\mathbb{K}=\operatorname{char}R$. When $\mathbb{K}=\mathbb{Q}$, this simply means that $[\sigma:\tau]^{\mathcal{M}}\neq 0$.

Let $\mathcal{I}$ be the set of connected components of $G^{\mathcal{M}}$ (computed ignoring the orientation of the edges). Recall that $v_{\varphi}(\sigma)$ is the maximal $k\in\mathbb{N}$ such that $\varphi^{k}$ divides $\mathit{w}(\sigma)$.

Definition 4.1**.**

The matching $\mathcal{M}$ is $\varphi$-precise (or simply precise) if, for any edge $\sigma\to\tau$ of $G^{\mathcal{M}}$, we have that $v_{\varphi}(\sigma)=v_{\varphi}(\tau)+1$.

In other words $\mathcal{M}$ is precise if, for any two simplices $\sigma$ and $\tau$ lying in the same connected component $i\in\mathcal{I}$, the following relation holds:

[TABLE]

Equivalently the quantity $|\sigma|-v_{\varphi}(\sigma)$, as a function of $\sigma$, is constant within a fixed connected component of $G^{\mathcal{M}}$. This definition is motivated by the fact that precise matchings exist in many cases of interest (we will see this in Section 5), and that the homology of the Morse complex is much simpler to compute (and takes a particularly nice form) when the matching is precise. The name “precise” has been chosen because for a generic matching one only has $v_{\varphi}(\sigma)\geq v_{\varphi}(\tau)$ (when $\sigma\to\tau$ is an edge of $G^{\mathcal{M}}$), and we require $v_{\varphi}(\sigma)$ to be precisely $v_{\varphi}(\tau)+1$.

Assume from now on in this section that $\mathcal{M}$ is a $\varphi$-precise matching. To simplify the notation, set $(A_{*},\partial)=((L_{*})_{\varphi}^{\mathcal{M}},\partial^{\mathcal{M}})$ and $(V_{*},\delta)=(C^{0}_{*}(K,\mathbb{K})^{\mathcal{M}},\delta^{\mathcal{M}})$. Our aim is to derive a formula for the homology of the Morse complex $A_{*}$. Since the differential $\delta$ vanishes between simplices in different connected components of $G^{\mathcal{M}}$, the complex $(V_{*},\delta)$ splits as follows:

[TABLE]

where

[TABLE]

and the boundary map $\delta^{i}\colon V^{i}_{*}\to V^{i}_{*}$ is the restriction of $\delta$ to $V^{i}_{*}$. The differential $\partial$ of $A_{*}$ is induced by $\delta$, and thus it also vanishes between simplices in different connected components. Therefore we have an analogous splitting for $(A_{*},\partial)$:

[TABLE]

where

[TABLE]

and the boundary map $\partial^{i}\colon A^{i}_{*}\to A^{i}_{*}$ is simply the restriction of $\partial$ to $A^{i}_{*}$.

Fix now a connected component $i\in\mathcal{I}$. Since $\mathcal{M}$ is precise, there exists some $k\in\mathbb{Z}$ (which depends on $i$) such that $v_{\varphi}(\sigma)=|\sigma|+k$ for all $\sigma\in i$. Therefore in degree $m$ we have

[TABLE]

where $\text{Cr}^{i}_{m}$ is the set of critical simplices $\sigma$ such that $\sigma\in i$ and $|\sigma|=m$. By construction the boundary $\partial^{i}_{m}\colon A^{i}_{m}\to A^{i}_{m-1}$ factors accordingly:

[TABLE]

where

[TABLE]

is the projection induced by the identity $R\to R$. Since $\text{im}\,\delta^{i}_{m+1}\subseteq\ker\delta^{i}_{m}$, each $V^{i}_{m}$ splits (as a vector space over $\mathbb{K}$) as a direct sum of linear subspaces:

[TABLE]

where $W^{i}_{m,1}=\text{im}\,\delta^{i}_{m+1}$ and $W^{i}_{m,1}\oplus W^{i}_{m,2}=\ker\delta^{i}_{m}$. Then

[TABLE]

Therefore the homology of $(A^{i}_{*},\delta^{i})$ is given, as an $R$-module, by

[TABLE]

In the last isomorphism we used the fact that $\dim W^{i}_{m,3}=\dim V^{i}_{m}-\dim\ker\delta^{i}_{m}=\operatorname{rk}\delta^{i}_{m}$.

Recall that the previous formula holds for a fixed connected component $i\in\mathcal{I}$, and $k$ depends on $i$. Since we now need to take the direct sum over the connected components, let $k_{i}$ be the value of $k$ for the component $i$.

Theorem 4.2**.**

The homology of $(L_{*})_{\varphi}$ is given, as an $R$-module, by

[TABLE]

Proof.

By Theorem 2.9, $H_{m}((L_{*})_{\varphi},\partial)\cong H_{m}((L_{*})_{\varphi}^{\mathcal{M}},\partial^{\mathcal{M}})$. Using what we have done in this section, we have that

[TABLE]

Let us see how the existence of a precise matching can be interpreted in terms of the spectral sequence associated to the weighted sheaf (see Theorem 2.6).

Proposition 4.3**.**

If a $\varphi$-precise matching $\mathcal{M}$ exists, then the spectral sequence $E_{p,q}^{r}$ associated to the weighted sheaf $\mathcal{S}_{\varphi}$ collapses at the $E^{2}$-page.

Proof.

By Theorem 2.12, the $E^{1}$-page can be computed through the Morse complex of our matching $\mathcal{M}$:

[TABLE]

and the differential $d^{1}_{p,q}$ is induced by the boundary of the Morse complex. The spectral sequence then splits as a direct sum over the connected components of $\smash{G^{\mathcal{M}}}$:

[TABLE]

where $E^{1,i}_{p,q}\cong H_{p+q}(\mathcal{F}^{p}A^{i}_{*})$. Since the matching is precise, for $m\neq p-k_{i}$ we have

[TABLE]

This means that the page $E^{0,i}_{p,q}\cong E^{1,i}_{p,q}$ is non-trivial only in the row $q=-k_{i}$, and the entire spectral sequence $E_{p,q}^{r}$ collapses at the $E^{2}$-page. ∎

What we have done so far in this section assumed $\varphi$ to be some fixed irreducible element of the PID $R$. In order to recover the full homology of $L_{*}$ we need to make $\varphi$ vary among all equivalence classes of irreducible elements of $R$ modulo the units. Suppose from now on to have a $\varphi$-precise matching $\mathcal{M}_{\varphi}$ on $\mathcal{S}_{\varphi}$, for each $\varphi$. The following results follows immediately from Theorem 4.2, provided that we add a “$\varphi$” subscript (or superscript) to all the quantities that depend on the matching $\mathcal{M}_{\varphi}$.

Theorem 4.4**.**

The homology of $L_{*}$ is given, as an $R$-module, by

[TABLE]

For later applications, we finally need to study how the isomorphism of Theorem 4.4 behaves with respect to the projection $\pi\colon C^{0}_{*}\to L_{*}$ of Remark 2.11. This projection is the direct sum over $\varphi$ of the projections

[TABLE]

Instead of studying the induced map $(\pi_{\varphi})_{*}\colon H_{m}(C^{0}_{*})\to H_{m}((L_{*})_{\varphi})$, we study the map

[TABLE]

between the Morse complexes (here $\mathcal{M}=\mathcal{M}_{\varphi}$ is a precise matching which depends on $\varphi$). For $i\in\mathcal{I}_{\varphi}$, let $\pi_{i}\colon(C^{0}_{*})^{\mathcal{M}}\to V^{\varphi,i}_{*}\otimes_{\mathbb{K}}R\subseteq(C^{0}_{*})^{\mathcal{M}}$ be the projection on the subcomplex corresponding to the connected component $i$, and let $(\pi_{i})_{*}$ be the map induced in homology. Let $[c]\in H_{m}((C^{0}_{*})^{\mathcal{M}})$, for some cycle $c\in\ker\delta^{\mathcal{M}}\subseteq(C^{0}_{m})^{\mathcal{M}}$. Applying the map $(\bar{\pi}_{\varphi})_{*}\colon H_{m}((C_{*}^{0})^{\mathcal{M}})\to H_{m}((L_{*})^{\mathcal{M}}_{\varphi})$ we obtain

[TABLE]

Applying the isomorphism of Theorem 4.2, this element is sent to

[TABLE]

We are going to use these computations to prove the following result which describes the kernel and the cokernel of $\pi_{*}$.

Proposition 4.5**.**

The cokernel of $\pi_{*}\colon H_{m}(C^{0}_{*})\to H_{m}(L_{*})$ is given by

[TABLE]

In addition, the kernel of $\pi_{*}$ is a free $R$-module isomorphic to $H_{m}(C^{0}_{*})$.

Proof.

Throughout the proof, consider the following $R$-modules identified one with each other, without explicitly mentioning the isomorphisms between them:

[TABLE]

Recall that the matching $\mathcal{M}$ depends on $\varphi$, although we write $\mathcal{M}$ instead of $\mathcal{M}_{\varphi}$ in order to make the notations more readable. Also recall that the isomorphisms $H_{m}((L_{*})_{\varphi})\cong H_{m}((L_{*})^{\mathcal{M}}_{\varphi})$ occur in the commutative diagram of Remark 2.11.

We want to show that the image of $\pi_{*}\colon H_{m}(C^{0}_{*})\to H_{m}(L_{*})$ is given by

[TABLE]

Let $\psi_{\varphi}\colon H_{*}(C^{0}_{*})\to H_{*}((L_{*})^{\mathcal{M}}_{\varphi})$ be the map defined as the composition

[TABLE]

By commutativity of the diagram of Remark 2.11, the image of $\pi_{*}=\bigoplus_{\varphi}(\pi_{\varphi})_{*}$ is the same as the image of

[TABLE]

We have already proved that, for any $[c]\in H_{m}((C^{0}_{*})^{\mathcal{M}})$,

[TABLE]

which means in particular that

[TABLE]

Therefore we immediately have the inclusion

[TABLE]

To prove the opposite inclusion, we show that any element of the form

[TABLE]

is in the image of $\pi_{*}$ (for any fixed $\varphi$ and $i$). To do so, choose $\alpha\in R$ such that $\alpha\equiv 1\;(\text{mod}\;\varphi^{m+k_{\varphi,i}})$ and $\alpha\equiv 0\;(\text{mod}\;\eta^{m+k_{\eta,j}})$ for any irreducible element $\eta\neq\varphi$ which divides some weight $\mathit{w}(\sigma)$, and for any connected component $j\in\mathcal{I}_{\eta}$ (there is only a finite number of such $\eta$ up to multiplication by units, because $K$ is finite). The element $c\otimes_{\mathbb{K}}\alpha$ is a cycle in $V_{m}^{\varphi,i}\otimes_{\mathbb{K}}R\subseteq C^{0}_{m}(K,\mathbb{K})^{\mathcal{M}}\otimes_{\mathbb{K}}R\cong(C^{0}_{m})^{\mathcal{M}}$. Then, if $[\tilde{c}]$ is the preimage of $[c\otimes_{\mathbb{K}}\alpha]$ under the isomorphism $H_{*}(C^{0}_{*})\xlongrightarrow{\cong}H_{*}((C^{0}_{*})^{\mathcal{M}})$, we have that:

[TABLE]

where $[c^{\prime}]$ is the image of $[c]$ under the isomorphism

[TABLE]

Therefore $[c]\otimes_{\mathbb{K}}[1]$ is in the image of $\pi_{*}$. We have thus proved that

[TABLE]

Then the cokernel of $\pi_{*}$ can be easily computed:

[TABLE]

The $R$-module $H_{m}(C^{0}_{*})\cong H_{m}(C^{0}_{*}(K;\mathbb{K}))\otimes_{\mathbb{K}}R$ is free and finitely generated, because $H_{m}(C^{0}_{*}(K;\mathbb{K}))$ is a finite-dimensional vector space over $\mathbb{K}$ (recall that $K$ is a finite simplicial complex). The kernel of $\pi_{*}$ is a submodule of $H_{m}(C^{0}_{*})$, so it is itself a free $R$-module with lower or equal rank. Let $[c_{1}],\dots,[c_{k}]$ be an $R$-base of $H_{m}(C^{0}_{*})$. Consider the non-zero ideal

[TABLE]

where $\varphi$ varies among the (finitely many) irreducible elements which divide some weight $\mathit{w}(\sigma)$ (for $\sigma\in K$). Fix any non-zero element $\alpha\in I$. Then the elements $\alpha[c_{1}],\dots,\alpha[c_{k}]$ generate a free submodule of $\ker\pi_{*}$ of rank $k=\operatorname{rk}H_{m}(C^{0}_{*})$. Therefore $\ker\pi_{*}$ and $H_{m}(C^{0}_{*})$ have the same rank, so they are isomorphic as $R$-modules. ∎

5. Precise matchings for Artin groups

Consider now the case of Artin groups, as in Section 2.4. For a Coxeter system $({\bf W},S)$ we have constructed a weighted sheaf $\mathcal{S}=(K,R,\mathit{w})$ with

[TABLE]

and $R=\mathbb{Q}[q^{\pm 1}]$. The associated weighted complex $L_{*}$ fits into the short exact sequence

[TABLE]

which gives rise to the long exact sequence (1):

[TABLE]

In order to compute $H_{*}(C_{*})=H_{*}(\mathbf{X}_{\mathbf{W}};R)$, we split this long exact sequence into the short exact sequences

[TABLE]

where on the left we have the cokernel of $\pi_{*}\colon H_{m+1}(C^{0}_{*})\to H_{m+1}(L_{*})$ and on the right we have the kernel of $\pi_{*}\colon H_{m}(C^{0}_{*})\to H_{m}(L_{*})$. Since $\ker\pi_{*}$ is a free $R$-module, these short exact sequences split:

[TABLE]

Recall that the only irreducible elements of $R$ that occur in the factorization of the weights are the cyclotomic polynomials $\varphi_{d}$ for $d\geq 2$. As in Section 4, suppose from now on that we have constructed a $\varphi$-precise matching $\mathcal{M}_{\varphi}$ for each cyclotomic polynomial $\varphi=\varphi_{d}$ (with $d\geq 2$). Then we have an explicit description of $\text{coker}\,\pi_{*}$ and $\ker\pi_{*}$ thanks to Proposition 4.5, and we obtain the following result.

Theorem 5.1**.**

Under the above hypothesis, the homology of $\mathbf{X}_{\mathbf{W}}$ with coefficients in the representation $R=\mathbb{Q}[q^{\pm 1}]$ is given by

[TABLE]

In particular the term $H_{m}(C^{0}_{*})$ gives the free part of the homology, and the other direct summands give the torsion part. The torsion part actually takes a very particular form, and we are going to highlight this in the following result.

Theorem 5.2**.**

Let ${\bf G_{W}}$ be an Artin group that admits a $\varphi$-precise matching for all cyclotomic polynomials $\varphi=\varphi_{d}$ (with $d\geq 2$). Then the homology $H_{*}(\mathbf{X}_{\mathbf{W}};R)$ does not have $\varphi^{k}$-torsion for $k\geq 2$.

∎

We are particularly interested in Artin groups of finite and affine type. When ${\bf G_{W}}$ is an Artin group of finite type with $n$ generators, $K$ is the full simplicial complex on $S\cong\{1,\dots,n\}$ and therefore $C^{0}_{*}$ has trivial homology in every dimension. Thus the formula of Theorem 5.1 reduces to

[TABLE]

When ${\bf G_{W}}$ is an Artin group of affine type with $n+1$ generators, $K$ is obtained from the full simplicial complex on $S\cong\{0,1,\dots,n\}$ by removing the single top-dimensional simplex. Then we have

[TABLE]

Remark 5.3**.**

When ${\bf G_{W}}$ is an Artin group of finite type, the corresponding reflection arrangement of hyperplanes $\mathcal{A}$ is finite. In this case it is well known that there is an $R$-module isomorphism between the twisted homology $H_{*}(\mathbf{X}_{\mathbf{W}};R)$ and the homology with constant coefficients $H_{*}(F;\mathbb{Q})$ of the Milnor fiber $F$ of $\mathcal{A}$ [Cal05]. The $q$-multiplication on the homology of $\mathbf{X}_{\mathbf{W}}$ corresponds to the action of the monodromy operator on the homology of $F$. If $N=|\mathcal{A}|$, the square of the defining polynomial of the arrangement is ${\bf W}$-invariant, thus the order of the monodromy of the Milnor fibration divides $2N$. It follows that the polynomial $q^{2N}-1$ must annihilate the homology. Since $q^{2N}-1$ is square-free in characteristic $0,$ the homology cannot have $\varphi^{k}$-torsion for $k\geq 2$. So the conclusion of Theorem 5.2 is not surprising in the case of Artin groups of finite type.

In the rest of this paper we are going to construct precise matchings for Artin groups of type $A_{n}$, $B_{n}$, $\smash{\tilde{A}_{n}}$ and $\smash{\tilde{C}_{n}}$. For each of these cases we are then going to: describe the critical simplices with respect to the constructed matching; find all alternating paths and incidence numbers between critical simplices; determine the ranks $\operatorname{rk}\delta^{\varphi}_{*}$ and use Theorem 5.1 to compute the homology $H_{*}(\mathbf{X}_{\mathbf{W}};R)$. The final results are stated in Theorem 3.1 (case $A_{n}$), Theorem 5.13 (case $B_{n}$), Theorem 5.16 (case $\smash{\tilde{A}_{n}}$), and Theorem 5.20 (case $\smash{\tilde{C}_{n}}$).

Let us first introduce some notation. We will have $S=\{1,2,\dots,n\}$ for the finite cases ($A_{n}$ and $B_{n}$), and $S=\{0,1,2,\dots,n\}$ for the affine cases ($\smash{\tilde{A}_{n}}$ and $\smash{\tilde{C}_{n}}$). A simplex $\sigma\subseteq S$ will be also represented as a string of bits $\epsilon_{i}\in\{0,1\}$ (for $i\in S$), where $\epsilon_{i}=1$ if $i\in S$ and $\epsilon_{i}=0$ if $i\not\in S$. For example, if $S=\{1,2,3,4\}$, the string representation of $\sigma=\{1,2,4\}$ is $1101$. Also, for $\sigma\subseteq S$ and $v\in S$, let

[TABLE]

(it can be regarded as the bitwise xor between the string representation of $\sigma$ and the string with $\epsilon_{v}=1$ and $\epsilon_{i}=0$ for $i\neq v$).

5.1. Case $A_{n}$

Many properties of the homology $H_{*}(\mathbf{X}_{\mathbf{W}};R)$ in the case $A_{n}$ have been thoroughly discussed in Section 3. Using precise matchings we are going to obtain a new proof of the formula for the homology of braid groups (Theorem 3.1).

Let $S=\{1,2,\dots,n\}$ be the set of vertices of the Coxeter graph of type $A_{n}$, as in Section 3, Figure 1. In this case $K$ is the full simplicial complex on $S$. Fix now an integer $d\geq 2$ and set $\varphi=\varphi_{d}$. The $\varphi$-weight of a simplex $\sigma\in K$ has been computed in Section 3 and is as follows:

[TABLE]

where $n_{i}$ is the size of the $i$-th (linear) connected component of the subgraph induced by $\sigma$, and $\omega_{\varphi}(A_{k})$ stands for the $\varphi$-weight of a connected component of type $A_{k}$, given by

[TABLE]

Fix also an integer $f$ with $0\leq f\leq d-1$. Let $K^{A}_{n,f}\subseteq K$ be as follows:

[TABLE]

Notice that $K^{A}_{n,0}=K$, and $K^{A}_{n,f}$ is not a subcomplex of $K$ for $f\geq 1$ (but it is still a subposet of $K$, so it makes sense to define a matching on it). We are going to construct a $\varphi$-precise matching on $K^{A}_{n,f}$. In particular, for $f=0$, we will get a $\varphi$-precise matching for $K$. The precise matchings on $\smash{K^{A}_{n,f}}$ for $f\geq 1$ will become useful when treating the cases $B_{n}$, $\smash{\tilde{A}_{n}}$ and $\smash{\tilde{C}_{n}}$.

For a fixed $d$, the matching will be constructed recursively in $n$ and $f$ for $n\geq 0$ and $0\leq f\leq d-1$. We will write $K_{n,f}$ for $K^{A}_{n,f}$ throughout Section 5.1. The matching is as follows.

1. (a)

If $\{1,\dots,d-1\}\subseteq\sigma$ then match $\sigma$ with $\sigma\veebar d$ (unless $n=d-1$, in which case $\sigma$ is critical). Here $\sigma$ is matched with a simplex which also occurs in case (a). Notice that for $f=d-1$ case (a) always applies, thus in the subsequent cases we can assume $f\leq d-2$. 2. (b)

Otherwise, if $n=f$ then $\sigma$ is critical. 3. (c)

Otherwise, if $f+1\in\sigma$ then match $\sigma$ with $\sigma\setminus\{f+1\}$. Notice that $\sigma\setminus\{f+1\}$ occurs in case (d). 4. (d)

Otherwise, if $\{f+2,\dots,d-1\}\nsubseteq\sigma$ then match $\sigma$ with $\sigma\cup\{f+1\}$. Notice that $\sigma\cup\{f+1\}$ occurs in case (c). 5. (e)

We are left with the simplices $\sigma$ such that $\{1,\dots,f,f+2,\dots,d-1\}\subseteq\sigma$ and $f+1\not\in\sigma$. If we ignore the vertices $1,\dots,f+1$ we are left with the simplices on the vertex set $\{f+2,\dots,n\}$ which contain $f+2,\dots,d-1$; relabeling the vertices, these are the same as the simplices on the vertex set $\{1,\dots,n-f-1\}$ which contain $1,\dots,d-2-f$. Then construct the matching recursively as in $K_{n-f-1,\;d-2-f}$.

Example 5.4**.**

For $n=5$, $d=3$ and $f=1$, $K_{n,f}$ contains $2^{4}=16$ simplices of which $14$ are matched and $2$ are critical. For instance, consider the simplex $\sigma=\{1,4,5\}$. Case (e) applies because $1\in\sigma$ and $2\not\in\sigma$; the recursion requires us to consider the new simplex $\sigma^{\prime}=\{2,3\}\in K_{3,0}$. Again case (e) applies because $1\not\in\sigma^{\prime}$ and $2\in\sigma^{\prime}$; the recursion requires us to consider the new simplex $\sigma^{\prime\prime}=\{1,2\}\in K_{2,1}$. Finally case (a) applies because $\{1,2\}\subseteq\sigma^{\prime\prime}$, and $\sigma^{\prime\prime}$ is critical because $\sigma^{\prime\prime}=\{1,2\}$. Therefore $\sigma=\{1,4,5\}\in K_{5,1}$ is also critical. See Table 1 for an explicit description of the matching for $K_{5,1}$, $d=3$.

Remark 5.5**.**

A peculiarity of this matching is that, if $\sigma^{\prime}\to\tau^{\prime}$ is in the matching, then $\sigma^{\prime}=\tau^{\prime}\cup\{v\}$ with $v\equiv f+1$ or $v\equiv 0\;(\text{mod}\;d)$. This can be easily checked by induction.

Lemma 5.6**.**

The matching described above is an acyclic weighted matching on $K_{n,f}$.

Proof.

Part 1: the matching is acyclic. The proof is by induction on $n$, the case $n=0$ being trivial. For $n\leq f$, in $K_{n,f}$ there are either $1$ or [math] simplices. Assume from now on $n>f$. Let $P=\{p_{\text{a}},p_{\text{e}},p_{\text{c,d}}\}$ be a three-element totally ordered poset, with the order given by $p_{\text{a}}>p_{\text{e}}>p_{\text{c,d}}$. Consider the map $\eta\colon K_{n,f}\to P$ which sends $\sigma\in K_{n,f}$ to the $p_{\text{x}}$ such that $\sigma$ occurs in case (x) (here cases (c) and (d) are united). For instance, $\eta(\sigma)=p_{\text{a}}$ if and only if $\{1,\dots,d-1\}\subseteq\sigma$. Notice that the map $\eta$ is compatible with the matching, i.e. two matched simplices lie in the same fiber of $\eta$.

Let us prove that $\eta$ is a poset map. Given two simplices $\sigma\geq\tau$, we want to prove that $\eta(\sigma)\geq\eta(\tau)$. If $\eta(\tau)=p_{\text{a}}$ then $\{1,\dots,d-1\}\subseteq\tau\subseteq\sigma$, so $\eta(\sigma)=p_{\text{a}}$ also. If $\eta(\tau)=p_{\text{e}}$ then $\{1,\dots,f,f+2,\dots,d-1\}\subseteq\tau\subseteq\sigma$, thus $\eta(\sigma)\in\{p_{\text{a}},p_{\text{e}}\}$. Finally, if $\eta(\tau)=p_{\text{c,d}}$ there is nothing to prove.

Since $\eta$ is a poset map, our matching is acyclic if and only if it is acyclic on each fiber of $\eta$. The restriction of the matching to $\eta^{-1}(p_{\text{a}})$ consists of edges $\sigma\to\tau$ with $\sigma=\tau\cup\{d\}$, so there is no alternating path of length $\geq 4$ and the matching is acyclic. The restriction to $\eta^{-1}(p_{\text{c,d}})$ is acyclic for the same reason (it is always the vertex $f+1$ which is added or removed). The restriction to $\eta^{-1}(p_{\text{e}})$ is acyclic by induction.

Part 2: the matching is weighted. The proof is by induction on $n$, the case $n=0$ being trivial. In case (a), $v_{\varphi}(\sigma)=v_{\varphi}(\sigma\veebar d)$ by what we have already said in Section 3 (see the proof of Theorem 3.6). In cases (c) and (d), both the simplices $\sigma$ and $\sigma\veebar(f+1)$ do not contain $\{1,\dots,d-1\}$. Thus the (possibly empty) connected component of $1$ has size $\leq d-2$ and does not contribute to the weight. Therefore $v_{\varphi}(\sigma)=v_{\varphi}(\sigma\veebar(f+1))$. Finally, suppose that $\sigma$ and $\tau$ are simplices that occur in case (e). Let $\hat{\sigma},\hat{\tau}\in K_{n-f-1,\;d-2-f}$ be the simplices obtained ignoring the vertices $1$, …, $f+1$, as described above. Since $f\leq d-2$ we have that

[TABLE]

so we are done by induction on $n$. ∎

The critical simplices of the matching on $K_{n,f}$ are quite simple to describe. If $f\leq d-2$, $n>f$ and $n\equiv f$ or $-1\;(\text{mod}\;d)$, there are 2 critical simplices, one of weight $1$ and one of weight [math]. If $n=f$ there is 1 critical simplex (in fact there is only one simplex in $K_{n,f}$). In all the other cases the matching has no critical simplices. We are going to prove this in the following theorem. See Table 2, and Figures 4 and 5, for an explicit description of the critical simplices. Notice in particular that, when there are 2 critical simplices, one is a face of the other in $K_{n,f}$.

Theorem 5.7** (Critical simplices in case $A_{n}$).**

The critical simplices for the matching on $K_{n,f}$ are those listed in Table 2. In particular the matching is always precise, and has no critical simplices if $n\not\equiv f,-1\;(\text{mod}\;d)$ or $n>f=d-1$. In addition, when there are two critical simplices there is only one alternating path between them (the trivial one).

Proof.

Part 1: critical simplices. The proof is by induction on $n$, the case $n=0$ being trivial (there is one critical simplex for $f=0$, namely the empty simplex $\varnothing$, and no critical simplices for $f>0$). Let $\sigma\in K_{n,f}$ be a simplex. Let us consider each of the five cases that can occur in the construction of the matching.

1. (a)

We have $\{1,\dots,d-1\}\subseteq\sigma$. Then $\sigma$ is critical if and only if $n=d-1$. If $n=d-1$ then $\sigma$ is indeed listed in Table 2, as the first of the two critical simplices in the case $n=kd-1$ (here $k=1$). Conversely, the only simplex of Table 2 which contains $\{1,\dots,d-1\}$ is the first one of the case $n=kd-1$ when $k=1$. 2. (b)

We have $n=f$ and $\sigma=\{1,\dots,f\}$. Then $\sigma$ is critical, and it is indeed listed as the second of the two critical simplices in the case $n=kd+f$ (here $k=0$). 3. (c)

We have $\{1,\dots,f+1\}\subseteq\sigma$. Then $\sigma$ is not critical. The only listed simplex which contains $\{1,\dots,f+1\}$ is the first one of the case $n=kd-1$ for $k=1$, but it is equal to $\{1,\dots,d-1\}$ and so it must be different from $\sigma$. 4. (d)

We have $f+1\not\in\sigma$ and $\{f+2,\dots,d-1\}\nsubseteq\sigma$. Then $\sigma$ is not critical. It is easy to check that all the listed simplices $\tau$ with $n>f$ and $f+1\not\in\tau$ satisfy $\{f+2,\dots,d-1\}\subseteq\tau$, so none of them can be equal to $\sigma$. 5. (e)

We have $\{1,\dots,f,f+2,\dots,d-1\}\subseteq\sigma$ and $f+1\not\in\sigma$. Then $\sigma$ is critical if and only if the simplex $\hat{\sigma}\in K_{n-f-1,\;d-2-f}$ is critical, where $\hat{\sigma}$ is constructed from $\sigma$ by ignoring the first $f+1$ vertices. By induction, $\hat{\sigma}$ is critical if and only if it is listed in Table 2. By taking the simplices of Table 2 for $K_{n-f-1,\;d-2-f}$ and adding $1^{f}0$ at the beginning, one obtains exactly the simplices of Table 2 for $K_{n,f}$ (but the two cases are exchanged). This concludes the induction argument.

For fixed $n$ and $f$, the quantity $|\sigma|-v_{\varphi}(\sigma)$ is constant among the critical simplices $\sigma$. More precisely, it is equal to $n-2k$ for $n=kd+f$ (except for $f=d-1$) and to $n-2k+1$ for $n=kd-1$. Therefore the matching is precise.

Part 2: alternating paths. We want to prove that the only alternating path between critical simplices is the trivial one. Consider the case $n=kd+f$ for $k\geq 1$ (the case $n=kd-1$ is analogous). Let $\sigma$ and $\tau$ be the two critical simplices (the ones of Figure 4). Notice that $\sigma=\tau\cup\{n-f\}$. To establish a contradiction suppose we have a non-trivial alternating path

[TABLE]

Since $\tau_{1}\neq\tau$, we must have $\sigma=\tau_{1}\cup\{v_{1}\}$ with $v_{1}\neq n-f$. Notice that $n-f$ is the only vertex $v\in\sigma$ which satisfies $v\equiv f+1$ or $v\equiv 0\;(\text{mod}\;d)$. Therefore, by Remark 5.5, the vertex $v_{1}$ will never be recovered throughout the alternating path. This is a contradiction since $v_{1}\in\tau$. ∎

As a consequence we can compute the homology $H_{*}({\bf G_{W}};R)$ for ${\bf G_{W}}$ of type $A_{n}$. This gives a proof of Theorem 3.1.

Proof of Theorem 3.1 (Homology in case $A_{n}$).

We simply need to apply the formula given by Theorem 5.1 using our precise matching on $K=K_{n,0}$. Since $f=0$, there are critical simplices for $n=kd$ or $n=kd-1$. The boundary $\delta_{m+1}^{\varphi}$ is non-trivial only for

[TABLE]

In both cases for $m=k(d-2)$ we have $\operatorname{rk}\delta_{m+1}^{\varphi}=1$, and all the other boundaries are trivial. Theorem 3.1 follows. ∎

Remark 5.8**.**

A relation with independence complexes can be found for $K_{n,f}$ also when $1\leq f\leq d-2$. Indeed, choosing a suitable weighted matching (similar to the one of Theorem 3.6), the set of critical simplices of positive weight in $K_{n,f}$ is

[TABLE]

5.2. Case $B_{n}$

Consider now a Coxeter graph of type $B_{n}$, as in Figure 6. In this case $K$ is again the full simplex on vertices $\{1,2,\dots,n\}$. The Poincaré polynomial of a Coxeter group of type $B_{k}$ is given by

[TABLE]

where, for a given cyclotomic polynomial $\varphi=\varphi_{d}$, we have

[TABLE]

(see for example [BB06]). Then we can compute the $\varphi$-weight of any simplex $\sigma\in K$ by looking at the subgraph $\Gamma(\sigma)$ induced by $\sigma$. Let $\Gamma_{1}(\sigma)$, …, $\Gamma_{m}(\sigma)$ be the connected components of $\Gamma(\sigma)$, with cardinality $n_{1}$, …, $n_{m}$ respectively, where $\Gamma_{1}(\sigma)$ is the (possibly empty) component that contains the vertex $1\in S$. Then

[TABLE]

The situation is quite different depending on the parity of $d$. If $d$ is odd, the $\varphi$-weight of a $B_{k+1}$ component is equal to the $\varphi$-weight of an $A_{k}$ component:

[TABLE]

For this reason it is possible to construct a very simple matching on $K$: match any simplex $\sigma\in K$ with $\sigma\veebar 1$.

Lemma 5.9**.**

The matching constructed above for $d$ odd is an acyclic weighted matching on $K$, with no critical simplices.

Proof.

Clearly the matching is acyclic and there are no critical simplices. Let us proof that the matching is weighted. Let $\sigma\in K$. The only difference between $\Gamma(\sigma)$ and $\Gamma(\sigma\veebar 1)$ is the leftmost connected component, which in one case is of type $B_{k+1}$ and in the other case is of type $A_{k}$. Therefore $v_{\varphi}(\sigma)=v_{\varphi}(\sigma\veebar 1)$. ∎

Suppose from now on that $d$ is even. The simplicial complex $K$ is partitioned as

[TABLE]

Here $\Gamma_{1}(\sigma)$ is the (possibly empty) connected component of $\Gamma(\sigma)$ which contains the vertex $1$. Notice that each $K_{q}$ is a subposet of $K$, but not a subcomplex in general. For a given simplex $\sigma\in K_{q}$, let $|\Gamma_{1}(\sigma)|=q\frac{d}{2}+r$ with $0\leq r<\frac{d}{2}$. The matching on $K_{q}$ is as follows.

1. (a)

If $r\geq 1$ (i.e. $q\frac{d}{2}+1\in\sigma$) then match $\sigma$ with $\sigma\setminus\left\{q\frac{d}{2}+1\right\}$. 2. (b)

If $r=0$ (i.e. $q\frac{d}{2}+1\not\in\sigma$) and $\left\{q\frac{d}{2}+2,\dots,(q+1)\frac{d}{2}\right\}\nsubseteq\sigma$, then match $\sigma$ with $\sigma\cup\left\{q\frac{d}{2}+1\right\}$ (unless $n=q\frac{d}{2}$, in which case $\sigma$ is critical). 3. (c)

We are left with the simplices $\sigma$ for which neither (a) nor (b) apply, i.e. with $r=0$ and $\smash{\left\{q\frac{d}{2}+2,\dots,(q+1)\frac{d}{2}\right\}\subseteq\sigma}$. Ignore the first $q\frac{d}{2}+1$ vertices and relabel the remaining ones from $1$ to $\smash{n-q\frac{d}{2}-1}$, so that we are left exactly with the simplices of $\smash{K^{A}_{n-q\frac{d}{2}-1,\;\frac{d}{2}-1}}$. Then construct the matching on $\smash{K^{A}_{n-q\frac{d}{2}-1,\;\frac{d}{2}-1}}$ as in Section 5.1.

Putting together the matchings on each $K_{q}$ we obtain a matching on the full simplicial complex $K$.

Example 5.10**.**

For $n=4$ and $d=4$, the simplicial complex $K$ contains $2^{4}=16$ simplices of which $12$ are matched and $4$ are critical. For instance, consider $\sigma=\{1,2\}\in K$. Then $q=1$ and $r=0$. Since $4\not\in\sigma$, case (b) occurs. Therefore $\sigma$ is matched with $\sigma\cup\{3\}=\{1,2,3\}$. See Table 3 for an explicit description of the matching in this case.

Lemma 5.11**.**

The matching constructed above for $d$ even is an acyclic weighted matching on $K$.

Proof.

Part 1: the matching is acyclic. The map $K\to(\mathbb{N},\leq)$ which sends $\sigma$ to $q=\left\lfloor\frac{|\Gamma_{1}(\sigma)|}{d/2}\right\rfloor$ is a poset map compatible with the matching, and its fibers are exactly the subsets $K_{q}$ for $q\in\mathbb{N}$. Therefore we only need to prove that the matching on each fiber $K_{q}$ is acyclic.

Let $P=\{p_{\text{c}},p_{\text{a,b}}\}$ be a two-element totally ordered poset with $p_{\text{c}}>p_{\text{a,b}}$. For a fixed $q\in\mathbb{N}$, consider the map $\eta\colon K_{q}\to P$ which sends $\sigma$ to the $p_{\text{x}}$ such that $\sigma$ occurs in case (x) (here cases (a) and (b) are united). Clearly $\eta$ is compatible with the matching. We want to prove that it is a poset map, and for this we only need to show that if $\eta(\tau)=p_{\text{c}}$ and $\tau\leq\sigma$ then $\eta(\sigma)=p_{\text{c}}$ also. We have that $\smash{\left\{1,\dots,q\frac{d}{2},q\frac{d}{2}+2,\dots,(q+1)\frac{d}{2}\right\}\subseteq\tau\subseteq\sigma}$. The simplex $\sigma$ cannot contain the vertex $\smash{q\frac{d}{2}+1}$, because otherwise we would have $\sigma\in K_{q+1}$. Therefore $\eta(\sigma)=p_{\text{c}}$.

On the fiber $\eta^{-1}(p_{\text{a,b}})$ the matching is acyclic because the same vertex $\smash{q\frac{d}{2}+1}$ is always added or removed. On the fiber $\eta^{-1}(p_{\text{c}})$ the matching is acyclic by Lemma 5.6. Therefore the entire matching on $K_{q}$ is acyclic.

Part 2: the matching is weighted. Let $\sigma\in K_{q}$ be a simplex which occurs either in case (a) or case (b). We want to show that $\smash{v_{\varphi}(\sigma)=v_{\varphi}\left(\sigma\veebar\left(q\frac{d}{2}+1\right)\right)}$. Suppose without loss of generality that $\sigma$ occurs in case (a), i.e. $r\geq 1$ and $q\frac{d}{2}+1\in\sigma$. Let $\tau=\sigma\setminus\left\{q\frac{d}{2}+1\right\}$. The only difference between $\Gamma(\sigma)$ and $\Gamma(\tau)$ is that $\Gamma(\sigma)$ has a $\smash{B_{q\frac{d}{2}+r}}$ component, whereas $\Gamma(\tau)$ has a $\smash{B_{q\frac{d}{2}}}$ component and an $A_{r-1}$ component instead. Since $\smash{1\leq r<\frac{d}{2}}$, we have that

[TABLE]

(the second equality holds because $r-1\leq\frac{d}{2}-1\leq d-2$). Therefore $v_{\varphi}(\sigma)=v_{\varphi}(\tau)$.

For simplices occurring in case (c), the matching only involves changes in the connected components not containing the first vertex (i.e. connected components of type $A_{k}$). Therefore Lemma 5.6 applies. ∎

Now we are going to describe the critical simplices. The matching has no critical simplices when $n$ is not a multiple of $\frac{d}{2}$. On the other hand, if $n=k\frac{d}{2}$, we have two families of critical simplices: $\sigma_{q}$ (for $0\leq q\leq k-2$) and $\sigma_{q}^{\prime}$ (for $0\leq q\leq k$). See Table 4 and Figure 7 for the definition of these simplices. For instance, in Example 3, the critical simplices are: $\sigma_{0}=\{2,3,4\}$, $\sigma_{2}^{\prime}=\{1,2,3,4\}$, $\sigma_{1}^{\prime}=\{1,2,4\}$ and $\sigma_{0}^{\prime}=\{1,3\}$. See also Table 5, where the critical simplices are listed by dimension.

Theorem 5.12** (Critical simplices in case $B_{n}$).**

The critical simplices for the matching on $K$ are those defined in Table 4. In particular the matching is always precise, and has no critical simplices if $d$ is odd or if $d$ is even and n\not\equiv 0\;\big{(}\text{mod}\;\frac{d}{2}\big{)}. In addition, if $d$ is even and n\equiv 0\;\big{(}\text{mod}\;\frac{d}{2}\big{)}, the incidence numbers between the critical simplices in the Morse complex are as follows:

[TABLE]

Proof.

Part 1: critical simplices. For $d$ odd there is nothing to prove. Suppose from now on that $d$ is even. For a given simplex $\sigma\in K$, let us consider each of the three cases that can occur in the construction of the matching.

1. (a)

We have $\smash{q\frac{d}{2}+1\in\sigma}$, and $\sigma$ is not critical. Indeed, none of the simplices of Table 4 contains the vertex $\smash{q\frac{d}{2}+1}$. 2. (b)

We have $\smash{q\frac{d}{2}+1\not\in\sigma}$ and $\smash{\left\{q\frac{d}{2}+2,\dots,(q+1)\frac{d}{2}\right\}\nsubseteq\sigma}$. In this case $\sigma$ is critical if and only if $\smash{n=q\frac{d}{2}}$. Indeed, the only simplex of this type in Table 4 is $\sigma_{k}^{\prime}$ (which occurs for $q=k$ i.e. $\smash{n=q\frac{d}{2}}$). 3. (c)

In the remaining case, we end up with the matching on $K^{A}_{n-q\frac{d}{2}-1,\;\frac{d}{2}-1}$. By Theorem 5.7, this matching admits critical simplices if and only if

[TABLE]

i.e. for \smash{n\equiv 0\;\big{(}\text{mod}\;\frac{d}{2}\big{)}}. Notice that if $\smash{f=\frac{d}{2}-1}$ then $\smash{d-2-f=\frac{d}{2}-1}$ also. Therefore, again by Theorem 5.7, for \smash{n\equiv 0\;\big{(}\text{mod}\;\frac{d}{2}\big{)}} the critical simplices are exactly the ones listed in Table 4.

For a fixed $\smash{n=k\frac{d}{2}}$, the quantity $|\sigma|-v_{\varphi}(\sigma)$ is constant among the critical simplices and is equal to $n-k$. Thus the matching is precise.

Part 2: incidence numbers. We are going to show that there is exactly one alternating path for each of the four pairs. Notice that, if $\sigma\to\tau$ is in the matching of $K_{q}$, then $\sigma=\tau\cup\{v\}$ with \smash{v\equiv 1\;\big{(}\text{mod}\;\frac{d}{2}\big{)}} and $\smash{v\geq q\frac{d}{2}+1}$. In particular, if at a certain point of an alternating path one removes a vertex $v$ with \smash{v\not\equiv 1\;\big{(}\text{mod}\;\frac{d}{2}\big{)}}, then that vertex will never be added again. But all the critical simplices contain every vertex $v$ with \smash{v\not\equiv 1\;\big{(}\text{mod}\;\frac{d}{2}\big{)}}, so any alternating path between critical simplices cannot ever drop a vertex $v$ with \smash{v\not\equiv 1\;\big{(}\text{mod}\;\frac{d}{2}\big{)}}.

Let us now consider each of the four pairs.

• •

$(\sigma_{q},\sigma_{q-1})$. We have that $\sigma_{q}=\sigma_{q-1}\cup\{v\}$ where $v=(q-1)\frac{d}{2}+1$, so there is the trivial alternating path $\sigma_{q}\to\sigma_{q-1}$ which contributes to the incidence number $[\sigma_{q}:\sigma_{q-1}]^{\mathcal{M}}$ by

[TABLE]

Suppose by contradiction that there exists some other (non-trivial) alternating path

[TABLE]

Let $\sigma_{q}=\tau_{1}\cup\{u\}$. By the previous considerations, we must have u\equiv 1\;\big{(}\text{mod}\;\frac{d}{2}\big{)}. If $u=(k-1)\frac{d}{2}+1$ then $\tau_{1}=\sigma_{q}^{\prime}$, but this is not possible since alternating paths stop at critical simplices. Similarly, if $u=(q-1)\frac{d}{2}+1$ then we would stop at $\tau_{1}=\sigma_{q-1}$. Therefore we must have $u\leq(q-2)\frac{d}{2}+1$. But then $\tau_{1}\in K_{q^{\prime}}$ with $q^{\prime}\leq q-2$, and by induction all the subsequent simplices in the alternating path must lie in

[TABLE]

In particular $\sigma_{q-1}\in K_{q^{\prime\prime}}$ for some $q^{\prime\prime}\leq q-2$, but this is a contradiction since $\sigma_{q-1}\in K_{q-1}$.

• •

$(\sigma_{q},\sigma_{q}^{\prime})$. This case is similar to the previous one, except for the fact that $\sigma_{q}=\sigma_{q}^{\prime}\cup\{v\}$ for $v=(k-1)\frac{d}{2}+1$. So the only alternating path is the trivial one, which contributes to the incidence number by

[TABLE]

• •

$(\sigma_{q}^{\prime},\sigma_{q-1}^{\prime})$. This case is also similar to the previous ones. Here we have $\sigma_{q}^{\prime}=\sigma_{q-1}^{\prime}\cup\{v\}$ with $v=(q-1)\frac{d}{2}+1$, so the contribution to the incidence number due to the trivial alternating path is

[TABLE]

• •

$(\sigma_{q}^{\prime},\sigma_{q-2})$. This case is more complicated because the only alternating path is non-trivial. Suppose we have an alternating path

[TABLE]

($m$ must be at least $1$ because $\sigma_{q-2}$ is not a face of $\sigma_{q}^{\prime}$ in $K$). Let $\sigma_{q}^{\prime}=\tau_{1}\cup\{v\}$, with v\equiv 1\;\big{(}\text{mod}\;\frac{d}{2}\big{)}. If $v=(q-1)\frac{d}{2}+1$ then $\tau_{1}=\sigma_{q-1}^{\prime}$ and we must stop because $\sigma_{q-1}^{\prime}$ is already critical. If $v\leq(q-3)\frac{d}{2}+1$ we fall into some $K_{q^{\prime}}$ with $q^{\prime}\leq q-3$ and it is not possible to reach $\sigma_{q-2}\in K_{q-2}$. Therefore necessarily $v=(q-2)\frac{d}{2}+1$. Then $\tau_{1}$ is matched with

[TABLE]

Then again, $\tau_{2}=\rho_{1}\setminus\{v_{2}\}$ and the only possibility for $v_{2}$ (in order to have $\tau_{2}\neq\tau_{1}$ and not to fall into some $K_{q^{\prime}}$ with $q^{\prime}\leq q-3$) is $v_{2}=(q-1)\frac{d}{2}+1$. Proceeding by induction we obtain that

[TABLE]

The path stops at $\tau_{k-q+1}=\sigma_{q-2}$, and its length is $m=k-q$. See Table 6 for an example. The contribution of this path to the incidence number is

[TABLE]

where

[TABLE]

(these formulas also hold for $\rho_{0}=\sigma_{q}^{\prime}$ and $\tau_{k-q+1}=\sigma_{q-2}$). Then

[TABLE]

Having a complete description of a precise matching on $K$, we can now compute the homology $H_{*}({\bf G_{W}};R)$ for ${\bf G_{W}}$ of type $B_{n}$. We recover the result of [DCPSS99].

Theorem 5.13** (Homology in case $B_{n}$ [DCPSS99]).**

For an Artin group ${\bf G_{W}}$ of type $B_{n}$, we have

[TABLE]

Proof.

For a fixed $\varphi=\varphi_{d}$, we need to compute the boundary $\delta_{m+1}^{\varphi}$. Assume that $n=k\frac{d}{2}$, otherwise there are no critical simplices. In top dimension ($m+1=n$) we have $\operatorname{rk}\delta_{n}^{\varphi}=1$, because $[\sigma_{k}^{\prime}:\sigma_{k-1}^{\prime}]\neq 0$. For $m\leq n-k-1$ the boundary $\delta_{m+1}^{\varphi}$ vanishes, because there are no critical simplices in dimension $\leq n-k-1$. For $m=n-k$ we have $\operatorname{rk}\delta_{m+1}^{\varphi}=1$, because $[\sigma_{1}^{\prime}:\sigma_{0}^{\prime}]\neq 0$. Finally, for $n-k+1\leq m\leq n-2$, we have (set $q=m-n+k$):

[TABLE]

because

[TABLE]

To summarize, we have: $\operatorname{rk}\delta_{m+1}^{\varphi}=1$ for $n-k\leq m\leq n-1$, and $\operatorname{rk}\delta_{m+1}^{\varphi}=0$ otherwise. We conclude applying Theorem 5.1. ∎

5.3. Case $\tilde{A}_{n}$

Consider now the case of affine Artin groups of type $\tilde{A}_{n}$ (see Figure 8 for a picture of the corresponding Coxeter graph). Recall that, for affine Artin groups, the simplicial complex $K$ consists of all the simplices $\sigma\subseteq S=\{0,\dots,n\}$ except for the full simplex $\sigma=\{0,\dots,n\}$. For any $\sigma\in K$, the induced subgraph $\Gamma(\sigma)$ consists only of connected components of type $A_{k}$. The weight $v_{\varphi}(\sigma)$ is then computed as in Section 5.1.

If $\sigma$ is a simplex of $K$ then $\sigma$ misses at least one vertex of $\{0,1,\dots,n\}$. Define $h$ to be the first vertex missing from $\sigma$ reading counterclockwise from [math]. Then $\sigma$ belongs to a unique

[TABLE]

In other words, the subsets $K_{h}$ form a partition of $K$:

[TABLE]

Construct a matching on a fixed $K_{h}$ as follows. Let $h=qd+r$ with $0\leq r\leq d-1$, and let $m=n-qd$. Consider the following clockwise relabeling of the vertices: $r-1$ becomes $1$, $r-2$ becomes $2$, …, [math] becomes $r$, $n$ becomes $r+1$, …, $h+2$ becomes $m-1$, $h+1$ becomes $m$ (the vertices $r$, $r+1$, …, $h$ are forgotten). This relabeling induces a poset isomorphism

[TABLE]

Then equip $K_{h}$ with the pull-back of the matching on $K^{A}_{m,r}$ defined in Section 5.1 (for $h=n$ the only simplex in $K_{h}$ is critical).

Lemma 5.14**.**

The matching defined above is an acyclic weighted matching on $K$.

Proof.

Consider the map $\eta\colon K\to(\mathbb{N},\leq)$ which sends $\sigma\in K$ to $\min\{h\in\mathbb{N}\mid h\not\in\sigma\}$. Then $\eta$ is a poset map with fibers $\eta^{-1}(h)=K_{h}$. The matching on each $K_{h}\cong K^{A}_{m,r}$ is acyclic by Lemma 5.6, therefore the whole matching on $K$ is acyclic.

Fix $h\in\{0,\dots,n\}$. The bijection $K_{h}\xrightarrow{\cong}K^{A}_{m,r}$ is such that, if $\sigma\mapsto\hat{\sigma}$, then $v_{\varphi}(\sigma)=v_{\varphi}(\hat{\sigma})+q$. Indeed, $\Gamma(\sigma)$ is obtained from $\Gamma(\hat{\sigma})$ by adding $qd$ vertices to one (possibly empty) connected component, and this increases the weight by $q$. Then, since the matching on $K^{A}_{m,r}$ is weighted, its pull-back also is. ∎

Theorem 5.15** (Critical simplices in case $\tilde{A}_{n}$).**

The critical simplices for the matching on $K$ are those listed in Table 7. The only non-trivial incidence numbers between critical cells are:

[TABLE]

In particular the matching is precise.

Proof.

Part 1: critical simplices. For $n\not\equiv-1\;(\text{mod}\;d)$, the matching on $K_{h}$ has critical simplices only for $h\equiv n\;(\text{mod}\;d)$. If we write $n=kd+r$ with $0\leq r\leq d-2$, then $K_{qd+r}\cong K^{A}_{n-qd,\,r}$ has two critical simplices for $0\leq q\leq k-1$ and one critical simplex for $q=k$. These simplices are the ones listed in Table 7.

Suppose now that $n=kd-1$. For any $h=qd+r$, with $0\leq r\leq d-2$, the complex $\smash{K_{qd+r}\cong K^{A}_{n-qd,\,r}}$ has two critical simplices because $n-qd\equiv-1\;(\text{mod}\;d)$. Moreover, for $h=n$ the complex $\smash{K_{n}\cong K^{A}_{d-1,\,d-1}}$ has one critical simplex. In the remaining cases the matching on $K_{h}$ has no critical simplices. Again, the critical simplices are those listed in Table 7.

Part 2: incidence numbers for $n=kd+r$. We want to find the incidence numbers between critical simplices of consecutive dimensions. We start with the case $n=kd+r$, with $0\leq r\leq d-2$. First, let us look for alternating paths from $\tau_{q}$ to $\tau_{q}^{\prime}$ (for $0\leq q\leq k-1$). Set $h=qd+r$. Suppose we have one such path:

[TABLE]

If at some point a vertex $v\in\{0,...,h-1\}$ is removed, then the path falls into some $K_{h^{\prime}}$ with $h^{\prime}<h$ and can never return in $K_{h}$. Therefore the path must be entirely contained in $\smash{K_{h}\cong K^{A}_{n-qd,\,r}}$, and by Theorem 5.7 it must be the trivial path $\tau_{q}\rhd\tau_{q}^{\prime}$. Thus $[\tau_{q}:\tau_{q}^{\prime}]=\pm 1$.

The other pairs of critical simplices in consecutive dimensions are $(\tau_{q+1}^{\prime},\tau_{q})$ for $0\leq q\leq k-1$. There is a trivial path $\tau_{q+1}^{\prime}\rhd\tau_{q}$ which consists in removing the vertex $h=qd+r$, and contributes to the incidence number by

[TABLE]

Suppose we have some other (non-trivial) alternating path:

[TABLE]

Let $\rho_{m}=\tau_{q}\cup\{v\}$. If $v=h$ then $\rho_{m}=\tau_{q+1}^{\prime}$, which is excluded. Then $v$ must be one of the other vertices that do not belong to $\tau_{q}$. They are of the form $v=n-sd$ (for $0\leq s\leq k-q-1$) or $v=n-(d-1-r)-sd$ (for $0\leq s\leq k-q-2$). If $v=n-(d-1-r)-sd$, then $\rho_{m}$ is matched with $\rho_{m}\cup\{n-(s+1)d\}$. This is impossible because $\rho_{m}$ must be matched with some $\zeta_{m}\lhd\rho_{m}$. Similarly, if $v=n-sd$ with $0\leq s\leq k-q-2$, then $\rho_{m}$ is matched with $\rho_{m}\cup\{n-(d-1-r)-sd\}$ and not with some $\zeta_{m}\lhd\rho_{m}$. The only remaining possibility is $v=n-sd$ with $s=k-q-1$, i.e. $v=n-(k-q-1)d=(q+1)d+r$. In this case $\rho_{m}$ is matched with $\zeta_{m}=\rho_{m}\setminus\{qd+2r+1\}$. Going on with the same argument, we have exactly one way to continue the alternating path (from right to left in (2)), and we eventually end up with $\tau_{q+1}^{\prime}$. From left to right, the obtained alternating path is the following:

[TABLE]

The length of the path is $m=2(k-q)$. Apart from $qd+r$, which is the vertex in $\tau_{q+1}^{\prime}\setminus\tau_{q}$, the other vertices are added and removed exactly once during the path. If a certain $v$ is added in some $\rho_{i}$ and removed in some $\zeta_{j}$ (the removal might possibly come before the addition), then

[TABLE]

This is true because, except for $v=n$, between the addition and the removal of $v$ exactly one vertex $u$ with $u<v$ has been added/removed. Namely, between the addition and the removal of a vertex $v=jd+2r+1$ (for $q\leq j\leq k-1$) the vertex $u=jd+r$ is added/removed, and between the addition of the removal of a vertex $v=jd+r$ (for $q+1\leq j\leq k-1$) the vertex $u=(j-1)d+2r+1$ is added. Therefore the alternating path (2) contributes to the incidence number by

[TABLE]

Finally, the incidence number is given by

[TABLE]

Part 3: incidence numbers for $n=kd-1$. Consider a generic alternating path starting from $\bar{\sigma}=\{0,\dots,n-1\}$:

[TABLE]

Let $\bar{\sigma}=\zeta_{1}\cup\{v\}$. If $n-d+1\leq v\leq n-1$ then $\zeta_{1}=\sigma_{k-1,r}^{\prime}$ for $r=n-1-v$. Therefore we have trivial alternating paths from $\bar{\sigma}$ to any of the $\sigma_{k-1,r}^{\prime}$. If $v\leq n-d$ then $\zeta_{1}\in K_{h}$ with $h\leq n-d=(k-1)d-1$. None of these $K_{h}$’s contains critical simplices $\zeta_{m+1}$ with $|\zeta_{m+1}|=n-1$, and the alternating path cannot return in any $K_{h^{\prime}}$ with $h^{\prime}\geq(k-1)d$. Thus there are no other alternating paths from $\bar{\sigma}$ to critical simplices of $K$. Then the non-trivial incidence numbers involving $\bar{\sigma}$ are:

[TABLE]

Consider now a generic alternating path from $\sigma_{q,r_{1}}$ to $\sigma_{q,r_{2}}^{\prime}$:

[TABLE]

Let $\rho_{m}=\sigma_{q,r_{2}}^{\prime}\cup\{v\}$. Adding $v$ to $\sigma_{q,r_{2}}^{\prime}$ causes the creation of a connected component $\Gamma_{i}$ with $|\Gamma_{i}|\equiv-1\;(\text{mod}\;d)$. This means that $\rho_{m}$ is matched with a simplex of higher dimension, or is not matched at all (this happens for $v=(q+1)d-1$). Therefore the alternating path must be trivial, and it occurs only for $r_{1}=r_{2}$. Then the non-trivial incidence numbers of the form $[\sigma_{q,r_{1}}:\sigma_{q,r_{2}}^{\prime}]^{\mathcal{M}}$ are:

[TABLE]

Finally consider a generic alternating paths from $\sigma_{q+1,r_{1}}^{\prime}$ to $\sigma_{q,r_{2}}$:

[TABLE]

As before, we work backwards. Let $\rho_{m}=\sigma_{q,r_{2}}\cup\{v\}$. Apart from the choices $v=qd+r$ and $v=(q+1)d+r$, in all other cases $\rho_{m}$ has a connected component of size $\equiv-1\;(\text{mod}\;d)$ and this prevents the continuation of the alternating path. For $v=qd+r$ we obtain $\rho_{m}=\sigma_{q+1,r_{2}}^{\prime}$, so we have a trivial alternating path from $\sigma_{q+1,r_{2}}^{\prime}$ to $\sigma_{q,r_{2}}$. For $v=(q+1)d+r$, iterating the same argument, we have exactly one way to continue the alternating path (from right to left in (3)) and we end up with $\sigma_{q+1,r_{2}}^{\prime}$. From left to right, the path is as follows (set $r=r_{2}$):

[TABLE]

The length of the path is $m=2(k-q)-1$. As happened in the case $n=kd+r$, the contribution of this alternating path to the incidence number is given by

[TABLE]

The contribution of the trivial path $\sigma_{q+1,r}^{\prime}\rhd\sigma_{q,r}$ is given by $(-1)^{qd+r}$. Therefore

[TABLE]

We are now able to recover the result of [CMS08b] about the homology $H_{*}({\bf G_{W}};R)$ when ${\bf G_{W}}$ is an Artin group of type $\smash{\tilde{A}_{n}}$.

Theorem 5.16** (Homology in case $\tilde{A}_{n}$ [CMS08b]).**

For an Artin group ${\bf G_{W}}$ of type $\tilde{A}_{n}$, we have

[TABLE]

where $0\leq r\leq d-2$ in the second case.

Proof.

We apply Theorems 5.1 and 5.15. For $n=kd+r$ ($0\leq r\leq d-2$), the boundary map $\delta_{m+1}^{\varphi}$ has rank $1$ when $m+1=n-2(k-q)+1$ (for $0\leq q\leq k-1$, due to $\tau_{q}$); it has rank [math] otherwise. For $n=kd-1$, the boundary map $\delta_{m+1}^{\varphi}$ has rank $d-1$ if $m+1=n-2(k-q)+2$ (for $0\leq q\leq k-2$, due to the simplices $\sigma_{q,r}$) or if $m+1=n$ (due to $\bar{\sigma}$); it has rank [math] otherwise. ∎

5.4. Case $\tilde{C}_{n}$

The last case we consider is that of Artin groups of type $\smash{\tilde{C}_{n}}$. The corresponding Coxeter graph $\Gamma$ is shown in Figure 9. As in the $\smash{\tilde{A}_{n}}$ case, the simplicial complex $K$ consists of all the simplices $\sigma\subseteq S=\{0,\dots,n\}$ except for the full simplex $\sigma=\{0,\dots,n\}$. For any $\sigma\in K$, the subgraph $\Gamma(\sigma)$ of $\Gamma$ splits as a union of connected components of type $B_{k}$ (those containing the first or the last vertex) and of type $A_{k}$ (the remaining ones).

For each $m$ with $1\leq m\leq n$, let $K^{B}_{m}$ be the full simplicial complex on $\{1,\dots,m\}$, endowed with the weight function of $B_{m}$ (see Section 5.2). We are going to construct a precise matching for the case $\tilde{C}_{n}$ using the precise matching on $K^{B}_{m}$.

For $h\in\{0,\dots,n\}$ set

[TABLE]

As in the $\tilde{A}_{n}$ case, since every simplex $\sigma\in K$ misses at least one vertex, the subsets $K_{h}$ form a partition of $K$:

[TABLE]

Construct a matching on a fixed $K_{h}$ as follows. Ignore the vertices [math], …, $h$ and relabel the remaining ones from right to left: $n$ becomes $1$, $n-1$ becomes $2$, …, $h+1$ becomes $n-h$. This induces a poset isomorphism

[TABLE]

Then equip $K_{h}$ with the pull-back of the matching on $K^{B}_{n-h}$ constructed in Section 5.2 (for $h=n$ the single simplex in $K_{h}$ is critical).

Remark 5.17**.**

For $d$ odd we are simply matching $\sigma$ with $\sigma\veebar n$ for all $\sigma$ except $\sigma=\{0,\dots,n-1\}$.

Lemma 5.18**.**

The matching defined above is an acyclic weighted matching on $K$.

Proof.

As in the proof of Lemma 5.14, the subsets $K_{h}$ are the fibers of a poset map and the matching is acyclic on each $K_{h}\cong K^{B}_{n-h}$.

If a simplex $\sigma\in K_{h}$ is sent to $\hat{\sigma}$ by the isomorphism $K_{h}\xrightarrow{\cong}K^{B}_{n-h}$, then $v_{\varphi}(\sigma)=v_{\varphi}(\hat{\sigma})+\omega_{\varphi}(B_{h})$. Since the matching on $\smash{K^{B}_{n-h}}$ is weighted, the matching on $K_{h}$ also is. ∎

Theorem 5.19** (Critical simplices in case $\tilde{C}_{n}$).**

The critical simplices for the matching on $K$ are those listed in Table 8. In particular the matching is precise. In addition the only non-trivial incidence numbers between critical simplices in the Morse complex are as follows (for $d$ even and $n=k\frac{d}{2}+r$):

[TABLE]

where \alpha=(k-q_{2})\big{(}\frac{d}{2}-1\big{)}+q_{1}+r+\frac{d}{2} and $\beta=q_{1}\frac{d}{2}+r$.

Proof.

As we have already said, for $d$ odd there is exactly one critical simplex. Suppose from now on that $d$ is even. Let $\smash{n=k\frac{d}{2}+r}$, with $0\leq r\leq\frac{d}{2}-1$.

Part 1: the critical simplices. In $K_{h}$ there are critical simplices if and only if \smash{n-h\equiv 0\;\big{(}\text{mod}\;\frac{d}{2}\big{)}}, i.e. when $\smash{h=q_{1}\frac{d}{2}+r}$ for some $q_{1}$ (with $0\leq q_{1}\leq k$). By Theorem 5.12 there are two families of critical simplices: $\sigma_{q_{1},q_{2}}$ (for $0\leq q_{1}+q_{2}\leq k-2$) and $\sigma_{q_{1},q_{2}}^{\prime}$ (for $0\leq q_{1}+q_{2}\leq k$), as shown in Table 8 and Figure 10. In a fixed dimension $m=n-k+l$ ($0\leq l\leq k$) there are $2l+1$ critical simplices if $l\leq k-1$ and $l$ critical simplices if $l=k$. See Table 9.

Part 2: paths ending in $\sigma_{q_{1},q_{2}}$. Consider a generic alternating path starting from any critical cell $\rho_{0}$ and ending in a critical cell of the form $\sigma_{q_{1},q_{2}}$:

[TABLE]

Let $\rho_{m}=\sigma_{q_{1},q_{2}}\cup\{v\}$. If $\smash{v=q_{1}\frac{d}{2}+r}$ then $\rho_{m}=\sigma_{q_{1}+2,\,q_{2}}^{\prime}$ and the alternating path stops. If $\smash{v=n-q_{2}\frac{d}{2}}$ then the alternating path stops at $\rho_{m}=\sigma_{q_{1},\,q_{2}+1}$. Suppose $q_{1}+q_{2}\leq k-3$, otherwise there are no more cases. If $q_{1}\frac{d}{2}+r+d<v<n-q_{2}\frac{d}{2}$, then $\Gamma(\rho_{m})$ has at least one connected component of size $d-1$ and therefore $\rho_{m}$ is matched with a simplex of higher dimension; thus the path stops without having reached a critical simplex. If $v=q_{1}\frac{d}{2}+r+d$ then $\rho_{m}$ is matched with

[TABLE]

in the binary string notation. From here the path can continue in many ways. Let $\rho_{m-1}=\tau_{m}\cup\{w\}$.

• •

If $w=q_{1}\frac{d}{2}+r$, we end up with $\rho_{m-1}=\sigma_{q_{1}+1,\,q_{2}}$.

• •

If $w=q_{1}\frac{d}{2}+r+\frac{d}{2}$ we would be going back to $\rho_{m}$.

• •

If $w>q_{1}\frac{d}{2}+r+\frac{3}{2}d$, then $\Gamma(\rho_{m-1})$ has at least one connected component of size $d-1$ and therefore $\rho_{m-1}$ is matched with a simplex of higher dimension.

• •

If $w=q_{1}\frac{d}{2}+r+\frac{3}{2}d$, then $\rho_{m-1}$ is matched with

[TABLE]

By induction, repeating the same argument as above, this path can be continued in exactly one way and it eventually arrives at the critical simplex $\sigma_{q_{1},\,q_{2}+2}^{\prime}$. The path has length $m=k-q_{1}-q_{2}-2$ and is as follows.

[TABLE]

Part 3: paths ending in $\sigma_{q_{1},q_{2}}^{\prime}$. Consider now a generic alternating path starting from any critical cell $\rho_{0}$ and ending in a critical cell of the form $\sigma_{q_{1},q_{2}}^{\prime}$:

[TABLE]

As usual, let $\rho_{m}=\sigma_{q_{1},q_{2}}^{\prime}\cup\{v\}$. For the same reasons as above, there are only three possibilities: $v=q_{1}\frac{d}{2}+r$, $v=q_{1}\frac{d}{2}+r+\frac{d}{2}$, $v=n-q_{2}\frac{d}{2}$; in all the other cases, $\rho_{m}$ is matched with a simplex of higher dimension. If $v=q_{1}\frac{d}{2}+r$ then the path ends (to the left in (4)) at $\rho_{m}=\sigma_{q_{1}+1,\,q_{2}}^{\prime}$. If $v=q_{1}\frac{d}{2}+r+\frac{d}{2}$ then the path ends at $\sigma_{q_{1},q_{2}}$. Finally, if $v=n-q_{2}\frac{d}{2}$ then the path ends at $\sigma_{q_{1},\,q_{2}+1}^{\prime}$.

Part 4: incidence numbers. We have seven families of incidence numbers to compute, each coming from one of the alternating paths we have found.

• •

From $\sigma_{q_{1}+2,\,q_{2}}^{\prime}$ to $\sigma_{q_{1},q_{2}}$. The alternating path is trivial and consists in removing the vertex $v=q_{1}\frac{d}{2}+r$, so

[TABLE]

• •

From $\sigma_{q_{1},\,q_{2}+1}$ to $\sigma_{q_{1},q_{2}}$. Again the path is trivial, and it consists in removing $v=n-q_{2}\frac{d}{2}$. Therefore

[TABLE]

• •

From $\sigma_{q_{1}+1,\,q_{2}}$ to $\sigma_{q_{1},q_{2}}$. The path consists in removing $q_{1}\frac{d}{2}+r$, adding $q_{1}\frac{d}{2}+r+\frac{d}{2}$, and removing $q_{1}\frac{d}{2}+r+d$. Therefore

[TABLE]

• •

From $\sigma_{q_{1},\,q_{2}+2}^{\prime}$ to $\sigma_{q_{1},q_{2}}$. The path is the one we explicitly wrote at the end of Part 2. Notice that from $\rho_{i-1}$ to $\tau_{i}$ one removes the vertex $v_{i}=n-(q_{2}-i+1)\frac{d}{2}$, and from $\tau_{i}$ to $\rho_{i}$ one adds the vertex $v_{i}^{\prime}=n-(q_{2}-i-1)\frac{d}{2}$. Therefore

[TABLE]

Then we can compute the incidence number in the Morse complex:

[TABLE]

• •

From $\sigma_{q_{1}+1,\,q_{2}}^{\prime}$ to $\sigma_{q_{1},q_{2}}^{\prime}$. The path is trivial and consists of removing $v=q_{1}\frac{d}{2}+r$. Then

[TABLE]

• •

From $\sigma_{q_{1},q_{2}}$ to $\sigma_{q_{1},q_{2}}^{\prime}$. The path is trivial and consists in removing $v=q_{1}\frac{d}{2}+r+\frac{d}{2}$. Then

[TABLE]

• •

From $\sigma_{q_{1},\,q_{2}+1}^{\prime}$ to $\sigma_{q_{1},q_{2}}^{\prime}$. The path is trivial and consists in removing $v=n-q_{2}\frac{d}{2}$. Then

[TABLE]

We can finally compute the homology $H_{*}({\bf G_{W}};R)$ for Artin groups of type $\tilde{C}_{n}$.

Theorem 5.20** (Homology in case $\tilde{C}_{n}$).**

Let ${\bf G_{W}}$ be an Artin group of type $\tilde{C}_{n}$. Then the $\varphi_{d}$-primary component of $H_{*}({\bf G_{W}};R)$ is trivial for $d$ odd, and for $d$ even is as follows:

[TABLE]

where $n=k\frac{d}{2}+r$.

Proof.

In order to apply Theorem 5.1 we need to find the rank the boundary maps $\delta_{m+1}^{\varphi}$ of the Morse complex. For $d$ odd there is only one critical cell, thus all the boundaries vanish. Suppose from now on that $d$ is even, and let $\smash{n=k\frac{d}{2}+r}$. In order to have a non-trivial boundary $\delta_{m+1}^{\varphi}$ we must have at least one critical simplex both in dimension $m$ and in dimension $m+1$, thus $m=n-k+l$ with $0\leq l\leq k-1$ (see Table 9).

Case 1: $l\leq k-2$. We are going to prove that, for $l\leq k-2$, a basis for the image of $\delta=\delta_{m+1}^{\varphi}$ is given by

[TABLE]

By Theorem 5.19 we obtain the following formula for $\delta\sigma_{q_{1},q_{2}}$:

[TABLE]

where \alpha=(k-q_{2})\big{(}\frac{d}{2}-1\big{)}+q_{1}+r+\frac{d}{2} and $\beta=q_{1}\frac{d}{2}+r$. In this formula the $\sigma_{q_{1},\,q_{2}-1}$ (resp. $\sigma_{q_{1}-1,\,q_{2}}$) term vanishes if $q_{2}=0$ (resp. $q_{1}=0$). The term $\sigma_{q_{1},q_{2}}^{\prime}$ appears in $\delta\sigma_{q_{1},q_{2}}$ but not in any other element of $\mathcal{B}$, thus $\mathcal{B}$ is a linearly independent set. In addition, if $q_{1}+q_{2}=l+1$, we have that

[TABLE]

Therefore $\mathcal{B}$ generates the image of $\delta_{m+1}^{\varphi}$. Thus $\operatorname{rk}\delta_{m+1}^{\varphi}=|\mathcal{B}|=l+1$ for $l\leq k-2$.

Case 2: $l=k-1$. For $l=k-1$, i.e. $m=n-1$, the situation is a bit different because there are no critical simplices of the form $\sigma_{q_{1},q_{2}}$ with $q_{1}+q_{2}=l$. However we can still define

[TABLE]

for $q_{1}+q_{2}=l$, and

[TABLE]

The term $\sigma_{q_{1},q_{2}}^{\prime}$ appears in $\epsilon_{q_{1},q_{2}}$ but not in any other element of $\mathcal{B}^{\prime}$, thus $\mathcal{B}^{\prime}$ is a linearly independent set. As above we have that, for $q_{1}+q_{2}=l+1$,

[TABLE]

Then $\mathcal{B}^{\prime}$ generates (and so it is a basis of) the image of $\delta_{n}^{\varphi}$.

We have proved that, for $0\leq l\leq k-1$, the rank of $\delta_{m+1}^{\varphi}$ is equal to $l+1=m+k-n+1$. Then we conclude applying Theorem 5.1. ∎

Concluding remarks

Future works will focus on other families of Artin groups. In particular it seems that precise matchings can be constructed in all finite and affine cases (see [Pao17b]), possibly allowing explicit homology computations.

The methods developed in this paper are particularly powerful when the coefficients are over a PID, but Artin groups with roots of different lengths (e.g. $B_{n}$, $\smash{\tilde{B}_{n}}$, and $\smash{\tilde{C}_{n}}$) also admit natural representations over polynomial rings with more than one variable. We believe that some of our theory can be extended to such cases, and this can also be the aim of future works.

Acknowledgements

The authors are grateful to the anonymous referee for his/her useful comments and suggestions. This work was partially supported by Ministero dell’Istruzione, dell’Università e della Ricerca, and by University of Pisa, Project no. PRA_67.