On the homology of the commutator subgroup of the pure braid group

TL;DR

This paper investigates the homology of the commutator subgroup of the pure braid group, revealing infinite rank in certain homology groups and determining its cohomological dimension.

Contribution

It provides new insights into the homological structure of the commutator subgroup of pure braid groups, including explicit calculations of homology and cohomological dimension.

Findings

Homology groups contain free abelian groups of infinite rank for specified degrees.

Cohomological dimension of the commutator subgroup is exactly n-2 for n ≥ 2.

Homological properties are explicitly characterized for all n ≥ 2.

Abstract

We study the homology of $[P_n,P_n]$, the commutator subgroup of the pure braid group on $n$ strands, and show that $H_l([P_n,P_n])$ contains a free abelian group of infinite rank for all $1\leq l\leq n-2$. As a consequence we determine the cohomological dimension of $[P_n,P_n]$: for $n\geq 2$ we have $\mathrm{cd}([P_n,P_n])=n-2$.

Equations

$$F_n=\left\{(z_1,\dots ,z_n)\in {\mathbb{C}}^n|z_i\ne z_j\forall i\ne j\right\}.$$

$$\mathrm{cd}([P_n,P_n])=n-2.$$

$$\mathbb{C}\setminus \left\{n\text{mbox}points\right\}$$

$$1\to {\mathbb{Z}}^{\ast n}\to P_{n+1}\to P_n\to 1.$$

$${\psi }_{ij}:P_n\to P_2\simeq \mathbb{Z}.$$

$$F_n^{\log }=\left\{\left({\left(z_i\right)}_{1\le i\le n},{\left(w_{ij}\right)}_{1\le i<j\le n}\right)|z_j-z_i=e^{w_{ij}}\forall 1\le i<j\le n\right\}.$$

$$1\to {\pi }_1(F_n^{\log })\to {\pi }_1(F_n)\to {\mathbb{Z}}^{\left(\genfrac{}{}{0ex}{}{n}{2}\right)}\to 1.$$

$$H_{\ast }(F_n^{\log }){\otimes }_{R(n)}\mathbb{K}(n)=0.$$

$$\mathrm{H}\left(((z_i),(w_{ij}));\theta \right)=\left((e^{\theta \sqrt{-1}}{z}_i),(w_{ij}+\theta \sqrt{-1})\right).$$

$$\prod _{1\le i<j\le n}{t}_{ij}:F_n^{\log }\to F_n^{\log },$$

$$\tilde{R}(n)=\mathbb{Z}\left[{\tilde{t}}_{ij}^{\pm 1}|1\le i<j\le n,(i,j)\ne (1,2)\right]$$

$$\mathrm{H}(\left((z_i),(w_{ij})\right);t)=\left((e^{-t{w}_{12}}\cdot (z_i-t{z}_1)),(w_{ij}-t{w}_{12})\right).$$

$$[\partial ({\pi }_r):({\pi }_r^{\prime })]=(-1{)}^{l+sgn({\pi }_l;{\pi }_l^{\prime })}=\pm 1,$$

$${\Delta }^{n-k}\times {\Delta }^{|{\pi }_1|}\times \cdots \times {\Delta }^{|{\pi }_{n-k}|}.$$

$$[\partial ({\pi }_r,W_{ij}):({\pi }_r^{\prime },W_{ij}^{\prime })]=(-1{)}^{l+sgn({\pi }_l;{\pi }_l^{\prime })}=\pm 1.$$

$$\tilde{e}({\pi }_r)\subset F_{n-2}(\mathbb{C}\setminus \left\{0,1\right\})$$

$${\mathfrak{F}}_p/{\mathfrak{F}}_{p-1}\mathcal{M}{\widetilde{\mathfrak{Ch}}}_{\bullet }^{\log }\cong \underset{S}{⨁}\left(\tilde{R}(n{)}^{p!}{\otimes }_{R(S)}{\mathfrak{Ch}}_{\bullet }^{\log ,S}\right),$$

$$\tilde{\mathbb{K}}(n){\otimes }_{\tilde{R}(n)}{H}_{\ast }([P_n,P_n])$$

$$E_{p,q}^1=H_{p+q}\left({\mathfrak{F}}_p/{\mathfrak{F}}_{p-1}\left(\tilde{\mathbb{K}}(n){\otimes }_{\tilde{R}(n)}\mathcal{M}{\widetilde{\mathfrak{Ch}}}_{\bullet }^{\log }\right)\right),$$

$$H_{n-2}\left({\mathfrak{F}}_{n-2}/{\mathfrak{F}}_{n-3}\left(\tilde{\mathbb{K}}(n){\otimes }_{\tilde{R}(n)}\mathcal{M}{\widetilde{\mathfrak{Ch}}}_{\bullet }^{\log }\right)\right)\simeq \tilde{\mathbb{K}}(n{)}^{(n-2)!}.$$

$$\tilde{\mathbb{K}}(n){\otimes }_{\tilde{R}(n)}\tilde{R}(n){\otimes }_{R(S)}{\mathfrak{Ch}}_S$$

$$\tilde{\mathbb{K}}(n){\otimes }_{\mathbb{K}(S)}\mathbb{K}(S){\otimes }_{R(S)}{\mathfrak{Ch}}_S$$

$${\tau }_S=\left[\left(\prod _{i,j\in S;i<j}{t}_{ij}\right)-1\right]\in \tilde{R}(n)\subset R(n).$$

$${\tau }_n=\prod _S{\tau }_S\in \tilde{R}(n)\subset R(n)$$

$$\tilde{R}(n)\left[{{\tau }_n}^{-1}\right]{\otimes }_{R(S)}{H}_{\ast }\left({\mathfrak{Ch}}_{\bullet }^{\log ,S}\right)=0.$$

$$\tilde{R}(n)\left[{\tau }_n^{-1}\right]{\otimes }_{\tilde{R}(n)}{H}_{\ast }([P_n,P_n])$$

$${\psi }_k^n(z_1,\dots ,z_n)=(z_1,\dots ,z_k)\in F_k.$$

$${\sigma }_n^k(z_1,\dots ,z_k)=(z_1,\dots ,z_k,M+1,\dots ,M+n-k)\in F_n.$$

$${\left({\sigma }_n^k\right)}_{\ast }:H_{k-2}([P_k,P_k])\to H_{k-2}([P_n,P_n])$$

$$1\to P_n\to B_n\to {\mathfrak{S}}_n\to 1.$$

Full text

This work was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy (EXC-2047/1, 390685813).

1. Introduction

Let $n\geq 2$ and denote by $F_{n}$ the ordered configuration space of $n$ points in the complex plane:

[TABLE]

The pure braid group on $n$ strands is defined as $P_{n}=\pi_{1}(F_{n})$.

In [3] David Recio-Mitter and the author posed the question of determining the cohomological dimension of $[P_{n},P_{n}]$, the commutator subgroup of the pure braid group, and conjectured that, for $n\geq 2$,

[TABLE]

In this work we prove this conjecture by computing a large part of the homology of $[P_{n},P_{n}]$; in particular we prove that $H_{*}([P_{n},P_{n}])$ contains a free abelian group of infinite rank in all degrees $1\leq*\leq n-2$ (see Theorem 6.1, Corollary 6.2 and Theorem 7.1).

To the best of the author’s knowledge there is no result in the literature concerning the homology of $[P_{n},P_{n}]$ for large values of $n$; on the contrary the homology of the commutator subgroup of Artin’s full braid group [2] has been extensively studied [12, 17, 6, 4], as well as the homology of Milnor fibers of discriminant fibrations associated with other hyperplane arrangements in $\mathbb{C}^{n}$ [7, 8, 5, 19].

Our strategy is the following. We consider the Salvetti complex $Sal_{n}$ associated with the $n$-th braid arrangement: the cell complex $Sal_{n}$ is a classifying space for $P_{n}$, and it has a covering $Sal^{\log}_{n}$ which is a classifying space for $[P_{n},P_{n}]$. The group $P_{n}^{\mathrm{ab}}\simeq\mathbb{Z}^{\binom{n}{2}}$ acts on $Sal^{\log}_{n}$ by deck transformations, and the action is cellular: hence the associated cellular chain complex $\mathfrak{Ch}^{\log}_{\bullet}$ is a chain complex of modules over the commutative ring $\mathbb{Z}[P_{n}^{\mathrm{ab}}]$, and consequently the homology $H_{*}\left([P_{n},P_{n}]\right)$ is also a $\mathbb{Z}[P_{n}^{\mathrm{ab}}]$-module.

We replace $\mathfrak{Ch}^{\log}_{\bullet}$ with a homotopy equivalent subcomplex $\tilde{\mathfrak{Ch}}^{\log}_{\bullet}$; the chain complex $\tilde{\mathfrak{Ch}}^{\log}_{\bullet}$ is only invariant for the action of a certain subgroup $\mathbb{Z}^{\binom{n}{2}-1}\subset P_{n}^{\mathrm{ab}}$, and we restrict this action also in homology, i.e. we consider $H_{*}\left([P_{n},P_{n}]\right)$ as a module over the commutative ring $\mathbb{Z}\left[\mathbb{Z}^{{\binom{n}{2}}-1}\right]$.

We define a filtration on $\tilde{\mathfrak{Ch}}^{\log}_{\bullet}$; the associated Leray spectral sequence, after localisation to the quotient field of $\mathbb{Z}\left[\mathbb{Z}^{{\binom{n}{2}}-1}\right]$, collapses on its first page: more precisely we have $E^{1}_{p,q}=0$ for all $(p,q)\neq(n-2,0)$. This proves the statement for $H_{n-2}([P_{n},P_{n}])$ (see Theorem 6.1).

To prove the statement in lower degrees we consider the interaction between commutator subgroups of different pure braid groups (see Theorem 7.1).

2. Preliminaries

We recall some classical constructions and results about configuration spaces and pure braid groups.

For all $1\leq i\leq n+1$ there is a map $\varphi_{i}\colon F_{n+1}\to F_{n}$, which forgets the $i-$th point of each configuration. This is a fiber bundle with fiber the punctured plane $\mathbb{C}\setminus\left\{n\mbox{ points}\right\}$, called the Fadell-Neuwirth fibration (see [10]):

[TABLE]

The space $\mathbb{C}\setminus\left\{n\mbox{ points}\right\}$ is a classifying space for the free group on $n$ generators $\mathbb{Z}^{*n}$, in particular it is an aspherical space. An induction argument shows that $F_{n}$ is also aspherical, and therefore $F_{n}$ is a classifying space for its fundamental group $P_{n}$. We obtain a short exact sequence

[TABLE]

Definition 2.1**.**

For all $1\leq i<j\leq n$ there is a forgetful map $\psi_{ij}\colon F_{n}\to F_{2}$, which forgets all points of a configuration except the $i$-th and the $j$-th. This map of spaces induces a map, that we still call $\psi_{ij}$, on fundamental groups:

[TABLE]

The collection of all these maps gives a homomorphism of groups $\psi\colon P_{n}\to\mathbb{Z}^{\binom{n}{2}}$.

A classical result by Arnold [1] states that $\psi$ is the abelianisation homomorphism, i.e. $P_{n}^{\mathrm{ab}}\simeq\mathbb{Z}^{\binom{n}{2}}$ along the map induced by $\psi$. In this article we focus on the group $[P_{n},P_{n}]=\ker\psi$, the commutator subgroup of the pure braid group.

3. Two classifying spaces for $[P_{n},P_{n}]$

We introduce two convenient models for the classifying space of $[P_{n},P_{n}]$.

Definition 3.1**.**

We define the space $F_{n}^{\log}$. A point in $F_{n}^{\log}$ is determined by a configuration $(z_{1},\dots,z_{n})\in F_{n}$ together with a choice $w_{ij}\in\mathbb{C}$ of a logarithm of $(z_{j}-z_{i})$, for all $i<j$:

[TABLE]

This space has a topology as subspace of $\mathbb{C}^{n}\times\mathbb{C}^{\binom{n}{2}}$.

There is a covering map $p\colon F_{n}^{\log}\to F_{n}$, which forgets the numbers $w_{ij}$. The fiber is isomorphic to $\mathbb{Z}^{\binom{n}{2}}$: to see this fix a point $\left((z_{i}),(\bar{w}_{ij})\right)$ lying over some point $(z_{i})\in F_{n}$. Let $\left((z_{i}),(w_{ij})\right)$ be any other point lying over $(z_{i})$: then there are integers $(k_{ij})_{1\leq i<j\leq n}$ such that $w_{ij}-\bar{w}_{ij}=2\pi\sqrt{-1}k_{ij}$ for all $1\leq i<j\leq n$. Viceversa given integers $(k_{ij})_{1\leq i<j\leq n}$ one can define a point $\left((z_{i}),(w_{ij})\right)$ in the fiber of $(z_{i})$ by setting $w_{ij}=\bar{w}_{ij}+2\pi\sqrt{-1}k_{ij}$ for all $1\leq i<j\leq n$.

The last construction gives a free action of $\mathbb{Z}^{\binom{n}{2}}$ on $F_{n}^{\log}$; this is an action by deck transformations of $p$ and is transitive on fibers of $p$: therefore $\mathbb{Z}^{\binom{n}{2}}$ is the whole group of deck transformations of $p$ and there is a short exact sequence

[TABLE]

We can then conclude that $[P_{n},P_{n}]\subseteq\pi_{1}(F_{n}^{\log})$, because $[P_{n},P_{n}]$ is contained in the kernel of any map from $P_{n}$ to an abelian group.

On the other hand the maps $\psi_{ij}\colon F_{n}\to F_{2}$ lift to maps $\psi_{ij}^{\log}\colon F_{n}^{\log}\to F_{2}^{\log}$: the map $\psi_{ij}^{\log}$ is defined by forgetting all data except $z_{i},z_{j}$ and $w_{ij}$.

The space $F_{2}^{\log}$ is contractible: this is a particular case of Lemma 3.7, and can be checked also directly. Therefore $\pi_{1}(F_{n}^{\log})$ is a subgroup of $P_{n}$ contained in the kernel of all maps $\psi_{ij}$, i.e. $\pi_{1}(F_{n}^{\log})\subseteq[P_{n},P_{n}]$. We obtain the following lemma.

Lemma 3.2**.**

The space $F_{n}^{\log}$ is a classifying space for the group $[P_{n},P_{n}]$.

The action of $P_{n}^{\mathrm{ab}}$ on $F_{n}^{\log}$ induces an action of the ring $\mathbb{Z}[P_{n}^{\mathrm{ab}}]$ on $H_{*}(F_{n}^{\log})$, so our first attempt is to study $H_{*}(F_{n}^{\log})=H_{*}([P_{n},P_{n}])$ as a module over this ring.

Definition 3.3**.**

Let $R(n)=\mathbb{Z}[P_{n}^{\mathrm{ab}}]$ be the ring of Laurent polynomials in $\binom{n}{2}$ variables $\mathbb{Z}\left[t_{ij}^{\pm 1}|1\leq i<j\leq n\right]$. The variable $t_{ij}$ corresponds to the generator of $P_{n}^{\mathrm{ab}}\simeq\mathbb{Z}^{\binom{n}{2}}$ which is dual to the map $\psi_{ij}\colon P_{n}\to P_{2}$, i.e. for all $i<j$ and $k<l$ we have $\psi_{ij}(t_{kl})=\delta_{ik}\delta_{jl}$.

The ring $R(n)$ is a domain and we call $\mathbb{K}(n)$ its quotient field.

The following lemma tells us that $H_{*}(F_{n}^{\log})$ cannot be too large.

Lemma 3.4**.**

[TABLE]

Proof.

Consider the following homotopy $\mathrm{H}\colon F_{n}^{\log}\times[0,2\pi]\to F_{n}^{\log}$ of the space $F_{n}^{\log}$ into itself. At time [math], the map $\mathrm{H}(\cdot;0)$ is the identity of $F_{n}^{\log}$; at time $\theta$ we rotate each configuration by an angle $\theta$ counterclockwise, adjusting logarithms:

[TABLE]

At time $2\pi$, the map $\mathrm{H}(\cdot;2\pi)$ preserves all $z_{i}$’s and shifts all $w_{ij}$’s by $2\pi\sqrt{-1}$: this last map is precisely the map

[TABLE]

i.e. the product of all deck transformations $t_{ij}\colon F_{n}^{\log}\to F_{n}^{\log}$.

Since $\prod_{1\leq i<j\leq n}t_{ij}$ is homotopic to the identity of $F_{n}^{\log}$, it induces the identity map on $H_{*}\left(F_{n}^{\log}\right)$.

Hence $H_{*}(F_{n}^{\log})$, as a $R(n)$-module, is $\left[(\prod_{\leq i<j\leq n}t_{ij})-1\right]$-torsion, in particular its $\mathbb{K}(n)$-localisation vanishes. ∎

The proof of the previous lemma tells us that the variable $t_{12}$ acts on $H_{*}([P_{n},P_{n}])$ as the product $\prod_{(i<j)\neq(1,2)}t_{ij}^{-1}$; therefore it seems convenient to replace $R(n)$ with a smaller ring, containing one variable less.

Definition 3.5**.**

We call

[TABLE]

the ring of Laurent polynomials in ${\binom{n}{2}}-1$ variables. $\tilde{R}(n)$ is naturally a subring of $R(n)$ by identifying each $\tilde{t}_{ij}$ with the corresponding $t_{ij}$, and therefore each $R(n)$-module is also a $\tilde{R}(n)$-module.

We can also identify $\tilde{R}(n)$ as the quotient of $R(n)$ by the ideal generated by the element $\left(\prod_{1\leq i<j\leq n}t_{ij}\right)-1$. The composition of maps of rings $\tilde{R}(n)\subset R(n)\to\tilde{R}(n)$ is the identity of $\tilde{R}(n)$.

The ring $\tilde{R}(n)$ is a domain, and we call $\tilde{\mathbb{K}}(n)$ its quotient field.

We want now to study $H_{*}([P_{n},P_{n}])$ as a $\tilde{R}(n)$-module. We introduce our second model of a classifying space for $[P_{n},P_{n}]$.

Definition 3.6**.**

The space $\tilde{F}_{n}^{\log}$ is defined as the subspace of $F_{n}^{\log}$ of configurations $\left((z_{i}),(w_{ij})\right)$ such that $z_{2}=1$, $z_{1}=0$ and $w_{12}=0$.

The space $\tilde{F}_{n}^{\log}$ is not invariant under the action of the whole group $P_{n}^{\mathrm{ab}}$ on $F_{n}^{\log}$: the action of $t_{12}$ consists in shifting $w_{12}$ by $2\pi\sqrt{-1}$, and this is not allowed inside $\tilde{F}_{n}^{\log}$. The other generators $\tilde{t}_{ij}$ of $P_{n}^{\mathrm{ab}}$ preserve $\tilde{F}_{n}^{\log}$; we conclude that $H_{*}(\tilde{F}_{n}^{\log})$ has a natural structure of $\tilde{R}(n)-$module, and the inclusion map $\tilde{F}_{n}^{\log}\subset F_{n}^{\log}$ induces a map of $\tilde{R}(n)$-modules in homology.

Lemma 3.7**.**

$\tilde{F}_{n}^{\log}$* is a deformation retract of $F_{n}^{\log}$, and therefore it is also a classifying space for $[P_{n},P_{n}]$.*

Proof.

We define a homotopy $\mathrm{H}\colon F_{n}^{\log}\times[0,1]\to F_{n}^{\log}$ starting with the identity of $F_{n}^{\log}$ and ending with a retraction onto $\tilde{F}_{n}^{\log}$; the space $\tilde{F}_{n}^{\log}$ will be fixed pointwise throughout the homotopy.

Let $\left((z_{i}),(w_{ij})\right)\in F_{n}^{\log}$. Then

[TABLE]

∎

4. Chain complexes

In this section let $n\geq 2$ be fixed. Our next aim is to describe explicitly a chain complex that computes the homology of $F_{n}^{\log}$. We first recall the classical chain complex computing the homology of $F_{n}$: it can be seen both as the dual of the reduced cochain complex of the one-point-compactification of $F_{n}$, in the spirit of Fuchs [13], or as chain complex associated with the Salvetti complex [18] of the $n$-th braid arrangement.

Definition 4.1**.**

An ordered partition of $\left\{1,\dots,n\right\}$ of degree $1\leq k\leq n$ is a partition of $\left\{1,\dots,n\right\}$ into $n-k$ non-empty subsets $(\pi_{1},\dots,\pi_{n-k})$, where each piece $\pi_{r}$ is endowed with a total order.

For $a,b\in\pi_{r}$ we write $a\prec b$ if $a$ precedes $b$ in the order associated with $\pi_{r}$, and we keep writing $a<b$ if $a$ is smaller than $b$ as natural numbers.

We define the chain complex $\mathfrak{Ch}_{\bullet}=\mathfrak{Ch}_{\bullet}(n)$. Let $\mathfrak{Ch}_{k}$ be the free abelian group with one generator (also called cell) for each ordered partition of $\left\{1,\dots,n\right\}$ of degree $k$.

In order to describe the boundary maps of $\mathfrak{Ch}_{\bullet}$ it is enough, for any two ordered partitions $(\pi_{r})_{1\leq r\leq n-k}$ and $(\pi^{\prime}_{r})_{1\leq r\leq n-k+1}$ of degree $k$ and $k-1$ respectively, to give a formula for the boundary index $[\partial(\pi_{r})\colon(\pi^{\prime}_{r})]$, i.e. the coefficient of $(\pi^{\prime}_{r})$ in $\partial(\pi_{r})$. There are two possibilities:

• •

$(\pi^{\prime}_{r})$ is obtained from $(\pi_{r})$ by

• –

splitting some piece $\pi_{l}$ into two pieces $\pi^{\prime}_{l}$ and $\pi^{\prime}_{l+1}$, each having as total order the restriction of the total order on $\pi_{l}$;

• –

setting $\pi^{\prime}_{r}=\pi_{r}$ for $r<l$ and $\pi^{\prime}_{r}=\pi_{r-1}$ for $r>l+1$, with the same total orders.

Then

[TABLE]

where, for an ordered set $(A,\prec)$ and a subset $B$, we define $sgn(A,B)$ as the parity of the number of couples $(a,b)$ of elements of $A$ with $b\prec a$, $b\in B$ and $a\not\in B$.

• •

$(\pi^{\prime}_{r})$ is not obtained from $(\pi_{r})$ as before. Then $[\partial(\pi_{r})\colon(\pi^{\prime}_{r})]=0$.

The chain complex $\mathfrak{Ch}_{\bullet}$ is the cellular chain complex of the Salvetti complex $Sal_{n}$: it is a finite cell complex contained in $F_{n}$, onto which $F_{n}$ deformation retracts [18].

Alternatively, in the spirit of Fuchs [13], one can consider the following stratification of $F_{n}$. For every ordered partition $(\pi_{r})_{1\leq r\leq n-k}$ of some degree $k$, we consider the subspace $e(\pi_{r})\subset F_{n}$ consisting of all configurations $(z_{1},\dots,z_{n})$ satisfying the following properties:

• •

there are exactly $n-k$ vertical lines in $\mathbb{C}$ passing through some of the $n$ points;

• •

for $1\leq r\leq n-k$, the $r$-th vertical line from left contains precisely the points $z_{i}$ with $i\in\pi_{r}$, and these points are assembled from the top to the bottom according to the total order $\prec$.

In particular for configurations $(z_{i})\in e(\pi_{r})$ the following properites hold:

• •

for all $i\neq j$, if $i\in\pi_{l}$ and $j\in\pi_{l^{\prime}}$ with $l<l^{\prime}$, the point $z_{i}$ lies on left of the point $z_{j}$, i.e. $\Re(z_{i})<\Re(z_{j})$;

• •

for all $i\neq j$, if both $i$ and $j$ belong to the same piece $\pi_{l}$ and if $i\prec j$, then $z_{i}$ lies above $z_{j}$, or equivalently $\Im(z_{i})>\Im(z_{j})$.

The one-point compactification $F_{n}^{+}$ of $F_{n}$ has a CW structure given by the subspaces $e(\pi_{r})$ together with the point at infinity $\infty$. The associated reduced cellular cochain complex is precisely the one described in Definition 4.1. Note that each cell $e(\pi_{r})$ is modeled on the interior of a product of simplices

[TABLE]

The local coordinates are the horizontal positions of the $n-k$ vertical lines and the vertical positions of the points $z_{i}$ on these lines. We regard $e(\pi_{r})$ as a manifold of dimension $2n-k$; an orientation can be given by declaring a total order on the simplicial local coordinates, and we choose the lexicographic order associated with the product structure written above.

With this convention, the boundary index $[\partial e(\pi^{\prime}_{r})\colon e(\pi_{r})]$ in the reduced cellular chain complex of $F_{n}^{+}$ equals the formula for $[\partial(\pi_{r})\colon(\pi^{\prime}_{r})]$ in Definition 4.1.

The space $F_{n}$ is a $2n$-dimensional manifold and its stratification by the subspaces $e(\pi_{r})$ gives rise to a Poincaré-dual cell complex, which is exactly the Salvetti complex $Sal_{n}$.

The space $Sal_{n}$ has a covering $Sal^{\log}_{n}$ corresponding to the subgroup $[P_{n},P_{n}]$ of its fundamental group $P_{n}$, and we can lift to $Sal^{\log}_{n}$ the cell complex structure on $Sal_{n}$. The group of deck transformations $P_{n}^{\mathrm{ab}}$ acts freely on the cells of $Sal^{\log}_{n}$; the associated chain complex is a chain complex of finitely generated, free $R(n)$-modules.

Definition 4.2**.**

We define a chain complex $\mathfrak{Ch}^{\log}_{\bullet}$. Let $\mathfrak{Ch}^{\log}_{k}$ be the free abelian group with one generator (called cell) for each choice of the following set of data:

• •

an ordered partition $(\pi_{r})_{1\leq r\leq n-k}$ of $\left\{1,\dots,n\right\}$ of degree $k$;

• •

integers $W_{ij}\in\mathbb{Z}$ for all $1\leq i<j\leq n$.

The boundary map has a similar formula as in Definition 4.1. Consider cells $(\pi_{r},W_{ij})_{1\leq r\leq n-k}$ and $(\pi^{\prime}_{r},W^{\prime}_{ij})_{1\leq r\leq n-k+1}$ in degrees $k$ and $k-1$ respectively.

• •

Suppose that the ordered partition $(\pi^{\prime}_{r})$ is obtained from $(\pi_{r})$ as in the first case in Definition 4.1, splitting some $\pi_{l}$ into $\pi^{\prime}_{l}$ and $\pi^{\prime}_{l+1}$. Suppose that for all $i<j$ satisfying

• –

$i,j\in\pi_{l}$;

• –

$i\prec j$ in $\pi_{l}$;

• –

$i\in\pi^{\prime}_{l}$ and $j\in\pi^{\prime}_{l+1}$

we have $W^{\prime}_{ij}=W_{ij}+1$. Finally, suppose that for all other couples of indices $i<j$ we have $W^{\prime}_{ij}=W_{ij}$. Then

[TABLE]

• •

If $(\pi^{\prime}_{r},W^{\prime}_{ij})$ cannot be obtained from $(\pi_{r},W_{ij})$ as before, then the boundary index is zero.

Similarly as before, we can stratify $F_{n}^{\log}$ as follows: for all $(\pi_{r},W_{ij})_{1\leq r\leq n-k}$ as in Definition 4.2, consider the subspace $e(\pi_{r},W_{ij})$ of $F_{n}^{\log}$ determined by the following properties:

• •

$e(\pi_{r},W_{ij})$ is a connected component of $p^{-1}(e(\pi_{r}))$, where $p\colon F_{n}^{\log}\to F_{n}$ is the usual covering map;

• •

for all $i<j$, there exists a configuration $((z_{i}),(w_{ij}))\in e(\pi_{r},W_{ij})$, depending on $i$ and $j$, such that one of the following four situations occurs, depending on the position of $i$ and $j$ in the ordered partition $(\pi_{r})$:

• –

$z_{j}=z_{i}+1$ and $w_{ij}=2\pi\sqrt{-1}\left(W_{ij}\right)$, assuming $i\in\pi_{l}$ and $j\in\pi_{l^{\prime}}$ for some $l<l^{\prime}$;

• –

$z_{j}=z_{i}+\sqrt{-1}$ and $w_{ij}=2\pi\sqrt{-1}\left(W_{ij}+\frac{1}{4}\right)$, assuming $i,j\in\pi_{l}$ for some $l$, and $j\prec i$;

• –

$z_{j}=z_{i}-1$ and $w_{ij}=2\pi\sqrt{-1}\left(W_{ij}+\frac{1}{2}\right)$, assuming $i\in\pi_{l}$ and $j\in\pi_{l^{\prime}}$ for some $l>l^{\prime}$;

• –

$z_{j}=z_{i}-\sqrt{-1}$ and $w_{ij}=2\pi\sqrt{-1}\left(W_{ij}+\frac{3}{4}\right)$, assuming $i,j\in\pi_{l}$ for some $l$, and $i\prec j$.

This stratification is the pull-back along $p$ of the stratification on $F_{n}$. We can add a point $\infty$ to $F_{n}^{\log}$ and obtain a space $\left(F_{n}^{\log}\right)^{+}$ with a CW structure with the cells $e(\pi_{r},W_{ij})$ together with the point $\infty$.

The space $\left(F_{n}^{\log}\right)^{+}$ is not the one-point compactification of $F_{n}^{\log}$, but it is universal among topological spaces satisfying the following properties:

• •

$\left(F_{n}^{\log}\right)^{+}$ is obtained from $F_{n}^{\log}$ by adding one point $\infty$;

• •

for every $X\subset F_{n}^{\log}$ meeting finitely many strata $e((\pi_{r}),(W_{ij}))$, the closure of $X$ in $\left(F_{n}^{\log}\right)^{+}$ is compact.

The genuine one-point compactification of $\left(F_{n}^{\log}\right)$ would have a coarser topology than $\left(F_{n}^{\log}\right)^{+}$, and in particular it would not have the topology of a CW complex.

The chain complex $\mathfrak{Ch}^{\log}$ coincides with the complex of reduced, compactly supported cochains of $\left(F_{n}^{\log}\right)^{+}$; the formulas for the indices are the same because we lift the canonical orientations of cells $e(\pi_{r})\subset F_{n}$ to their preimages along $p$. The manifold $F_{n}^{\log}$ is stratified by the spaces $e((\pi_{r}),W_{ij})$ and there is a Poincaré dual cell complex, which is precisely the covering $Sal^{\log}_{n}$ of the Salvetti complex $Sal_{n}$.

Putting together all $\mathbb{Z}$-summands generated by cells $(\pi_{r},W_{ij})\in\mathfrak{Ch}^{\log}$ for fixed $(\pi_{r})$ and varying $(W_{ij})$ we obtain one $R(n)$-summand of $\mathfrak{Ch}^{\log}_{\bullet}$: the action of $P^{\mathrm{ab}}_{n}$ on this summand is analogous to the one discussed for the space $F_{n}^{\log}$ (see the discussion preceding Lemma 3.2): multiplication times $t_{kl}$ consists in shifting the number $W_{kl}$ by 1, while keeping the other numbers $W_{ij}$ as well as the ordered partition $(\pi_{r})$.

We note that $\mathfrak{Ch}^{\log}_{\bullet}$ is a chain complex of finitely generated, free $R(n)$-modules; a $R(n)$-basis is given by those elements $(\pi_{r},W_{ij})\in\mathfrak{Ch}^{\log}_{\bullet}$ with $W_{ij}=0$ for all $i,j$; we call these basis elements $(\pi_{r},0)\in\mathfrak{Ch}^{\log}_{\bullet}$ to distinguish them from the elements $(\pi_{r})\in\mathfrak{Ch}_{\bullet}$ generating $\mathfrak{Ch}_{\bullet}$ over $\mathbb{Z}$.

The differentials of $\mathfrak{Ch}^{\log}_{\bullet}$ with respect to the basis of the elements $(\pi_{r},0)$ are expressed in a similar way as in Definition 4.2, but boundary indices are no longer always equal to [math] or $\pm 1$, rather they can take the form of a product of some variables $t_{ij}^{\pm 1}$, with a sign $\pm 1$ determined in the same way as in Definition 4.2. It is however still true that all boundary indices of $\mathfrak{Ch}^{\log}_{\bullet}$ are either [math] or invertible elements of $R(n)$.

There is a natural map $\mathfrak{Ch}^{\log}_{\bullet}\to\mathfrak{Ch}_{\bullet}$ of chain complexes of abelian groups, mapping the generator $(\pi_{r},W_{ij})$ to the generator $(\pi_{r})$: this map is induced by the covering map $Sal^{\log}_{n}\to Sal_{n}$, which by construction is a cellular map.

Definition 4.3**.**

The chain complex $\mathfrak{Ch}^{\log}_{\bullet}$ contains a subcomplex $\tilde{\mathfrak{Ch}}^{\log}_{\bullet}$ of free abelian groups generated by cells $(\pi_{r},W_{ij})$ such that:

• •

there are indices $l<l^{\prime}$ with $1\in\pi_{l}$ and $2\in\pi_{l^{\prime}}$;

• •

$W_{12}=0$.

Note that $\tilde{\mathfrak{Ch}}^{\log}_{\bullet}$ is a subcomplex of abelian groups, and in particular is closed along boundary maps: if $(\pi_{r},W_{ij})$ is a generator of $\tilde{\mathfrak{Ch}}^{\log}_{\bullet}$, then $1$ and $2$ already belong to different pieces of the partition $(\pi_{r})$, so that $W_{12}$ cannot change along boundaries, according to Definition 4.2. The degrees of cells in $\tilde{\mathfrak{Ch}}^{\log}_{\bullet}$ range from [math] to $n-2$, because there are always at least two pieces in the partition.

Lemma 4.4**.**

The chain complex $\tilde{\mathfrak{Ch}}^{\log}_{\bullet}$ computes the homology of $\tilde{F}_{n}^{\log}$.

Proof.

The space $\tilde{F}_{n}^{\log}$ can be also defined as follows. Let $F_{n-2}(\mathbb{C}\setminus\left\{0,1\right\})$ be the subspace of $F_{n}$ of configurations $(z_{1},\dots,z_{n})$ with $z_{1}=0$ and $z_{2}=1$. The space $F_{n-2}(\mathbb{C}\setminus\left\{0,1\right\})$ is the ordered configuration space of $n-2$ points in the $2$-punctured plane, so it is the fiber over $(z_{1}=0,z_{2}=1)$ of the bundle map $\psi_{12}\colon F_{n}\to F_{2}$ forgetting all points but the first two (see Definition 2.1).

The space $F_{n-2}(\mathbb{C}\setminus\left\{0,1\right\})$ is aspherical, and its fundamental group is the kernel of the map induced by $\psi_{12}$ on fundamental groups; moreover $H_{1}(F_{n-2}(\mathbb{C}\setminus\left\{0,1\right\}))\simeq\mathbb{Z}^{{\binom{n}{2}}-1}$, where the isomorphism is exhibited by the collection of maps $\psi_{ij}$ with $(i,j)\neq(1,2)$.

The commutator subgroup of $\pi_{1}\left(F_{n-2}(\mathbb{C}\setminus\left\{0,1\right\})\right)$ can be identified with

$[P_{n},P_{n}]$, and $\tilde{F}_{n}^{\log}$ is the covering of $F_{n-2}(\mathbb{C}\setminus\left\{0,1\right\})$ corresponding to this group.

The space $F_{n-2}(\mathbb{C}\setminus\left\{0,1\right\})$ is the complement in $\mathbb{C}^{n-2}$ of a hyperplane arrangement: using $z_{3},\dots,z_{n}$ as coordinates of $\mathbb{C}^{n-2}$ we are considering the following hyperplanes with real equations

• •

$z_{i}=0$, for $3\leq i\leq n$;

• •

$z_{i}=1$, for $3\leq i\leq n$;

• •

$z_{i}=z_{j}$, for $3\leq i<j\leq n$.

Hence also $F_{n-2}(\mathbb{C}\setminus\left\{0,1\right\})$ deformation retracts onto a Salvetti complex, that we call $\tilde{S}al_{n}\subset F_{n-2}(\mathbb{C}\setminus\left\{0,1\right\})$. Using the definition of the Salvetti complex [18] it is straightforward to check that the cellular chain complex $\tilde{\mathfrak{Ch}}_{\bullet}$ of $\tilde{S}al_{n}$ is isomorphic to the subcomplex of $\mathfrak{Ch}_{\bullet}$ generated by cells $(\pi_{r})$ satisfying the first condition of Definition 4.3.

Another possibility is the following. For every ordered partition $(\pi_{r})$ satisfying the first condition of Definition 4.3, we can consider the subspace

[TABLE]

containing configurations $(z_{3},\dots,z_{n})$ such that the point $(0,1,z_{3},\dots,z_{n})\in F_{n}$ belongs to the subspace $e(\pi_{r})$. The subspaces $\tilde{e}(\pi_{r})$, together with the point at infinity $\infty$, give a CW structure of the one-point compactification $F_{n-2}(\mathbb{C}\setminus\left\{0,1\right\})^{+}$ of the manifold $F_{n-2}(\mathbb{C}\setminus\left\{0,1\right\})$. The reduced cellular cochain complex $\tilde{\mathfrak{Ch}}_{\bullet}$ of the space $F_{n-2}(\mathbb{C}\setminus\left\{0,1\right\})^{+}$ is by construction isomorphic to the subcomplex of $\mathfrak{Ch}_{\bullet}$ generated by cells $(\pi_{r})$ satisfying the first condition of Definition 4.3, up to a shift in dimension due to the fact that $F_{n}$ has (real) dimension $2n$, whereas $F_{n-2}(\mathbb{C}\setminus\left\{0,1\right\})$ has dimension $2n-4$. The Salvetti complex $\tilde{S}al_{n}$ is the Poincaré dual of the cell decomposition of $F_{n-2}(\mathbb{C}\setminus\left\{0,1\right\})^{+}$, and its cellular chain complex is also isomorphic to $\tilde{\mathfrak{Ch}}_{\bullet}$.

We can now restrict the covering $p\colon F_{n}^{\log}\to F_{n}$ first to a connected covering $p\colon\tilde{F}_{n}^{\log}\to F_{n-2}(\mathbb{C}\setminus\left\{0,1\right\})$, and then to a connected covering $\tilde{S}al^{\log}_{n}\to\tilde{S}al_{n}$. Note that $\tilde{F}_{n}^{\log}$ is only one connected component of $p^{-1}\left(F_{n-2}(\mathbb{C}\setminus\left\{0,1\right\})\right)\subset F_{n}^{\log}$: there is indeed one connected component for any fixed value of $w_{12}\in 2\pi\sqrt{-1}\mathbb{Z}$.

We pull back the cell structure on $\tilde{S}al_{n}$ along $p$ to a cell structure on $\tilde{S}al^{\log}_{n}$; thus the chain complex associated with $\tilde{S}al^{\log}_{n}$ is precisely $\tilde{\mathfrak{Ch}}^{\log}_{\bullet}$. ∎

We define filtrations on the chain complexes that we have introduced.

Definition 4.5**.**

For each generator $(\pi_{r})_{1\leq r\leq n-k}$ of $\mathfrak{Ch}_{\bullet}$ there is an index $l$ such that $1\in\pi_{l}$: we denote $\iota(\pi_{r})=l$.

We filter $\mathfrak{Ch}_{\bullet}$ in the following way: a generator $(\pi_{r})_{1\leq r\leq n-k}$ in some degree $k$ has height $p$, with $0\leq p\leq n-1$, if there are exactly $p$ indices $i\in\pi_{\iota(\pi_{r})}$ such that $i\prec 1$. Note that by Definition 4.1 the height can only decrease along boundaries in $\mathfrak{Ch}_{\bullet}$.

In the same way we can filter the chain complex $\mathfrak{Ch}^{\log}_{\bullet}$: a generator $(\pi_{r},W_{ij})$ has the same height as the corresponding generator $(\pi_{r})$ of $\mathfrak{Ch}_{\bullet}$. Note that we obtain a $P_{n}^{\mathrm{ab}}$-invariant filtration on $\mathfrak{Ch}^{\log}_{\bullet}$: in other words $\mathfrak{Ch}^{\log}_{\bullet}$ becomes a filtered chain complex of $R(n)$-modules.

The chain complex $\tilde{\mathfrak{Ch}}^{\log}_{\bullet}$ has a natural action of the group $H_{1}(F_{n-2}(\mathbb{C}\setminus\left\{0,1\right\}))\simeq\mathbb{Z}^{{\binom{n}{2}}-1}$; as we have already seen, the group $H_{1}(F_{n-2}(\mathbb{C}\setminus\left\{0,1\right\}))$ can be identified with the kernel of the map $\psi_{12}\colon P^{\mathrm{ab}}_{n}\to\mathbb{Z}$, and is generated by elements $\tilde{t}_{ij}$ for $1\leq i<j\leq n$ with $(i,j)\neq(1,2)$. Hence $\tilde{\mathfrak{Ch}}^{\log}_{\bullet}$ can be seen as a chain complex of free $\tilde{R}(n)-$modules.

Definition 4.6**.**

We consider $\tilde{\mathfrak{Ch}}^{\log}_{\bullet}$ as a chain complex of free $\tilde{R}(n)-$modules and call $\Omega$ the basis containing those elements $(\pi_{r},0)\in\mathfrak{Ch}^{\log}_{\bullet}$ that lie in $\tilde{\mathfrak{Ch}}^{\log}_{\bullet}$.

The chain complex $\tilde{\mathfrak{Ch}}^{\log}_{\bullet}$ inherits a filtration from $\mathfrak{Ch}^{\log}_{\bullet}$, with heights $p$ ranging from [math] to $n-2$: this is a filtration in $\tilde{R}(n)-$modules.

We call $\mathfrak{F}_{p}\tilde{\mathfrak{Ch}}^{\log}_{\bullet}\subset\tilde{\mathfrak{Ch}}^{\log}_{\bullet}$ the subcomplex generated by cells of height $\leq p$, and $\mathfrak{F}_{p}/\mathfrak{F}_{p-1}\tilde{\mathfrak{Ch}}^{\log}_{\bullet}$ the $p-$th filtration stratum.

Note that $\Omega$ is a filtered basis for $\tilde{\mathfrak{Ch}}^{\log}_{\bullet}$.

5. Morse flows

In this section we simplify the complex $\tilde{\mathfrak{Ch}}^{\log}_{\bullet}$ to a chain complex with fewer generators: we use Forman’s discrete Morse theory, which was first introduced in [11]; see [14] or [15] for an introduction to discrete Morse theory. The Morse complex that we present has already appeared in a similar way in [9] and [16].

Definition 5.1**.**

Recall from Definition 4.6 that $\Omega$ is a basis for $\tilde{\mathfrak{Ch}}^{\log}_{\bullet}$ as a chain complex of finitely generated, free $\tilde{R}(n)$-modules. For a cell $\mathfrak{e}=(\pi_{r},0)\in\Omega$, the index $\iota(\mathfrak{e})$ was introduced in Definition 4.5. We define a matching $\mathcal{M}$ on $\Omega$:

• •

a cell $\mathfrak{e}=(\pi_{r},0)$ is critical if $\iota(\mathfrak{e})=1$ (i.e. $1\in\pi_{1}$), and if $1$ is the last element of $\pi_{1}$ according to $\prec$ (i.e. $i\prec 1$ for all $i\in\pi_{1}$ with $i\neq 1$);

• •

a cell $\mathfrak{e}=(\pi_{r},0)$ is collapsible if $1$ is not the last element of $\pi_{\iota(e)}$. In this case the redundant partner of $\mathfrak{e}$ is $\mathfrak{e}^{\prime}=(\pi^{\prime}_{r},0)$, where $(\pi^{\prime}_{r})$ is obtained from $(\pi_{r})$ by splitting $\pi_{\iota(\mathfrak{e})}$ into $\pi^{\prime}_{\iota(\mathfrak{e})}=\left\{i\in\pi_{l}\,|\,1\prec i\right\}$ and $\pi^{\prime}_{\iota(\mathfrak{e})+1}=\left\{i\in\pi_{l}\,|\,i\preceq 1\right\}$, as in Definition 4.2 with $l=\iota(\mathfrak{e})$, and $\prec$ is restricted to the two pieces. Informally, we push all elements $i$ lying below $1$ to the left. Note that $\iota(\mathfrak{e}^{\prime})=\iota(\mathfrak{e})+1\geq 2$. We write $\mathfrak{e}^{\prime}\nearrow\mathfrak{e}$, meaning that the couple $(\mathfrak{e}^{\prime},\mathfrak{e})$ is in $\mathcal{M}$.

• •

a cell $\mathfrak{e}=(\pi_{r},0)$ is redundant if $\iota(\mathfrak{e})\geq 2$ and $1$ is the last element of $\pi_{\iota(\mathfrak{e})}$ according to $\prec$. In this case the collapsible partner of $\mathfrak{e}$ is $\mathfrak{e}^{\prime}=(\pi^{\prime}_{r},0)$, where $(\pi^{\prime}_{r})$ is obtained from $(\pi_{r})$ by concatenating $\pi_{\iota(\mathfrak{e})}$ and $\pi_{\iota(\mathfrak{e})-1}$ into $\pi^{\prime}_{\iota(\mathfrak{e})-1}$: on the new set $\pi^{\prime}_{\iota(\mathfrak{e})-1}$ the order $\prec$ is defined by extending $\prec$ on $\pi_{\iota(\mathfrak{e})}$ and $\pi_{\iota(\mathfrak{e})-1}$ with the rule $i\prec j$ for all $i\in\pi_{\iota(\mathfrak{e})}$ and $j\in\pi_{\iota(\mathfrak{e})-1}$. In particular $1\prec j$ for all $j\in\pi_{\iota(\mathfrak{e})-1}$. Informally, we push the column on left of 1 underneath 1. We write $\mathfrak{e}\nearrow\mathfrak{e}^{\prime}$.

By Definition 4.2 if two cells $\mathfrak{e}\nearrow\mathfrak{e}^{\prime}$ are matched, then $[\partial\mathfrak{e}^{\prime}\colon\mathfrak{e}]$ is invertible in $\tilde{R}(n)$.

To check that $\mathcal{M}$ is acyclic, note first that $\mathcal{M}$ is compatible with the filtration of the chain complex $\tilde{\mathfrak{Ch}}^{\log}_{\bullet}$, hence it suffices to check that $\mathcal{M}$ is acyclic on each filtration stratum $\mathfrak{F}_{p}/\mathfrak{F}_{p-1}\tilde{\mathfrak{Ch}}^{\log}_{\bullet}$.

Let $\mathfrak{e}=(\pi_{r},0)\nearrow\mathfrak{e}^{\prime}=(\pi^{\prime}_{r},0)\searrow\mathfrak{e}^{\prime\prime}=(\pi^{\prime\prime}_{r},0)$ be an alternating path of three distinct cells of degrees $k,k+1,k$, all having the same height $p$. This means that the redundant cell $\mathfrak{e}$ is matched with the collapsible cell $\mathfrak{e}^{\prime}$, and that $[\partial\mathfrak{e}^{\prime}\colon\mathfrak{e}^{\prime\prime}]\neq 0$. Suppose also that $\mathfrak{e}^{\prime\prime}$ is redundant.

Then both $\mathfrak{e}$ and $\mathfrak{e}^{\prime\prime}$ are obtained from $\mathfrak{e}^{\prime}$ by splitting precisely the piece $\pi^{\prime}_{\iota(\mathfrak{e}^{\prime})}$ as in Definition 4.2: indeed $1$ is not the last element in $\pi^{\prime}_{\iota(\mathfrak{e}^{\prime})}$, but is the last element of both $\pi_{\iota(\mathfrak{e})}$ and $\pi^{\prime\prime}_{\iota(\mathfrak{e}^{\prime\prime})}$.

Moreover there are exactly two ways to split $\pi^{\prime}_{\iota(\mathfrak{e}^{\prime})}$ in two pieces, so that the following conditions hold:

• •

$1$ becomes the last element of its piece;

• •

the height $p$ doesn’t decrease, i.e. all elements preceding $1$ in $\pi^{\prime}_{\iota(\mathfrak{e}^{\prime})}$ still belong to the same piece as $1$ and precede $1$.

The two pieces must be, in some order, $\left\{i\in\pi^{\prime}_{\iota(\mathfrak{e}^{\prime})}\,|\,i\preceq 1\right\}$ and $\left\{i\in\pi^{\prime}_{\iota(\mathfrak{e}^{\prime})}\,|\,1\prec i\right\}$, and we can only choose which piece is split to the left and which to the right.

If $\left\{i\in\pi^{\prime}_{\iota(\mathfrak{e}^{\prime})}\,|\,1\prec i\right\}$ is split to the left, then we get the redundant partner of $\mathfrak{e}^{\prime}$, that is, $\mathfrak{e}$; in the other case we must get $\mathfrak{e}^{\prime\prime}$.

We conclude that $\iota(\mathfrak{e})=\iota(\mathfrak{e}^{\prime})+1$, and $\iota(\mathfrak{e}^{\prime\prime})=\iota(\mathfrak{e}^{\prime})$; in particular $\iota(\mathfrak{e}^{\prime\prime})>\iota(\mathfrak{e})$. This shows that the matching is acyclic on each stratum $p$, because the index $\iota$ strictly increases along alternating paths.

Definition 5.2**.**

We call $\mathcal{M}\tilde{\mathfrak{Ch}}^{\log}_{\bullet}$ the Morse complex associated with the acyclic matching $\mathcal{M}$: it is a chain complex of finitely generated, free $\tilde{R}(n)$-modules, with basis $\Omega^{\mathcal{M}}$ given by $\mathcal{M}$-critical cells in $\Omega$. The chain complex $\mathcal{M}\tilde{\mathfrak{Ch}}^{\log}_{\bullet}$ is also a filtered chain complex of $\tilde{R}(n)$-modules: the subcomplex $\mathfrak{F}_{p}\mathcal{M}\tilde{\mathfrak{Ch}}^{\log}_{\bullet}$ is generated by $\mathcal{M}$-critical cells of height $\leq p$, and the $p$-th filtration stratum is denoted by $\mathfrak{F}_{p}/\mathfrak{F}_{p-1}\mathcal{M}\tilde{\mathfrak{Ch}}^{\log}_{\bullet}$.

We conclude this section by analysing more carefully the structure of the filtration strata.

Definition 5.3**.**

Let $S$ be a subset of $\left\{2,\dots,n\right\}$ containing $2$. We denote by $R(S)$ the ring $\mathbb{Z}[t_{ij}^{\pm 1}]_{i,j\in S,i<j}$. This is a domain and is naturally contained in $\tilde{R}(n)$; its quotient field is denoted by $\mathbb{K}(S)$, and there is an inclusion $\mathbb{K}(S)\subset\tilde{\mathbb{K}}(n)$. In the particular case $S=\left\{2\right\}$ we have $R(S)=\mathbb{Z}$.

Let $\mathfrak{Ch}_{\bullet}^{S}$ and $\mathfrak{Ch}_{\bullet}^{\log,S}$ be defined in analogy with Definitions 4.1 and 4.2 but using, instead of the set of indices $\left\{1,\dots,n\right\}$, its subset $S$. In particular generators of $\mathfrak{Ch}_{\bullet}^{S}$ are given by ordered partitions $(\pi_{r})_{1\leq r\leq\left|S\right|-k}$ of $S$; generators of $\mathfrak{Ch}_{\bullet}^{\log,S}$ are given by an ordered partition of $S$ together with a choice of integers $(W_{ij})$ for all $i<j$ with $i,j\in S$.

Note that $\mathfrak{Ch}_{\bullet}^{\log,S}$ is a chain complex of finitely generated, free $R(S)$-modules, supported in degrees ranging from [math] to $|S|-1$. In the particular case $S=\left\{2\right\}$ we have that $\mathfrak{Ch}_{\bullet}^{\log,S}$ consists of a copy of $\mathbb{Z}$ in degree 0.

Lemma 5.4**.**

Let $0\leq p\leq n-2$; then there is an isomorphism of chain complexes of $\tilde{R}(n)$-modules

[TABLE]

where the sum is taken over all sets $S\subset\left\{2,\dots,n\right\}$ with $|S|=n-p-1$ and $2\in S$. This isomorphism shifts degrees by $-p$.

Proof.

Recall that the differential in the chain complex $\mathcal{M}\tilde{\mathfrak{Ch}}^{\log}_{\bullet}$ is defined as follows: for two $\mathcal{M}$-critical cells $\mathfrak{e}=(\pi_{r},0)$ and $\mathfrak{e}^{\prime}=(\pi^{\prime}_{r},0)$ in $\Omega^{\mathcal{M}}$ the boundary index $[\partial\mathfrak{e}^{\prime}\colon\mathfrak{e}]$ is the sum of the weights of all alternating paths from $\mathfrak{e}^{\prime}$ to $\mathfrak{e}$.

If $\mathfrak{e}$ and $\mathfrak{e}^{\prime}$ have the same height $p$, then an alternating path $\mathfrak{e}^{\prime}=\mathfrak{e}^{\prime}_{0}\searrow\mathfrak{e}_{0}\nearrow\mathfrak{e}^{\prime}_{1}\searrow\dots\searrow\mathfrak{e}_{l}=\mathfrak{e}$ must contain only cells of height $p$. Since $\mathfrak{e}^{\prime}_{0}$ is critical, $1$ is the last element of $\pi^{\prime}_{1}$, and splitting in two pieces $\pi^{\prime}_{1}$ would let the height $p$ of $\mathfrak{e}^{\prime}_{0}$ decrease to a smaller height in $\mathfrak{e}_{0}$: hence $\mathfrak{e}_{0}$ is obtained from $\mathfrak{e}^{\prime}$ by splitting some other piece $\pi^{\prime}_{l}$ with $l\geq 2$, and therefore $\mathfrak{e}_{0}$ is already critical, hence $\mathfrak{e}_{0}=\mathfrak{e}$.

Thus the differential in the chain complex $\mathfrak{F}_{p}/\mathfrak{F}_{p-1}\mathcal{M}\tilde{\mathfrak{Ch}}^{\log}_{\bullet}$ is isomorphic to the differential obtained from Definition 4.2 by allowing only a splitting in two pieces of some piece of the partition $\pi_{l}$ with $l\geq 2$.

In particular we can split our chain complex $\mathfrak{F}_{p}/\mathfrak{F}_{p-1}\mathcal{M}\tilde{\mathfrak{Ch}}^{\log}_{\bullet}$ into many subcomplexes according to which $p$ elements, all different from $2$, appear in $\pi_{1}$ and in which order $\prec$, provided that $1$ is the last element of $\pi_{1}$.

To determine one of these subcomplexes we can equivalently choose a set $S\subset\left\{2,\dots,n\right\}$ of $n-p-1$ elements, with $2\in S$, and declare that the other $p+1$ elements $i\in\left\{1,\dots,n\right\}$, including 1, are the elements of $\pi_{1}$. Moreover there are exactly $p!$ ways to order these $p+1$ elements inside $\pi_{1}$, if we require $1$ to be the last in the order: each of these possible choices of $\prec$ on $\pi_{1}$ gives rise to a different subcomplex.

Finally we note that each of these subcomplexes is isomorphic to the chain complex $\tilde{R}(n)\otimes_{R(S)}\mathfrak{Ch}_{\bullet}^{\log,S}$, where the isomorphism is given by mapping the $\mathcal{M}$-critical cell $(\pi_{r},0)_{1\leq r\leq n-k}$ to the cell $1\otimes(\pi_{r},0)_{2\leq r\leq n-k}$: this map has degree $-p$. ∎

6. The spectral sequence with coefficients in $\tilde{\mathbb{K}}(n)$

In this section we prove that $H_{n-2}([P_{n},P_{n}])\neq 0$. More precisely we prove the following theorem.

Theorem 6.1**.**

For $n\geq 2$ the graded $\tilde{\mathbb{K}}(n)$-vector space

[TABLE]

has dimension $(n-2)!$ in degree $n-2$ and vanishes in all other degrees.

This means, in particular, that $H_{n-2}([P_{n},P_{n}])$ contains an embedded copy of $\tilde{R}(n)^{(n-2)!}$, which for $n\geq 3$ is a free abelian group of infinite rank.

The following is an immediate consequence of Theorem 6.1.

Corollary 6.2**.**

For $n\geq 2$ the cohomological dimension of $[P_{n},P_{n}]$ is $n-2$.

Proof.

We have $\operatorname{cd}([P_{n},P_{n}])\geq n-2$ because $H_{n-2}([P_{n},P_{n}])\neq 0$. Moreover, as already seen in the proof of Lemma 4.4, the space $\tilde{F}_{n}^{\log}$ deformation retracts onto the space $\tilde{S}al^{\log}_{n}$, which is a cell complex of dimension $n-2$; hence $\operatorname{cd}([P_{n},P_{n}])\leq n-2$. ∎

Proof of Theorem 6.1.

We consider the filtered chain complex $\mathcal{M}\tilde{\mathfrak{Ch}}^{\log}_{\bullet}$. Since localisation is exact we can compute $H_{*}([P_{n},P_{n}])\otimes_{\tilde{R}(n)}\tilde{\mathbb{K}}(n)$ as the homology of the chain complex $\tilde{\mathbb{K}}(n)\otimes_{\tilde{R}(n)}\mathcal{M}\tilde{\mathfrak{Ch}}^{\log}_{\bullet}$, which is a filtered chain complex of $\tilde{\mathbb{K}}(n)$-vector spaces.

The first page of the associated Leray spectral sequence is

[TABLE]

and our aim is to show that the latter groups are all trivial, except for $p=n-2$ and $q=0$, where we have

[TABLE]

Once this statement is proved, Theorem 6.1 follows immediately because the spectral sequence collapses on its first page.

By Lemma 5.4 the chain complex $\mathfrak{F}_{n-2}/\mathfrak{F}_{n-3}\left(\mathcal{M}\tilde{\mathfrak{Ch}}^{\log}_{\bullet}\right)$ is isomorphic to the chain complex $\tilde{R}(n)^{(n-2)!}\otimes_{R_{\left\{2\right\}}}\mathfrak{Ch}_{\bullet}^{\log,\left\{2\right\}}$. Since the ring $R\left({\left\{2\right\}}\right)$ is just $\mathbb{Z}$, and since the chain complex $\mathfrak{Ch}_{\bullet}^{\log,\left\{2\right\}}$ is just a copy of $\mathbb{Z}$ in degree [math], we have that the filtration stratum $\mathfrak{F}_{n-2}/\mathfrak{F}_{n-3}\left(\mathcal{M}\tilde{\mathfrak{Ch}}^{\log}_{\bullet}\right)$ is concentrated in degree $n-2$ and its homology is $\tilde{R}(n)^{(n-2)!}$, also concentrated in degree $n-2$.

Tensoring with $\tilde{\mathbb{K}}(n)$ we have that $E_{n-2,0}\simeq\tilde{\mathbb{K}}(n)^{(n-2)!}$, and $E_{n-2,q}=0$ for all $q\neq 0$.

We want now to show that the chain complex $\mathfrak{F}_{p}/\mathfrak{F}_{p-1}\left(\tilde{\mathbb{K}}(n)\otimes_{\tilde{R}(n)}\mathcal{M}\tilde{\mathfrak{Ch}}^{\log}_{\bullet}\right)$ is acyclic for all $0\leq p\leq n-3$. By Lemma 5.4 it suffices to prove that, for any set $S\subset\left\{2,\dots,n\right\}$ containing $2$, the chain complex

[TABLE]

is acyclic. We note that $\tilde{\mathbb{K}}(n)$ contains $\mathbb{K}(S)$, so we can equally consider

[TABLE]

and the latter is acyclic because $\mathbb{K}(S)\otimes_{R(S)}\mathfrak{Ch}_{S}$ is acyclic by Lemma 3.4, and extending the field $\mathbb{K}(S)\subset\tilde{\mathbb{K}}(n)$ is exact. ∎

We note that it was not necessary to localise $\tilde{R}(n)$ with respect to all non-zero elements, i.e. passing from $\tilde{R}(n)$ to its quotient field $\tilde{\mathbb{K}}(n)$.

Definition 6.3**.**

Let $S$ be a finite subset of $\left\{2,\dots,n\right\}$ containing $2$. We call

[TABLE]

Define also

[TABLE]

where the product is extended over all subsets $S\subset\left\{2,\dots,n\right\}$ containing $2$.

Then the same argument of the proof of Lemma 3.4 tells us that, for all subsets $2\in S\subset\left\{2,\dots,n\right\}$ with $S\neq\left\{2\right\}$, we have

[TABLE]

Therefore we can repeat the proof of Theorem 6.1 to show that

[TABLE]

is concentrated in degree $n-2$, where it is equal to $\tilde{R}(n)\left[\tau_{n}^{-1}\right]^{(n-2)!}$.

7. Homology in lower degrees

In this section we prove non-triviality of $H_{*}([P_{n},P_{n}])$ in all degrees $*\leq n-2$. More precisely, we prove the following theorem.

Theorem 7.1**.**

For all $1\leq*\leq n-2$ the group $H_{*}([P_{n},P_{n}])$ contains a free abelian group of infinite rank.

Proof.

By Theorem 6.1 we know that $H_{n-2}([P_{n},P_{n}])$ contains a free abelian group of infinite rank. In the following we fix $3\leq k\leq n-1$ and prove that $H_{k-2}([P_{n},P_{n}])$ has the same property.

Consider the map $\psi^{n}_{k}\colon F_{n}\to F_{k}$ that forgets the last $n-k$ points of a configuration (compare with the maps $\psi_{ij}$ from Definition 2.1):

[TABLE]

The map $\psi^{n}_{k}$ is a fibration (see [10]) and there is a section $\sigma^{k}_{n}\colon F_{k}\to F_{n}$ given by adjoining $n-k$ points far on the right: formally we set $M(z_{1},\dots,z_{k})=\max_{i=1}^{k}\left|z_{i}\right|$ and then we define

[TABLE]

We have induced maps on fundamental groups $\psi^{n}_{k}\colon P_{n}\to P_{k}$ and $\sigma^{k}_{n}\colon P_{k}\to P_{n}$; the composition $\psi^{n}_{k}\circ\sigma^{k}_{n}\colon P_{k}\to P_{k}$ is the identity of $P_{k}$.

The maps $\psi^{n}_{k}$ and $\sigma^{k}_{n}$ restrict to maps between commutator subgroups; in particular the composition $\psi^{n}_{k}\circ\sigma^{k}_{n}\colon[P_{k},P_{k}]\to[P_{k},P_{k}]$ is the identity of $[P_{k},P_{k}]$.

This implies that the induced map in homology

[TABLE]

is injective, and again by Theorem 6.1 we know that $\colon H_{k-2}([P_{k},P_{k}])$ contains a free abelian group of infinite rank. ∎

8. Future directions

Computing the homology of $[P_{n},P_{n}]$ as a $R(n)$-module seems a difficult task, in particular because $R(n)$ is not a principal ideal domain and we lack a good classification of finitely generated modules over $R(n)$. We only observe that $H_{*}([P_{n},P_{n}])$ is finitely generated over $R(n)$: indeed the chain complex $\mathfrak{Ch}^{\log}_{\bullet}$ is finitely generated over $R(n)$, and $R(n)$ is a noetherian ring.

Computing $H_{*}([P_{n},P_{n}])$ directly as an abelian group seems not to be easy either. In Theorems 6.1 and 7.1 we have proved that $H_{k}([P_{n},P_{n}])$ contains a free abelian group of infinite rank for $1\leq k\leq n-2$; we conjecture that $H_{*}([P_{n},P_{n}])$ is indeed a free abelian group, and in particular is torsion-free. Our conjecture is related to a conjecture by Denham [8] on the structure of the homology of the Milnor fibre of a complexified real arrangement; this conjecture was investigated also by Settepanella [19]. Note that for $n=3$ our conjecture holds, as $[P_{n},P_{n}]$ is a free group.

Finally, it would also be interesting to study $H_{*}([P_{n},P_{n}])$ as a representation. Denote by $B_{n}$ Artin’s braid group on $n$ strands [2], and by $\mathfrak{S}_{n}$ the $n$-th symmetric group. There is a short exact sequence

[TABLE]

In particular $P_{n}$ is a normal subgroup of $B_{n}$; since $[P_{n},P_{n}]$ is a characteristic subgroup of $P_{n}$, $[P_{n},P_{n}]$ is also normal in $B_{n}$ and we have a short exact sequence

[TABLE]

It would be interesting to understand $H_{*}([P_{n},P_{n}])$ as a representation of

$B_{n}/[P_{n},P_{n}]$.

Acknowledgments

The author would like to thank Carl-Friedrich Bödigheimer, Filippo Callegaro, Florian Kranhold, Mark Grant, Davide Lofano, Martin Palmer, Giovanni Paolini, Oscar Randal-Williams, David Recio-Mitter and Mario Salvetti for useful discussions and precious comments during the preparation of this paper.