# On $\tau$-tilting subcategories

Javad Asadollahi, Somayeh Sadeghi and Hipolito Treffinger  Department of Pure
Mathematics, Faculty of Mathematics and Statistics, University of Isfahan,
P.O.Box: 81746-73441, Isfahan, Iran<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>Université de Paris, Bâtiment
Sophie Germain 5, rue Thomas Mann 75205, Paris Cedex 13, FRANCE
<EMAIL_ADDRESS>
###### Abstract.

The main theme of this paper is to study $\tau$-tilting subcategories in an
abelian category $\mathscr{A}$ with enough projective objects. We introduce
the notion of $\tau$-cotorsion torsion triples and show a bijection between
the collection of $\tau$-cotorsion torsion triples in $\mathscr{A}$ and the
collection of $\tau$-tilting subcategories of $\mathscr{A}$, generalizing the
bijection by Bauer, Botnan, Oppermann and Steen between the collection of
cotorsion torsion triples and the collection of tilting subcategories of
$\mathscr{A}$. General definitions and results are exemplified using
persistent modules. If $\mathscr{A}={\rm{Mod\mbox{-}}}R$, where $R$ is an
unitary associative ring, we characterize all support $\tau$-tilting, resp.
all support $\tau^{-}$-tilting, subcategories of ${\rm{Mod\mbox{-}}}R$ in term
of finendo quasitilting, resp. quasicotilting, modules. As a result, it will
be shown that every silting module, respectively every cosilting module,
induces a support $\tau$-tilting, respectively support $\tau^{-}$-tilting,
subcategory of ${\rm{Mod\mbox{-}}}R$. We also study the theory in ${\rm
Rep}(Q,\mathscr{A})$, where $Q$ is a finite and acyclic quiver. In particular,
we give an algorithm to construct support $\tau$-tilting subcategories in
${\rm Rep}(Q,\mathscr{A})$ from certain support $\tau$-tilting subcategories
of $\mathscr{A}$ and present a systematic way to construct $(n+1)$-tilting
subcategories in ${\rm Rep}(Q,\mathscr{A})$ from $n$-tilting subcategories in
$\mathscr{A}$.

###### Key words and phrases:

Abelian category, ($\tau$-)tilting subcategory, torsion theory, silting
module, quiver representation

###### 2020 Mathematics Subject Classification:

18E10, 18E40, 16S90, 16E30, 18G15

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries

##  1\. Introduction

Tilting theory is one of the most prominent tools in representation theory of
artin algebras. The classical tilting modules were introduced by Brenner and
Butler [BB], and Happel and Ringel [HR] as an axiomatisation of the reflection
functors of Bernstein, Gelfand and Ponomarev [BGP] and Auslander, Platzek and
Reiten [APR]. It has been shown by Bongartz [Bon] that every partial tilting
module can be completed to a tilting module and by Happel and Unger
[HappelUnger] that every almost complete tilting module can be completed in at
most two ways. However, there are examples of almost complete tilting modules
that have exactly one complement.

Several years later, and inspired by the cluster algebras defined by Fomin and
Zelevnisky in [FZ1], Adachi, Iyama and Reiten introduced $\tau$-tilting theory
[AIR]. This is a generalization of classical tilting theory in which every
almost complete support $\tau$-tilting module has exactly two complements,
allowing to introduce a notion of mutation among these objects. The success of
$\tau$-tilting theory was immediate, offering an explanation for several
phenomena in the module category of artin algebras and offering new
connections between representation theory and other areas of mathematics (see
[TreffingerSurvey]).

Due to the effectiveness of $\tau$-tilting theory for the study of the
categories of finitely presented modules, many mathematicians have introduced
theories generalizing $\tau$-tilting theory, and its dual, to other contexts.
For instance, there are the works of Angeleri-Hügel, Marks and Vitoria [AMV]
and Breaz and Pop [BP] for the module category of rings; Iyama, Jørgensen and
Yang [IJY] for functor categories; or Liu and Zhou for Hom-finite abelian
categories with enough projective objects [LZh].

In this paper we are interested in studying $\tau$-tilting theory in arbitrary
abelian categories with enough projective objects. Since in general there is
no notion of Auslander-Reiten translation $\tau$ in such general categories,
the definition of support $\tau$-tilting subcategories needs to be made with
no mention to it. We follow [IJY, LZh] and define support $\tau$-tilting
subcategories as follows.

######  Definition 1.1.

Let $\mathscr{A}$ be an abelian category with enough projective objects. Let
$\mathscr{T}$ be an additive contravariantly finite full subcategory of
$\mathscr{A}$. Then $\mathscr{T}$ is called a support $\tau$-tilting
subcategory if

  * $1.$

${\rm{Ext}}^{1}_{\mathscr{A}}(T_{1},{\rm{Fac}}(T_{2}))=0,$ for all
$T_{1},T_{2}\in\mathscr{T}$.

  * $2.$

For any projective $P$ in $\mathscr{A}$, there exists an exact sequence

$P\stackrel{{\scriptstyle f}}{{\longrightarrow}}T^{0}\longrightarrow
T^{1}\longrightarrow 0$

such that $T^{0}$ and $T^{1}$ are in $\mathscr{T}$ and $f$ is a left
$\mathscr{T}$-approximation of $P$.

The idea of generalizing a well-behaved theory from the category of finitely
presented modules of an artin algebras to more general abelian categories is
not exclusive to $\tau$-tilting theory. Indeed, Beligiannis introduced tilting
theory for arbitrary abelian categories in [B] at the beginning of the
millenium, see also [BR]. In recent years, the work of Bauer, Botnan,
Oppermann and Steen [BBOS] has found a striking application of this theory in
persistence theory and topological data analysis (TDA). One of the main
results in [BBOS] is a bijection between tilting subcategories of an abelian
category with enough projective objects and the collection of cotorsion
torsion triples in the same category. The motivation that started this
collaboration is to find a generalization of [BBOS, Theorem 2.29] to support
$\tau$-tilting subcategories, but for that we first need an adequate notion of
triple. Inspired by [BR, Lemma V.3.3] we give the definition of
$\tau$-cotorsion torsion triple as follows.

######  Definition 1.2.

Let $\mathscr{A}$ be an abelian category with enough projective objects. A
triple of full subcategories $(\mathscr{C},\mathscr{D},\mathscr{F})$ of
$\mathscr{A}$ is called a $\tau$-cotorsion torsion triple if

  * $1.$

$\mathscr{C}={}^{\perp_{1}}\mathscr{D}$.

  * $2.$

For every projective object $P\in\mathscr{A}$, there exists an exact sequence

$P\stackrel{{\scriptstyle f}}{{\longrightarrow}}{D}\longrightarrow
C\longrightarrow 0,$

where $D\in\mathscr{C}\cap\mathscr{D}$, $C\in\mathscr{C}$ and $f$ is a left
$\mathscr{D}$-approximation.

  * $3.$

$\mathscr{C}\cap\mathscr{D}$ is a contravariantly finite subcategory of
$\mathscr{A}$.

  * $4.$

$(\mathscr{D},\mathscr{F})$ is a torsion pair in $\mathscr{A}$.

Having the notion of $\tau$-cotorsion torsion triple we are able to show the
desired bijection in an abelian category with enough projective objects.

######  Theorem 1.3 (Theorem LABEL:Bijection).

Let $\mathscr{A}$ be an abelian category with enough projective objects. Then
there are bijections

$\displaystyle\Phi:\\{\mbox{support $\tau$-tilting subcategories}\\}$
$\displaystyle\longrightarrow\\{\mbox{$\tau$-cotorsion torsion triples}\\}$
$\displaystyle\mathscr{T}$
$\displaystyle\longmapsto({}^{\perp_{1}}{(\rm{Fac}(\mathscr{T}))},\rm{Fac}(\mathscr{T}),\mathscr{T}^{\perp_{0}})$
$\displaystyle\Psi:\\{\mbox{$\tau$-cotorsion torsion triples}\\}$
$\displaystyle\longrightarrow\\{\mbox{support $\tau$-tilting
subcategories}\\}$ $\displaystyle(\mathscr{C},\mathscr{D},\mathscr{F})$
$\displaystyle\longmapsto\mathscr{C}\cap\mathscr{D}$

which are mutually inverse. Moreover these bijections restrict to bijections
between the collection of tilting subcategories of $\mathscr{A}$ and the
collection of cotorsion torsion triples in $\mathscr{A}$.

We note that Buan and Zhou [BZ] introduced the notion of left weak cotorsion
torsion triple in order to provide a version of [BBOS, Theorem 2.29] for
support $\tau$-tilting modules in ${\rm{{mod\mbox{-}}}}\Lambda$, where
$\Lambda$ is an artin algebra. In Theorem LABEL:lwcotorsion we show that a
triple $(\mathscr{C},\mathscr{D},\mathscr{F})$ of subcategories of
${\rm{{mod\mbox{-}}}}\Lambda$ is a left weak cotorsion torsion triple if and
only if it is a $\tau$-cotorsion torsion triple.

One of the main properties of support $\tau$-tilting modules is that they
generate functorially finite torsion classes in module categories. As a
consequence of our results, we obtain the following corollary, see Theorem
LABEL:Bijection and Corollary LABEL:Cor-ff.

######  Corollary 1.4.

Let $\mathscr{A}$ be an abelian category with enough projective objects and
let $\mathscr{T}$ be a support $\tau$-tilting subcategory of $\mathscr{A}$.
Then ${\rm{Fac}}(\mathscr{T})$ is a functorially finite torsion class in
$\mathscr{A}$.

For ${\rm{{mod\mbox{-}}}}\Lambda$, using results of [AIR], it is shown that a
module $T$ in ${\rm{{mod\mbox{-}}}}\Lambda$ is a support $\tau$-tilting module
if and only if ${\rm{add}}(T)$ is a support $\tau$-tilting subcategory of
${\rm{{mod\mbox{-}}}}\Lambda$. On the other hand, $T$ is a support
$\tau^{-}$-tilting module if and only if ${\rm{add}}(T)$ is a support
$\tau^{-}$-tilting subcategory of ${\rm{{mod\mbox{-}}}}\Lambda$, see
Propositions LABEL:EquivalentDefinitions and LABEL:DualEquivalentDefinitions.

Suppose that $\mathscr{A}$ is an abelian category with enough injective
objects. Then in this category all the dual definitions and results can be
stated and proved, see Section LABEL:Sec:Dual. Using this symmetry, when
$\mathscr{A}$ is an abelian category with enough projective and enough
injective objects, we introduce the notion of a $\tau$-$\tau^{-}$-quadruple
and use it to create a map from support $\tau$-tilting to support
$\tau^{-}$-tilting subcategories of $\mathscr{A}$. We show in Proposition
LABEL:daggermap that this map is exactly the dual of the dagger map defined in
[AIR, Theorem 2.15] when $\mathscr{A}={\rm{{mod\mbox{-}}}}\Lambda$.

Besides ${\rm{{mod\mbox{-}}}}\Lambda$, another prominent example of an abelian
category with enough projective objects is ${\rm{Mod\mbox{-}}}R$, the category
of modules over an unitary associative ring $R$.

The theory of support $\tau$-tilting modules of [AIR] is generalized to
${\rm{Mod\mbox{-}}}R$ by Angeleri Hügel, Marks and Vitoria [AMV], where they
introduced silting modules showing that finitely presented silting
$\Lambda$-modules coincide with the support $\tau$-tilting modules of [AIR].
As the categorical dual of silting modules, Breaz and Pop introduced the
notion of cosilting modules in ${\rm{Mod\mbox{-}}}R$ and showed that finitely
copresented cosilting $\Lambda$-modules coincide with the support
$\tau^{-}$-tilting modules.

We study the connection between silting, resp. cosilting, modules in
${\rm{Mod\mbox{-}}}R$ with the $\tau$-tilting, resp. $\tau^{-}$-tilting
subcategories of ${\rm{Mod\mbox{-}}}R$. Furthermore, we characterize all
support $\tau$-tilting subcategories of ${\rm{Mod\mbox{-}}}R$. In particular,
we provide a bijection between the equivalence classes of all support
$\tau$-tilting subcategories of ${\rm{Mod\mbox{-}}}R$ and the collection of
all equivalent classes of certain $R$-modules, the so-called finendo
quasitilting $R$-modules, see Theorem LABEL:Bijection-Finendo. It is known
that all silting modules are finendo quasitilting. Dually, we also
characterize all support $\tau^{-}$-tilting subcategories of
${\rm{Mod\mbox{-}}}R$ by providing a bijection between the equivalence classes
of all support $\tau^{-}$-tilting subcategories of ${\rm{Mod\mbox{-}}}R$ and
the collection of all equivalent classes of the quasicotilting $R$-modules.

Towards the end of the paper, we provide applications of this theory to the
category of the representations of a finite and acyclic quiver in an abelian
category $\mathscr{A}$ with enough projective objects. Let $Q$ be such a
quiver. It is known that the category ${\rm Rep}(Q,\mathscr{A})$ of
representations of $Q$ over $\mathscr{A}$ is again an abelian category with
enough projective objects. See Section LABEL:Sec:Quivers for more details. We
give a recipe to construct support $\tau$-tilting subcategories in ${\rm
Rep}(Q,\mathscr{A})$ from certain support $\tau$-tilting subcategories of
$\mathscr{A}$. More explicitly, we have the following result.

######  Theorem 1.5 (Theorem LABEL:ProduceTauTilting).

Let $\mathscr{A}$ be an abelian category with enough projective objects and
$Q$ be a finite acyclic quiver. Let $\mathscr{T}$ be a support $\tau$-tilting
subcategory of $\mathscr{A}$ such that ${\rm{Fac}}(\mathscr{T})$ is closed
with respect to the kernels of epimorphisms. Then

$\mathbb{T}={\rm{add}}\\{e_{i}^{\rho}(T)|~{}i\in Q_{0},T\in\mathscr{T}\\}$

is a support $\tau$-tilting subcategory of ${\rm Rep}(Q,\mathscr{A})$.

Let ${\rm{Prj}}(\mathscr{A})$ denote the category of all projective objects in
$\mathscr{A}$. In [BBOS, Proposition 3.9] the authors have shown that

$\mathbb{T}={\rm{add}}\\{e_{i}^{\rho}(P)|~{}i\in
Q_{0},P\in{\rm{Prj}}(\mathscr{A})\\}$

is a tilting subcategory of $\mathscr{A}$. Then the previous result can be
seen as a generalization of this result, since
$\mathscr{A}={\rm{Fac}}({\rm{Prj}}(\mathscr{A}))$ is clearly closed under
kernels of epimorphisms.

There are other generalisations of tilting theory that we have not mentioned
yet. One of these generalizations consists on defining $n$-tilting
subcategories, where $1$-tilting subcategories correspond to classical tilting
subcategories, see Definition 2.1. By convention we consider
${\rm{Prj}}(\mathscr{A})$ as the $0$-tilting subcategory of $\mathscr{A}$.
Using similar methods as in the proof of Theorem LABEL:ProduceTauTilting we
were able to show another generalization of [BBOS, Proposition 3.9], which we
believe might have interesting implications in the study of higher homological
algebra, introduced by Iyama in [Iyama1, Iyama2].

######  Theorem 1.6 (Theorem LABEL:ThHigherTilt).

Let $\mathscr{A}$ be an abelian category with enough projective objects. Let
$n$ be a non-negative integer and let $Q$ be a finite and acyclic quiver. For
an $n$-tilting subcategory $\mathscr{T}$ of $\mathscr{A}$ put

$\mathscr{T}^{\prime}={\rm{add}}(\\{e_{i}^{\rho}(T)|~{}i\in
Q_{0},T\in\mathscr{T}\\}).$

Then $\mathscr{T}^{\prime}$ is an $(n+1)$-tilting subcategory of ${\rm
Rep}(Q,\mathscr{A})$.

Throughout the paper, several examples are provided using the theory of
persistent modules, which are central objects of study in topological data
analysis (TDA). We refer the reader to the survey [Ca] by Carlsson, or the
book [Ou] by Oudot, for an introduction to topological data analysis and
connections to quiver representations.

The paper is structured as follows. In Section 2 we fix notation and give the
necessary background for the rest of the paper. In Section
LABEL:Sec:TauTiltSubcategories we define support $\tau$-tilting subcategories
in abelian categories with enough projective objects and we give some of its
basic properties. Then, in Section LABEL:Sec:TauCotorsionTorsionTriples we
introduce and study the notion of $\tau$-cotorsion torsion triples. We also
compare $\tau$-cotorsion torsion triples with the left weak cotorsion torsion
triples of [BZ]. In Section LABEL:Sec:Bijection we show the bijection between
support $\tau$-tilting subcategories and $\tau$-cotorsion torsion triples.
Section LABEL:Sec:Dual is a compilation of dual definitions and results.
Later, in Section LABEL:Sec:Quadruples we relate support $\tau$-tilting
subcategories with support $\tau^{-}$-tilting subcategories via
$\tau$-$\tau^{-}$-quadruples. In Section LABEL:Sec:Connection we study support
$\tau$\- and $\tau^{-}$\- tilting subcategories in ${\rm{Mod\mbox{-}}}R$,
where $R$ is a unitary associative ring, and show that how they are connected
to the silting and cosilting theory. We end the paper in Section
LABEL:Sec:Quivers where we study support $\tau$-tilting subcategories and
$n$-tilting subcategories in the category ${\rm Rep}(Q,\mathscr{A})$ of quiver
representations over $\mathscr{A}$.

##  2\. Preliminaries

Let $\mathscr{A}$ be an abelian category. In this paper by subcategory we
always mean a full subcategory. Let $\mathscr{X}$ be a subcategory of
$\mathscr{A}$. A morphism $\varphi:X\longrightarrow A$, where $A$ is an object
of $\mathscr{A}$, is called a right $\mathscr{X}$-approximation of $A$ if,
$X\in\mathscr{X}$ and for every $X^{\prime}\in\mathscr{X}$, the induced
morphism
$\mathscr{A}(X^{\prime},X)\longrightarrow\mathscr{A}(X^{\prime},A)\longrightarrow
0$ of abelian groups is exact. We say that $\mathscr{X}$ is a contravariantly
finite subcategory of $\mathscr{A}$ if every object $A$ of $\mathscr{A}$
admits a right $\mathscr{X}$-approximation. Dually, the notions of left
$\mathscr{X}$-approximations and covariantly finite subcategories are defined.
Moreover we say that $\mathscr{X}$ is a functorially finite subcategory of
$\mathscr{A}$ if it is both a contravariantly finite and a covariantly finite
subcategory of $\mathscr{A}$. If $\mathscr{X}$ is closed under taking finite
direct sums and direct summands we say that it is additively closed.

Let $n$ be a non-negative integer. We define

$\mathscr{X}^{\perp_{n}}:=\\{A\in\mathscr{A}~{}|~{}{\rm{Ext}}^{n}_{\mathscr{A}}(\mathscr{X},A)=0\\},$
${}^{\perp_{n}}\mathscr{X}:=\\{A\in\mathscr{A}~{}|~{}{\rm{Ext}}^{n}_{\mathscr{A}}(A,\mathscr{X})=0\\}.$

Note that ${\rm{Ext}}^{0}$ is just the usual Hom-functor.

Moreover, we set

$\mathscr{X}^{\perp}:=\\{A\in\mathscr{A}~{}|~{}{\rm{Ext}}^{i}_{\mathscr{A}}(\mathscr{X},A)=0,~{}~{}~{}\forall~{}i\geq
1\\},$
${}^{\perp}\mathscr{X}:=\\{A\in\mathscr{A}~{}|~{}{\rm{Ext}}^{i}_{\mathscr{A}}(A,\mathscr{X})=0,~{}~{}~{}\forall~{}i\geq
1\\}.$

######  Definition 2.1.

Let $\mathscr{A}$ be an abelian category with enough projective objects. Let
$n$ be a non-negative integer. An additively closed subcategory $\mathscr{T}$
of $\mathscr{A}$ is called an $n$-tilting subcategory if

  * $(i)$

$\mathscr{T}$ is a contravariantly finite subcategory of $\mathscr{A}$.

  * $(ii)$

${\rm{Ext}}^{i}_{\mathscr{A}}(T_{1},T_{2})=0$, for all
$T_{1},T_{2}\in\mathscr{T}$ and all $i\geq 1$.

  * $(iii)$

Every object $T\in\mathscr{T}$ has projective dimension at most $n$.

  * $(iv)$

For every projective object $P$ in $\mathscr{A}$, there exists a short exact
sequence

$0\longrightarrow P\stackrel{{\scriptstyle
f}}{{\longrightarrow}}T^{0}\longrightarrow
T^{1}\longrightarrow\cdots\longrightarrow T^{n}\longrightarrow 0$

with $T^{i}\in\mathscr{T}$.

If $\mathscr{T}$ only satisfies the conditions $(ii)-(iv)$, it is called a
weak $n$-tilting subcategory of $\mathscr{A}$.

######  Remark 2.2.

It follows, by breaking out the exact sequence of $(iv)$ to short exact
sequences and using the vanishing of Ext-groups in $(ii)$ of the previous
definition, that the map $f:P\longrightarrow T^{0}$ is a left
$\mathscr{T}$-approximation of $P$. Moreover, it is easy to see that the
category ${\rm{Prj}}(\mathscr{A})$ of all projective objects in $\mathscr{A}$
is the only $0$-tilting subcategory of $\mathscr{A}$. In this paper, when we
say tilting subcategory we mean a $1$-tilting subcategory.

######  2.3.

Cotorsion torsion triple. Here we recall the notion of torsion pairs,
cotorsion pairs and cotorsion torsion triples. Let us begin by the following
classical definition of Dickson [Dickson].

######  Definition 2.4.

Let $\mathscr{A}$ be an abelian category. A pair $(\mathscr{T},\mathscr{F})$
of full subcategories of $\mathscr{A}$ is called a torsion pair if
${\rm{Hom}}_{\mathscr{A}}(\mathscr{T},\mathscr{F})=0$ and for every
$A\in\mathscr{A}$ there is a short exact sequence

$0\longrightarrow tA\longrightarrow A\longrightarrow fA\longrightarrow 0$

such that $tA\in\mathscr{T}$ and $fA\in\mathscr{F}$.

It is well known that $\mathscr{T}$ and $\mathscr{F}$ determine each others
and the short exact sequence is functorial, see for instance [BR, §I.1].

######  Definition 2.5.

(see [Sa]) Let $\mathscr{A}$ be an abelian category with enough projective
objects. A pair $(\mathscr{C},\mathscr{D})$ of full subcategories of
$\mathscr{A}$ is called a cotorsion pair if
$\mathscr{C}={}^{\perp_{1}}\mathscr{D}$, $\mathscr{D}=\mathscr{C}^{\perp_{1}}$
and for every object $A\in\mathscr{A}$, there are short exact sequences

$0\longrightarrow D\longrightarrow C\longrightarrow A\longrightarrow 0$
$0\longrightarrow A\longrightarrow D^{\prime}\longrightarrow
C^{\prime}\longrightarrow 0$

where $C$ and $C^{\prime}$ are in $\mathscr{C}$ and $D$ and $D^{\prime}$ are
in $\mathscr{D}$.

This notion sometimes called complete cotorsion pair in the literature. See
[BBOS, Remark 2.7] for more details.

######  Remark 2.6.

It is immediate from the definition that in a cotorsion pair
$(\mathscr{C},\mathscr{D})$ both $\mathscr{C}$ and $\mathscr{D}$ are closed
under extensions.

Let $\mathscr{A}$ be an abelian category with enough projective and enough
injective objects. In view of [BR, Lemma V.3.3], a pair
$(\mathscr{C},\mathscr{D})$ of full subcategories of $\mathscr{A}$ is a
cotorsion pair if and only if the following conditions are satisfied:

  * $1.$

$\mathscr{C}={}^{\perp_{1}}\mathscr{D}$,

  * $2.$

for every object $A\in\mathscr{A}$, there exists a short exact sequence

$0\longrightarrow A\stackrel{{\scriptstyle
f}}{{\longrightarrow}}D\longrightarrow C\longrightarrow 0,$

where $D\in\mathscr{D}$ and $C\in\mathscr{C}$.

Following proposition shows that when $\mathscr{D}$ is closed under factors it
is enough, in the above definition, to check the Condition $2$ only for
projective objects.

######  Proposition 2.7.

Let $\mathscr{A}$ be an abelian category with enough projective and enough
injective objects. Let $(\mathscr{C},\mathscr{D})$ be a pair of full
subcategories of $\mathscr{A}$ such that $\mathscr{D}$ is closed under
factors. Then $(\mathscr{C},\mathscr{D})$ is a cotorsion pair if and only if

  * $1.$

$\mathscr{C}={}^{\perp_{1}}\mathscr{D}$,

  * $2.$

for every projective object $P\in\mathscr{A}$, there exists a short exact
sequence

$0\longrightarrow P\stackrel{{\scriptstyle
f}}{{\longrightarrow}}D\longrightarrow C\longrightarrow 0,$

where $D\in\mathscr{D}\cap\mathscr{C}$ and $C\in\mathscr{C}$.

###### Proof.

We start by showing the necessity. Let $A$ be an arbitrary object of
$\mathscr{A}$. Since $\mathscr{A}$ has enough projective objects, we have an
epimorphism $P\longrightarrow A\longrightarrow 0$. By assumption $P$ fits into
the short exact sequence

$0\longrightarrow P\stackrel{{\scriptstyle
f}}{{\longrightarrow}}D\longrightarrow C\longrightarrow 0.$

The pushout diagram

|
---|---
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{D\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{U}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0,}$$\textstyle{0}$$\textstyle{0}$

induces the short exact sequence $0\longrightarrow A\longrightarrow
U\longrightarrow C\longrightarrow 0$, where $U\in{\rm{Fac}}(\mathscr{D})$.
Since $\mathscr{D}$ is closed under factors, we deduce that $U\in\mathscr{D}$
and hence this is the desired short exact sequence for $A$.

For the sufficiency, let $P$ be a projective object in $\mathscr{A}$. Then
there exists a short exact sequence

$0\longrightarrow P\stackrel{{\scriptstyle
f}}{{\longrightarrow}}D\longrightarrow C\longrightarrow 0$

where $D\in\mathscr{D}$ and $C\in\mathscr{C}$. Since $P$ is projective, then
$P\in{}^{\perp_{1}}\mathscr{D}=\mathscr{C}$. Hence $D\in\mathscr{C}$ because
$\mathscr{C}$ is closed under extensions and the result follows. ∎

Following lemma shows that if the subcategory $\mathscr{C}$ of a cotorsion
pair $(\mathscr{C},\mathscr{D})$ of $\mathscr{A}$ is closed with respect to
factors, then $\mathscr{C}\cap\mathscr{D}$ is contravariantly finite in
$\mathscr{A}$.

######  Lemma 2.8.

Let $(\mathscr{C},\mathscr{D})$ be a cotorsion pair in $\mathscr{A}$ such that
$\mathscr{D}$ is closed under factors. Then $\mathscr{C}\cap\mathscr{D}$ is a
contravariantly finite subcategory of $\mathscr{A}$.

###### Proof.

Since $(\mathscr{C},\mathscr{D})$ is a cotorsion pair, it follows easily that
$\mathscr{D}$ is closed under extensions and products. So $\mathscr{D}$ is
indeed a torsion class. Now, since $(\mathscr{C},\mathscr{D})$ is a cotorsion
pair, there exists a short exact sequence

$0\longrightarrow D\longrightarrow
C\stackrel{{\scriptstyle\imath}}{{\longrightarrow}}tA\longrightarrow 0$

such that $D\in\mathscr{D}$ and $C\in\mathscr{C}$. Here, since
$D,tA\in\mathscr{T}$ and $\mathscr{T}$ is closed under extensions, we get
$C\in\mathscr{C}\cap\mathscr{D}$. We claim that the composition
$C\stackrel{{\scriptstyle\imath}}{{\longrightarrow}}tA\stackrel{{\scriptstyle\jmath}}{{\longrightarrow}}A$
is a right $\mathscr{C}\cap\mathscr{D}$-approximation. To show the claim, let
$X\stackrel{{\scriptstyle\ell}}{{\longrightarrow}}A$ be a morphism with
$X\in\mathscr{C}\cap\mathscr{D}$. Since $X\in\mathscr{D}$ and every map from
$\mathscr{D}$ to $A$, factors through $tA$, the morphism $\ell$ factors
through $\jmath$. Now since $X\in\mathscr{C}$ and ${\rm{Ext}}^{1}_{A}(X,D)=0$,
the morphism $\ell$ factors through $\jmath\imath$. Hence the claim is proved.
∎

######  Definition 2.9.

Let $\mathscr{A}$ be an abelian category with enough projective objects. A
triple $(\mathscr{C},\mathscr{T},\mathscr{F})$ of full subcategories in
$\mathscr{A}$ is called a cotorsion torsion triple, if
$(\mathscr{C},\mathscr{T})$ is a cotorsion pair and
$(\mathscr{T},\mathscr{F})$ is a torsion pair.

It has been recently shown in [BBOS] that cotorsion torsion triples and
tilting subcategories are closely related. The explicit relation between these
two concepts is as follows.

######  Theorem 2.10.

[BBOS, Theorem 2.29] Let $\mathscr{A}$ be an abelian category with enough
projective objects. Then there is a bijection

$\\{\mbox{tilting
subcategories}\\}~{}~{}~{}~{}\longleftrightarrow~{}~{}~{}~{}\\{\mbox{cotorsion
torsion triples}\\}$

$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \
\mathscr{T}~{}~{}~{}~{}\longrightarrow~{}~{}~{}~{}~{}({}^{\perp_{1}}{(\rm{Fac}(\mathscr{T}))},\rm{Fac}(\mathscr{T}),\mathscr{T}^{\perp_{0}})$

$\mathscr{C}\cap\mathscr{T}~{}~{}\longleftarrow~{}~{}~{}(\mathscr{C},\mathscr{T},\mathscr{F})$

where $\rm{Fac}(\mathscr{T})$ is the full subcategory of $\mathscr{A}$
consisting of factors of objects in $\mathscr{T}$.


# Homotopy braid groups are torsion-free

Emmanuel Graff

###### Abstract.

We show that, for any number of components, the group of braids up to link-
homotopy is torsion-free. This generalizes a result of Humphries up to six
components, and provides an explicit solution to a question posed by Lin and
addressed by Linell and Schick regarding the existence of non-abelian torsion-
free quotients of the braid group. The proof relies on the diagrammatic theory
of welded braids and uses the Artin representation. As a corollary, we obtain
yet another proof that braid groups themselves are torsion-free.

## Introduction

Artin in [Art47] is the first author to mention the notion of _link-homotopy_
, in the context of braids.111For links, this theory was deeply investigated
by [Mil54, Lev88, HL90]. This is an equivalence relation that allows
continuous deformations during which two distinct components remain disjoint
at all times, but each component can self-intersect. Subsequently, numerous
authors have explored braids up to link-homotopy.

Among them, Goldsmith, in [Gol74], answers a question of Artin about the
distinction between isotopy and link-homotopy of braids, by providing an
example of a non-trivial braid, up to isotopy, that is trivial up to link-
homotopy. She also provided a presentation of the _homotopy braid group_ ,
which emerges as a quotient of the standard braid group. Habegger and Lin, in
[HL90], examined pure braids as reduced free group automorphisms, through a
homotopy version of _Artin’s representation_. Additionally, Humphries in
[Hum01] addressed the question of torsion in the homotopy braid group, proving
torsion-freeness for six components or less.

This _torsion problem_ also arises in [BVW22], where the authors mention the
broader question posed by Lin, formulated in [Lin96] and addressed in the
Kourovka notebook [MK14]: ‘Is there a non-trivial epimorphism of the braid
group onto a non-abelian group without torsion?’. Linnell and Schick, in
[LT07], provide a complete solution by showing that the braid group is
residually torsion-free nilpotent-by-finite, hence in particular has plenty of
non-trivial torsion-free quotients. However, they only give an existence
proof, and explicit examples are not known for more than six components. Our
main result is a complete solution to the torsion problem for any number of
components.

######  Theorem.

(Theorem 2.9) The homotopy braid group is torsion-free for any number of
components.

Additionally, as a corollary of this result, we recover the well-known result
that the braid group is torsion-free for any number of components (Corollary
2.11). This fact, originally due to Fadell and Neuwirth in [FN62, Thm. 8],
also follows from the stronger property of left-orderability established by
Dehornoy in [Deh94]. The property of left-orderability for the homotopy braid
group is not known to this day and constitutes an interesting open question,
as discussed in Remark 2.12.

Our result relies heavily on the theory of _welded braids_ , which are a
diagrammatic generalization of braids, allowing virtual crossings in addition
to the classical ones. They are regarded up to certain local deformations that
generalize the usual Reidemeister moves. Welded braids can be defined through
various equivalent definitions that have been investigated by numerous authors
across different contexts [FRR97, Sav96, McC86, BWC07, BH13, ABMW17a]. While
we won’t discuss these various perspectives here, we direct interested readers
to Damiani’s survey [Dam17] for further details. Instead, our focus lies on
the notion of link-homotopy for welded objects, alongside its interpretation
in terms of _arrow calculus_.

The notion of _$w$ -arrows_, and more generally, _$w$ -trees_, developed in
[MY19], is a welded version of Habiro’s claspers [Hab00]. These are
diagrammatic tools upon which surgery operations can be performed. Their
manipulation up to link-homotopy is described in [MY19, §9] in what is
referred to as homotopy arrow calculus. This homotopy arrow calculus, will be
central to our study.

The paper is divided into two sections. In the first preparatory section, we
review the notion of homotopy welded braids along with its associated arrow
calculus. Additionally, we recall the definition of the Artin representation
in this context. Then, in the second section, we prove our main theorem in two
steps: first, a topological characterization of torsion elements in the
homotopy braid group using the Artin representation; then, an algebraic step
where we show that the previously established characterization is never
satisfied for classical braids.

_Acknowledgement_ : The author thanks Jean–Baptiste Meilhan and Akira Yasuhara
for their careful reading and pertinent corrections. The author also thanks
Luis Paris for an insightful comment that led to one of the key ideas.

##  1\. Preliminaries

In this section, we review the notion of braids and their welded extension,
focusing specifically on their study up to link-homotopy. Additionally, we
present the theory of homotopy arrow calculus. Finally we recall the Artin
representation in this context.

###  1.1. Welded Braids

Let us take $n$ fixed points, in the unit interval $[0,1]$, denoted by
$p_{1}<p_{2}<\cdots<p_{n}$.

######  Definition 1.1.

An _$n$ -component welded braid diagram_ $\beta=(\beta_{1},\ \ldots,\
\beta_{n})$ is the oriented222In this paper, the adopted convention for the
orientation of welded braid diagrams is from top to bottom. image of an
immersion

$(\beta_{1},\ \ldots,\ \beta_{n}):\underset{i\leq
n}{\bigsqcup}[0,1]\to[0,1]\times[0,1]$

such that, for some permutation of $\\{1,\ \ldots,\ n\\}$ associated to
$\beta$ and denoted by $\pi(\beta)$, we have $\beta_{i}(0)=(p_{i},\ 0)$ and
$\beta_{i}(1)=(p_{\pi(\beta)(i)},\ 1)$ for each $i$. We require the
singularities to be a finite number of transverse double points, labeled
either as classical crossings or as virtual crossings, as illustrated in
Figure 1. Additionally, we require the immersion to be monotonic, which means
that $\beta_{i}(t)\in[0,1]\times\\{t\\}$ for any $t\in[0,\ 1]$ and any $i$. We
call the image of $\beta_{i}$, the _$i$ -th component_ of $\beta$.

Figure 1. A classical and a virtual crossing.

A diagram whose associated permutation is the identity is said to be _pure_. A
diagram with no virtual crossings is called _classical_.

######  Definition 1.2.

_Welded link-homotopy_ is the equivalence relation generated by planar
isotopies and the following local moves:

  * •

classical Reidemeister moves,

  * •

virtual Reidemeister moves,

  * •

mixed Reidemeister move, $\hskip 113.81102pt\bullet$ OC move,

  * •

self-virtualization move.

Generally, welded braid diagrams are studied up to _welded isotopy_ , which is
defined by the same set of moves except for the self-virtualization move.

Note that we do not require welded link-homotopy to preserve the monotonic
property during the deformation. Indeed, in order to apply the self-
virtualization move, we need self-crossings, which are prohibited by
monotonicity. Furthermore, if a diagram is not monotonic, we can apply a
welded link-homotopy to convert it into a monotonic diagram (see [ABMW17b,
Thm. 4.1]). Therefore, since we are working up to link-homotopy, we can freely
forget the monotonic condition, which will be useful later, as _arrow surgery_
typically does not preserve this property.

Moreover, note that the self-virtualization move generates the _self-crossing
change_ , where a classical crossing involving two strands from the same
component is changed into its opposite. This proves, in particular, that the
usual _link-homotopy_ 333Link-homotopy, in classical braid theory, is the
equivalence relation defined by planar isotopy, classical Reidemeister moves,
and self-crossing changes. implies the welded link-homotopy.

The set of welded braid diagrams up to welded link-homotopy, equipped with the
stacking operation, forms a group: the _homotopy welded braid group_ , denoted
by $hWB_{n}$. Elements of $hWB_{n}$ are called _homotopy welded braids_. The
set of pure welded braid diagrams up to welded link-homotopy forms a subgroup
of $hWB_{n}$ denoted by $hWP_{n}$ and called the _pure homotopy welded braid
group_. Similarly, the set of classical braid diagrams up to link-homotopy,
equipped with the stacking operation, forms a group called the _homotopy braid
group_ and denoted by $hB_{n}$. Elements of $hB_{n}$ are called _homotopy
braids_. Finally, The set of pure classical braid diagrams up to link-homotopy
forms a subgroup of $hB_{n}$ denoted by $hP_{n}$ and called the _pure homotopy
braid group_.

In the following proposition, we recall the know fact that the set-theoretic
inclusion of $hB_{n}$ in $hWB_{n}$ is injective; a proof can be found in
[Gra23, Prop. 4.2.23].

######  Proposition 1.3.

The homotopy braid group $hB_{n}$ injects into the homotopy welded braid group
$hWB_{n}$.

###  1.2. Arrow calculus

Let us now review the _arrow calculus_ developed by Meilhan and Yasuhara in
[MY19]. More precisely, let us consider the _homotopy arrow calculus_ , which
deals with welded link-homotopy and will prove to be a central tool in our
study.

######  Definition 1.4.

[MY19, Def. 3.1] A _$w$ –tree_ for a welded braid diagram $\beta$ is an
immersed connected uni-trivalent tree $T$, such that

  * •

The univalent vertices of $T$ are pairwise disjoint and are contained in
$\beta\setminus\\{$crossings of $\beta\\}$.

  * •

There are finitely many singularities that are transverse double points of
only two possible kinds:

    * –

virtual crossings between edges of $T$,

    * –

virtual crossings between strands of $\beta$ and edges of $T$.

  * •

All edges of $T$ are oriented, such that each trivalent vertex has two ingoing
edges and one outgoing edge.

  * •

Each edge of $T$ is assigned a number (possibly zero) of involutive444By
involutive, we mean that twists on a same edge cancel pairwise. decorations,
called _twists_ , which are disjoint from all vertices and crossings.

A $w$–tree with a single edge is called a _$w$ –arrow_. We define the _degree_
of $T$, denoted by $\deg(T)$, as its number of trivalent vertices plus one.

Figure 2. Example of a degree $3$ $w$-tree for a welded braid diagram.

The unique univalent vertex with an ingoing edge is called the _head_ of the
$w$-tree. By graphic convention, it is represented by an arrow in the figures.
The other univalent vertices are called _tails_. When we do not need to
distinguish between tails and head, we simply call all univalent vertices,
_endpoints_. In the figures, portions of the diagram are represented by black
lines and $w$-trees edges by red lines. Finally, twists are represented
graphically by red dots $\bullet$. See Figure 2 for an example.

Given a union of $w$-trees $F$ for a welded braid diagram $\beta$, there is a
procedure called _surgery_ detailed in [MY19] to construct a new diagram
denoted $\beta_{F}$. We illustrate in Figure 3 the surgery along a $w$-arrow.

Figure 3. Surgery on a $w$-arrow.

Note that the orientation of the strand containing the tail, needs to be
specified to define the surgery move. In the case where a $w$-arrow contains
some twist, surgery introduces a virtual crossing, as shown on the left-hand
side of Figure 4. Moreover, if the edge of the $w$-arrow intersects the
diagram $\beta$, or another edge of a $w$-arrow, then the surgery introduces
virtual crossings as indicated on the right-hand side of Figure 4.

Figure 4. Surgeries near a twist and crossings.

Now if $F$ contains some $w$-trees with degree higher than one, we first apply
the _expanding rule_ shown in Figure 5555Here and in the following figures, we
use the diagrammatic convention adopted in [MY19, Convention 5.1]. at each
trivalent vertex: this breaks up $F$ into a union of $w$-arrows, on which we
can perform surgery.

Figure 5. The expanding rule.

Welded braid diagram composition yields a notion of _product_ for $w$-trees,
as follows. We say that a union $F$ of $k$ $w$-trees ($k\geq 2$) for the
trivial braid $\mathbf{1}$, is a product if

$\mathbf{1}_{F}=\prod_{i=1}^{k}\mathbf{1}_{T_{i}},$

where $T_{i}$ is a single $w$-tree for each $i$.

_Homotopy arrow calculus_ refers to the set of operations on unions of welded
braid diagrams with some $w$-trees, which yield welded link-homotopic surgery
results. These operations are defined and developed in [MY19, §9]. In the rest
of this section, we describe some of them, that will be used later.

######  Definition 1.5.

_Arrow isotopy_ is the equivalence relation generated by planar isotopies,
virtual Reidemeister moves involving edges of $w$-trees and/or strands of
diagrams, and the following local moves:

here the vertical strands are either edges of $w$-trees or strands of
diagrams.

######  Lemma 1.6.

[MY19, Lem. 5.6] Two $w$-trees related by an arrow isotopy have welded
isotopic, hence link-homotopic surgery results.

######  Definition 1.7.

A $w$-tree for a welded braid diagram $\beta$ is _repeated_ if it intersects a
component of $\beta$ in at least two endpoints.

######  Lemma 1.8.

[MY19, Lem. 9.2 ] Surgery along a repeated $w$-tree does not change the welded
link-homotopy class of the welded braid diagram.

The following lemma describes how to manipulate endpoints up to welded link-
homotopy.

######  Lemma 1.9.

[MY19] Let $T$ and $S$ denote two $w$-trees for a given welded braid diagram
$\beta$. We have the following local moves up to welded link-homotopy.666By
the notation $T=S$ we mean that $\beta_{T}$ and $\beta_{S}$ are equal in
$hWB_{n}$

  * •

Head/Tail reversal. If $T^{\prime}$ is obtained from $T$ by modifying its
head, resp. one of its tail, as shown on the left, resp. right of the figure,
then $T=T^{\prime}$.

  * •

_Tails exchange_. If $T$ and $S$ have two adjacent tails and if
$T^{\prime}\cup S^{\prime}$ is obtained from $T\cup S$ by exchanging these
tails, then $T\cup S=T^{\prime}\cup S^{\prime}$ as shown on the left of the
figure.

  * •

_Heads exchange_. If the heads of $T$ and $S$ are adjacent and if
$T^{\prime}\cup S^{\prime}$ is obtained from $T\cup S$ by exchanging these
heads as depicted in the middle of the figure, then $T\cup S=T^{\prime}\cup
S^{\prime}\cup\tilde{T}$, where $\tilde{T}$ is as shown in the figure.

  * •

_Head/Tail exchange_. If the head of $T$ is adjacent to a tail of $S$ and if
$T^{\prime}\cup S^{\prime}$ is obtained from $T\cup S$ by exchanging these
endpoints as depicted on the right of the figure, then $T\cup S=T^{\prime}\cup
S^{\prime}\cup\tilde{T}$, where $\tilde{T}$ is as shown in the figure.

We remark that the Head/Tail reversal, the Tails exchange and the Heads
exchange moves are already valid up to welded isotopy.

###  1.3. Artin representation

Originally, the Artin representation was defined in [Art47] within the
framework of classical braids, up to isotopy. Subsequently, it has been
extended and extensively studied in the contexts of link-homotopy and/or
welded braids. Below, we focus on its welded version, up to link-homotopy.

Recall the Artin braid generators $\sigma_{i}$ for $i\in\\{1,\ \ldots,\
n-1\\}$ illustrated in Figure 7 and the virtual braid generators $\rho_{i}$
for $i\in\\{1,\ \ldots,\ n-1\\}$, illustrated in Figure 7.

Figure 6. The Artin generator $\sigma_{i}$.

Figure 7. The virtual braid generator $\rho_{i}$.

We also need the _reduced free group_ denoted $\mathcal{R}F_{n}$, given by $n$
generators $x_{1}$, $x_{2}$, $\ldots$, $x_{n}$, subject to the relations
$[\omega x_{i}\omega^{-1},x_{i}]=1$ for any $1\leq i\leq n$ and any word
$\omega$ in $x_{1},\,\ldots,x_{n}$.

Let us now define the _homotopy welded Artin representation_.

######  Definition 1.10.

We call _homotopy welded Artin representation_ the homomorphism denoted by
$\phi:hWB_{n}\rightarrow Aut(\mathcal{R}F_{n})$. It is defined for
$i\in\\{1,\,\ldots,\ n-1\\}$ as follows:

$\phi(\rho_{i}):\left\\{\begin{array}[]{llll}x_{i}&\mapsto&x_{i+1},&\\\
x_{i+1}&\mapsto&x_{i},&\\\ x_{k}&\mapsto&x_{k}&\mbox{if }k\notin\\{i,\
i+1\\},\end{array}\right.$

and

$\phi(\sigma_{i}):\left\\{\begin{array}[]{llll}x_{i}&\mapsto&x_{i+1},&\\\
x_{i+1}&\mapsto&x_{i+1}^{-1}x_{i}x_{i+1},&\\\ x_{k}&\mapsto&x_{k}&\mbox{if
}k\notin\\{i,\ i+1\\}.\end{array}\right.$

######  Example 1.11.

For any $1\leq i\neq j\leq n$, consider the pure homotopy welded braid
generators:

$\chi_{ij}=\rho_{i}\cdots\rho_{j-2}\sigma_{j-1}\rho_{j-1}\cdots\rho_{i}$ if
$i<j$,

$\chi_{ij}=\rho_{i-1}\cdots\rho_{j}\sigma_{j}\rho_{j+1}\cdots\rho_{i-1}$ if
$j<i$.

A direct computation gives

$\Phi(\chi_{ij})(x_{k})=\left\\{\begin{array}[]{ll}x_{k}&\mbox{if }k\neq i,\\\
x_{j}^{-1}x_{i}x_{j}&\mbox{if }k=i.\end{array}\right.$

This defines the ‘pure’ part of the representation, which already appears in
[ABMW17a]. There, the authors consider two equivalent definitions of the
representation: a geometric one, in terms of ribbon tubes, and a combinatorial
one using Gauss diagrams.

######  Proposition 1.12.

The homotopy welded Artin representation $\phi$ is injective.

###### Proof.

Let us first note that for any $\beta\in hWB_{n}$ the homomorphism
$\phi(\beta)$ sends a generator $x_{i}$ to a conjugates of
$x_{\pi^{-1}(\beta)(i)}$. In particular, if $\beta\in\ker(\phi)$ then $\beta$
is a pure braid. Therefore, we conclude with [ABMW17a, Thm. 2.34], which
states that the representation $\phi$ restricted to $hWP_{n}$ is injective. ∎

######  Lemma 1.13.

Let $F$ be a $w$-tree for the trivial braid $\mathbf{1}\in hWB_{n}$ not having
its head on the $k$-th component. The action of $\phi(\mathbf{1}_{F})$ on
$x_{k}\in\mathcal{R}F_{n}$, is given by:

$\phi(\mathbf{1}_{F})(x_{k})=x_{k}.$

###### Proof.

By using the expanding rule of Figure 5 iteratively, we can break up $F$ into
a union of $w$-arrows, none of which has its head on the $k$-th component.
Then, using arrow isotopies, we can rearrange such a union into a product. It
is therefore enough to prove the result when $F$ is of degree one. In this
case, up to arrow isotopy and the Head/Tail reversal move of Lemma 1.9, we can
assume that $F$ is simply a horizontal $w$-arrow. Then, $\mathbf{1}_{F}$ is
either $\chi_{ij}$ or $\chi_{ij}^{-1}$ for some $1\leq i\neq j\leq n$ with
$k\neq i$, and the result follows from Example 1.11. ∎

##  2\. Proof of Torsion-Freeness

This section contains the main result of our paper: the proof that homotopy
braid groups are torsion-free (Theorem 2.9). The proof is decomposed into two
steps. In the first step, we show that torsion elements in $hWB_{n}$ appear as
conjugates of specific braids, having only virtual crossings. Then, in a
second step, using an algebraic obstruction, we show that such conjugates
never correspond to classical braids in $hB_{n}$.

###  2.1. A characterization of torsion

Let us denote by $\lambda_{n}\in hWB_{n}$ the homotopy welded braid,
illustrated in Figure 8, given by

$\lambda_{n}=\rho_{1}\rho_{2}\cdots\rho_{n-1}.$

Figure 8. The homotopy welded braid $\lambda_{n}$.

Notice that $\lambda_{n}$ is a torsion element of order $n$. We denote by
$\tau_{n}$ the $n$-cycle $(n\ n-1\ \cdots\ 2\ 1)=\pi(\lambda_{n})$ associated
to $\lambda_{n}$. When the value of $n$ is clear from the context, it will be
omitted in the notation. Let us describe in the following lemma the action of
conjugation by $\lambda$ on $w$-trees.

######  Lemma 2.1.

Let $F$ be a product of $w$-trees for the trivial braid $\mathbf{1}\in
hWB_{n}$, all having their heads on the $k$-th component. Then, the conjugate
$\lambda\mathbf{1}_{F}\lambda^{-1}$ is link-homotopic to
$\mathbf{1}_{F^{\prime}}$, where $F^{\prime}$ is a product of $w$-trees, all
having their heads on the component $\tau^{-1}(k).$

###### Proof.

Since the conjugate of a product is the product of the conjugates, it is
enough to prove the result when $F$ is a single $w$-tree. Consider, therefore,
the conjugate $\lambda\mathbf{1}_{F}\lambda^{-1}$ where $F$ is a single
$w$-tree having its head on the $k$-th component. We use an arrow isotopy to
slide $F$ through $\lambda$. This operation turns $F$ into a new $w$-tree
whose head is on component $\tau^{-1}(k)$. Finally, simplify $\lambda$ with
$\lambda^{-1}$ by applying a welded isotopy. ∎

In the following lemma, we use arrow calculus to describe specific conjugates
of homotopy welded braids. Later on Lemma 2.6, we will use this lemma to
characterize torsion elements in $hB_{n}$ up to conjugation.

######  Lemma 2.2.

Let $\beta\in hWB_{n}$ be a homotopy welded braid, whose associated
permutation is an $n$-cycle. Then $\beta$ is conjugate to
$\mathbf{1}_{F}\cdot\lambda$, where $F$ is a product of $w$-trees all having
their heads on the $n$-th component.

###### Proof.

Let us show by induction on $k\in\\{1,\ \ldots,\ n\\}$ that $\beta$ is
conjugate to a braid of the form $\mathbf{1}_{F_{k}}\cdot\lambda$, where
$F_{k}$ is a product of $w$-trees having their heads on components numbered
from $k$ to $n$.

For the base case, we can assume that $\beta$ is conjugated with $\beta_{1}$
satisfying $\pi(\beta_{1})=\tau=(n\ n-1\ \cdots\ 2\ 1)$. We can then write

$\beta_{1}=\theta\lambda,$

where $\theta=\beta_{1}\lambda^{-1}$ is a pure homotopy welded braid.
Additionally, it is clear that a virtualization move can be achieved by
surgery on a $w$-arrow (see [MY19, Fig. 6]). This implies that any pure braid
can be obtained by $w$-arrow surgeries on the trivial braid. We can thus write

$\beta_{1}=\mathbf{1}_{F_{1}}\cdot\lambda,$

where $F_{1}$ is a product of $w$-arrows.

Now, assume by induction that for some $1\leq k<n$ the braid $\beta$ is
conjugate with

$\beta_{k}=\mathbf{1}_{F_{k}}\cdot\lambda,$

where $F_{k}$ is a product of $w$-trees with their heads on components
numbered from $k$ to $n$. Note, on one hand, that Lemma 1.9 allows us to
exchange the relative position of two consecutive factors in a product, up to
higher degree $w$-trees. On the other hand, note that by Lemma 1.8, $w$-trees
of degree higher than $n$ are repeated and are therefore trivial up to link-
homotopy. Combining these two facts,777Note that a similar sorting process
appears in the proof of [MY19, Thm. 9.4]; see also [Yas09, Thm. 4.3] for an
analogous argument using claspers. we rearrange $w$-trees degree by degree to
obtain the following claim.

######  Claim 2.3.

The pure homotopy welded braid $\mathbf{1}_{F_{k}}$ decomposes as follows:

$\mathbf{1}_{F_{k}}=\mathbf{1}_{F^{\prime}_{k}}\mathbf{1}_{F^{\prime\prime}_{k}},$

where $F^{\prime}_{k}$ is a product of $w$-trees having their heads on the
$k$-th component and $F_{k}^{\prime\prime}$ is a product of $w$-trees having
their heads on components numbered from $k+1$ to $n$.

We then consider and compute the conjugate,

$\displaystyle\beta_{k+1}:=$
$\displaystyle\,(\mathbf{1}_{F^{\prime}_{k}})^{-1}\beta_{k}\mathbf{1}_{F^{\prime}_{k}},$
$\displaystyle=$
$\displaystyle\,(\mathbf{1}_{F^{\prime}_{k}})^{-1}\mathbf{1}_{F^{\prime}_{k}}\mathbf{1}_{F^{\prime\prime}_{k}}\cdot\lambda\mathbf{1}_{F^{\prime}_{k}}\lambda^{-1}\cdot\lambda,$
$\displaystyle=$
$\displaystyle\,\mathbf{1}_{F^{\prime\prime}_{k}}\cdot\lambda\mathbf{1}_{F^{\prime}_{k}}\lambda^{-1}\cdot\lambda.$

Finally, by Lemma 2.1, $\lambda\mathbf{1}_{F^{\prime}_{k}}\lambda^{-1}$ is a
product of $w$-trees having their heads on the $(k+1)$-th component which
concludes the induction as well as the proof. ∎

Let us now tackle the question of torsion, starting with a preliminary result
for pure braids. This proposition is well-known and can be found, for example,
in [HL90] and [Hum01].

######  Proposition 2.4.

The pure homotopy braid group $hP_{n}$ is torsion-free for any positive
integer $n$.

With the following lemma, we reduce our study to prime numbers of components.

######  Lemma 2.5.

If there is torsion in $hB_{n}$, then for some prime number $p\leq n$, there
exists a torsion element of order $p$ in $hB_{p}$.

###### Proof.

Let $\beta\in hB_{n}$ be a torsion element of prime order $p$, and let
$\pi(\beta)$ be its associated permutation. By Proposition 2.4, the subgroup
$hP_{n}$ is torsion-free. Consequently, $\pi(\beta)\neq 1$, implying that
$\pi(\beta)$ is a torsion element of order $p$ in the symmetric group $S_{n}$.
Specifically, $\pi(\beta)$ can be expressed as a product of disjoint
$p$-cycles, where $p\leq n$. Let us denote one such $p$-cycle as
$(i_{1},\ldots,i_{p})$. Now, consider the subgroup $G$ of $hB_{n}$ generated
by elements whose associated permutation sends the set
$\\{i_{1},\ldots,i_{p}\\}$ to itself. Next, define the homomorphism from $G$
to $hB_{p}$, which retains only the components $\\{i_{1},\ldots,i_{p}\\}$.
This homomorphism sends $\beta$ to a torsion element of order $p$ in $hB_{p}$,
thereby completing the proof. ∎

Recall from Definition 1.1 that a homotopy welded braid is _classical_ if it
belongs to the subgroup $hB_{n}$ of $hWB_{n}$ generated by the Artin
generators $\sigma_{i}$ (see Proposition 1.3). Note also that the welded braid
$\lambda\in hWB_{n}$ is a torsion element of order $n$. In the following
lemma, we give a characterization of torsion elements in $hB_{n}$ using this
braid $\lambda$ as well as the notion of classical braid.

######  Lemma 2.6.

There is torsion in $hB_{n}$ if and only if for some prime number $p\leq n$
the braid $\lambda\in hWB_{p}$ given by
$\lambda=\rho_{1}\rho_{2}\cdots\rho_{p-1}$ is conjugate to a classical braid.

###### Proof.

According to Lemma 2.5, if there is torsion in $hB_{n}$, we can find a torsion
element $\beta$ of order $p$ in $hB_{p}$ for some prime number $p$.
Consequently, $\pi(\beta)^{p}=\mathrm{Id}$, and by Proposition 2.4,
$\pi(\beta)\neq\mathrm{Id}$. Thus, $\pi(\beta)$ is a torsion element of order
$p$ and therefore it is a $p$-cycle. Now, by Proposition 1.3, we can consider
the classical braid $\beta$ as an element of $hWB_{p}$. Therefore, by Lemma
2.2, $\beta$ is conjugate to $\mathbf{1}_{F}\cdot\lambda$, where $F$ is a
product of $w$-trees, all having their heads on the $p$-th component.
Moreover, by Lemma 2.1, for any integer $k\in\\{1,\ \ldots,\ p-1\\}$, the
conjugates $\lambda^{k}\mathbf{1}_{F}\lambda^{-k}$ are products of $w$-trees,
none of which have their heads on the $p$-th component. Hence, by Lemma 1.13,

$\phi\left(\lambda^{k}\mathbf{1}_{F}\lambda^{-k}\right)(x_{p})=x_{p}$

for any $k\in\\{1,\ \ldots,\ p-1\\}$. In particular:

$\displaystyle\phi\big{(}(\mathbf{1}_{F}\lambda)^{p}\big{)}(x_{p})$
$\displaystyle=\phi\Big{(}\mathbf{1}_{F}\big{(}\lambda\mathbf{1}_{F}\lambda^{-1}\big{)}\big{(}\lambda^{2}\mathbf{1}_{F}\lambda^{-2}\big{)}\cdots\big{(}\lambda^{p-1}\mathbf{1}_{F}\lambda^{1-p}\big{)}\lambda^{p}\Big{)}(x_{p}),$
$\displaystyle=\phi(\mathbf{1}_{F})\circ\phi\big{(}\lambda\mathbf{1}_{F}\lambda^{-1}\big{)}\circ\phi\big{(}\lambda^{2}\mathbf{1}_{F}\lambda^{-2}\big{)}\circ\cdots\circ\phi\big{(}\lambda^{p-1}\mathbf{1}_{F}\lambda^{1-p}\big{)}(x_{p}),$
$\displaystyle=\phi(\mathbf{1}_{F})(x_{p}).$

On the other hand, since $\beta$ is a torsion element,
$\beta^{p}=(\mathbf{1}_{F}\lambda)^{p}=\mathbf{1}$, which implies that

$\phi\big{(}(\mathbf{1}_{F}\lambda)^{p}\big{)}(x_{p})=\phi(\mathbf{1})(x_{p})=x_{p}.$

By combining the two previous equalities, we deduce that
$\phi(\mathbf{1}_{F})(x_{p})=x_{p}$. Moreover, by Lemma 1.13 again, we also
have that $\phi(\mathbf{1}_{F})(x_{k})=x_{k}$ for any $k<p$. Thus,
$\phi(\mathbf{1}_{F})=\mathrm{Id}$ and by injectivity of $\phi$ (Proposition
1.12), the braid $\mathbf{1}_{F}$ is trivial. Therefore, the classical braid
$\beta$ is conjugate to $\lambda$, thus showing the ‘only if’ part of the
statement.

To show the converse implication, we use the fact that any conjugate of
$\lambda$ is a torsion element of order $p$ in $hWB_{p}$ and that,
consequently, a braid given by the same expression in $hB_{n}$ is also a
torsion element. ∎

###  2.2. An algebraic obstruction

By Lemma 2.6, the final part of the proof of our main result consists in
showing that for any integer $n$, the braid $\lambda$ has no classical braid
as conjugate. To achieve this, we need to determine whether a welded braid is
also a classical one. This is done in the following lemma, drawing inspiration
from [HL90, Thm. 1.7].

######  Lemma 2.7.

Let $\beta\in hWB_{n}$ be a homotopy welded braid. If $\beta$ is a classical
braid then

$\phi(\beta)(x_{1}x_{2}\cdots x_{n})=x_{1}x_{2}\cdots x_{n}.$

###### Proof.

This follows from a simple verification on the classical generators
$\sigma_{i}$. ∎

Recall now the _reduced Magnus expansion_. This is the homomorphism

$M:\mathcal{R}F_{n}\to\mathcal{A}_{n}$

from the reduced free group into the polynomial algebra $\mathcal{A}_{n}$ in
non-commuting variables $X_{1},\ \ldots,\ X_{n}$ in which monomials
$X_{\alpha_{1}}X_{\alpha_{2}}\cdots X_{\alpha_{k}}$ vanish if
$\alpha_{i}=\alpha_{j}$ for some $i\neq j$. The image of a generator $x_{i}$
is defined by the polynomial $M(x_{i}):=1+X_{i}$.

In the following lemma, we describe elements not belonging to the fixed points
of $\phi(\lambda)$.

######  Lemma 2.8.

For any $\beta\in hWB_{n}$, $\phi(\lambda)\circ\phi(\beta)(x_{1}x_{2}\cdots
x_{n})\neq\phi(\beta)(x_{1}x_{2}\cdots x_{n})$.

###### Proof.

Let us define the map $F:\mathcal{A}_{n}\to\mathbf{Z}$ which sends a
polynomial to the sum of the coefficients of its monomials of degree $n$. On
the one hand, we observe that

$F\Big{(}M\big{(}\phi(\rho_{i})(x)\big{)}\Big{)}=F\big{(}M(x)\big{)}\quad\text{and}\quad
F\Big{(}M\big{(}\phi(\sigma_{i})(x)\big{)}\Big{)}=F\big{(}M(x)\big{)},$

for any $x\in\mathcal{R}F_{n}$ and any $i$. The first equality is clear since
$M\big{(}\phi(\rho_{i})(x)\big{)}$ is obtained from $M(x)$ by permuting the
variables $X_{i}$ and $X_{i+1}$. The second equality is less obvious: indeed,
$M\big{(}\phi(\sigma_{i})(x)\big{)}$ is obtained from $M(x)$ by first
permuting the variables $X_{i}$ and $X_{i+1}$ and then substituting $X_{i}$
with $X_{i}+X_{i}X_{i+1}-X_{i+1}X_{i}$, potentially introducing new monomials
of degree $n$. However, these extra monomials appear in pairs and with
opposite signs and thus do not change the value of $F$. Therefore, we have
that

$F\big{(}M\big{(}\phi(\beta)(x)\big{)}\big{)}=F\big{(}M(x)\big{)},$

for any $\beta\in hWB_{n}$ and any $x\in\mathcal{R}F_{n}$. Moreover note that,

$\displaystyle F\big{(}M(x_{1}x_{2}\cdots x_{n})\big{)}$
$\displaystyle=F\big{(}(1+X_{1})(1+X_{2})\cdots(1+X_{n})\big{)},$
$\displaystyle=1,$

hence $F\big{(}M\big{(}\phi(\beta)(x_{1}x_{2}\cdots x_{n})\big{)}\big{)}=1$
for any $\beta\in hWB_{n}$. But on the other hand, for any
$\omega\in\mathcal{R}F_{n}$, $M\big{(}\phi(\lambda)(\omega)\big{)}$ is
obtained from $M(\omega)$ by permuting the variables $X_{i}$ cyclically. Hence
a necessary condition for having
$M\big{(}\phi(\lambda)(\omega)\big{)}=M(\omega)$ is that

$F\big{(}M(\omega)\big{)}\equiv 0\quad(\mathrm{mod}\ n).$

Therefore, $\phi(\lambda)\circ\phi(\beta)(x_{1}x_{2}\cdots
x_{n})\neq\phi(\beta)(x_{1}x_{2}\cdots x_{n})$ for any $\beta\in hWB_{n}$. ∎

We can finally prove the main theorem of this paper.

######  Theorem 2.9.

The homotopy braid group $hB_{n}$ is torsion-free for any number of components
$n$.

###### Proof.

Suppose by contradiction that there is a torsion element in $hB_{n}$. By Lemma
2.6 there exists a prime number $p\leq n$ and some braid $\beta\in hWB_{p}$
such that $\beta^{-1}\lambda\beta$ is a classical braid. According to Lemma
2.7 this conjugate must satisfy,

$\phi\big{(}\beta^{-1}\lambda\beta\big{)}\big{(}x_{1}x_{2}\cdots
x_{p})=x_{1}x_{2}\cdots x_{p},$

or equivalently,

$\phi(\lambda)\circ\phi(\beta)(x_{1}x_{2}\cdots
x_{p})=\phi(\beta)(x_{1}x_{2}\cdots x_{p}).$

This yields a contradiction by Lemma 2.8. ∎

It follows from Theorem 2.9 that the standard braid group $B_{n}$ is torsion-
free for all $n$. To prove this corollary, we need the following well-known
proposition, which essentially dates back to Artin [Art47].

######  Proposition 2.10.

The pure braid group $P_{n}$ is torsion-free for any number of components $n$.

We recover in this way a result of Fadell and Neuwirth (see Remark 2.12).

######  Corollary 2.11.

The braid group $B_{n}$ is torsion-free for any number of components $n$.

###### Proof.

Let us consider the projection $p:B_{n}\to hB_{n}$. Since $hB_{n}$ is torsion-
free (Theorem 2.9), any torsion element in $B_{n}$ must belong to the kernel
$K:=\ker(p)$. However, it is clear from [Gol74] that $K\subset P_{n}$, thus
$K$ is torsion-free by Proposition 2.10 and the proof is complete. ∎

######  Remark 2.12.

The study of torsion in braid groups dates back to Fadell and Neuwirth in
1962.888We mention however a more geometric proof, given in [GM11, §2] and
based on the earlier works of [Nie42, vK19, Eil34]. Building upon topological
methods, they show in [FN62, Thm. 8] that $B_{n}$ is torsion-free for every
$n$. One may wonder whether a similar methods may provide alternative proof of
the torsion-freeness in homotopy braid groups. However, to date, there is no
known classifying space for $hB_{n}$ that would enable such a demonstration.
Moreover, another common proof of torsion-freeness in $B_{n}$ appears in
[Deh94], where Dehornoy established the stronger property of left-
orderability. One may wonder again if a similar proof applies for $hB_{n}$,
but the question of orderability in this group is still open.

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Yidi Jiang1, Ruijie Tao1, Zexu Pan1, Haizhou Li1,2

# Target Active Speaker Detection with Audio-visual Cues

###### Abstract

In active speaker detection (ASD), we would like to detect whether an on-
screen person is speaking based on audio-visual cues. Previous studies have
primarily focused on modeling audio-visual synchronization cue, which depends
on the video quality of the lip region of a speaker. In real-world
applications, it is possible that we can also have the reference speech of the
on-screen speaker. To benefit from both facial cue and reference speech, we
propose the Target Speaker TalkNet (TS-TalkNet), which leverages a pre-
enrolled speaker embedding to complement the audio-visual synchronization cue
in detecting whether the target speaker is speaking. Our framework outperforms
the popular model, TalkNet on two datasets, achieving absolute improvements of
1.6% in mAP on the AVA-ActiveSpeaker validation set, and 0.8%, 0.4%, and 0.8%
in terms of AP, AUC and EER on the ASW test set, respectively. Code is
available at https://github.com/Jiang-Yidi/TS-TalkNet/.

Index Terms: Active speaker detection, target speaker, audio-visual, speaker
recognition

##  1 Introduction

The study of active speaker detection (ASD) is to determine whether an on-
screen person is speaking in each video frame of an audio-visual scene [1]. It
plays a critical frontend role for various speech processing tasks, such as
audio-visual speaker localization [2], speaker verification [3, 4, 5], speaker
extraction [6, 7], speech recognition [8, 9], among others [10, 11].

Typically, ASD methods rely on modeling the synchronization between the audio
and visual modalities to detect whether a person is talking [12, 13, 14].
Early approaches focused on short temporal segments of speech activity [15,
16], lip movement [17] and audio-visual synchronization [18, 19, 20] to
identify the active speaker. These methods have an efficient structure, but
lack the ability to capture the long-term temporal association between the
audio and visual signals. The recent work, TalkNet [21], incorporated long-
term temporal information and demonstrated strong performance on the ASD task.
Following this pipeline, other works attempted to model audio-visual
synchronization in a more efficient way, such as Graph Neural Networks (GNN)
[22, 23]. However, obtaining high-resolution video of the lip regions is not
always feasible, which can hinder the accurate determination of audio-visual
synchronization.

Human has an inherent ability to focus on a known speaker in complex social
environments, known as selective auditory attention in a ``cocktail party''
[24, 25, 26]. In other words, when we are already familiar with a person's
voice, i.e., the target speaker, we can perform selective listening by
comparing the heard voice with the reference voice.

In target speaker extraction task, the voice of a specific speaker is pre-
enrolled in form of a speaker embedding. This speaker embedding is then used
as a reference for target speaker extraction [27, 28]. Similarly, target
speaker voice activity detection maps unknown speech to extracted target
speaker embeddings to perform speaker diarization task [15, 29]. In [30],
speaker diarization task was reformulated to a supervised classification
problem by leveraging audio-visual correlation and speaker models for each
speaker. These studies highlight the benefits of understanding a target
speaker's voice characteristic in speech processing. Motivated by these
studies, we believe that the reference speech could be informative in ASD task
as well, in particular, in challenging acoustic environments.

In this work, we propose the Target Speaker TalkNet (TS-TalkNet) framework as
an extension of TalkNet [21], introducing the concept of the ``target speaker"
to the ASD task. TS-TalkNet first enrolls the on-screen speaker as the target
speaker and then performs auditory attention with the corresponding pre-
enrolled speaker embedding for the ASD task. To achieve this, we construct a
face-speaker library where each target speaker's face is associated with their
pre-enrolled reference speech (if exists), providing the target speaker
embedding. TS-TalkNet explores the voice characteristic cue to determine
whether the audio signals resemble the target speaker embedding, which
complements the audio-visual synchronization cue. Noted that if reference
speech does not exist, TS-TalkNet only utilizes the synchronization cue. We
also explore various multi-modal fusion methods to investigate the efficacy of
leveraging speaker embedding in TS-TalkNet. The contributions of this paper
can be summarised as follows:

  1. 1.

We explore the use of reference speech to assist active speaker detection.

  2. 2.

We propose the TS-TalkNet framework that fully makes use of both audio and
visual cues, i.e., voice characteristic signal and audio-visual
synchronization information.

  3. 3.

We conduct experiments on the AVA-ActiveSpeaker (AVA) and the Active Speakers
in the Wild (ASW) datasets with improvements: 1.6% in mAP on the AVA
validation set; 0.8%, 0.4%, and 0.8% in AP, AUC and EER, respectively, on the
ASW test set, confirming the superiority of TS-TalkNet.

Figure 1: The overview framework of our TS-TalkNet. It consists of a feature
representation frontend and a speaker detection backend classifier. The
feature representation frontend includes audio and visual temporal encoders,
and speaker encoder. The speaker detection backend comprises a cross-attention
and a fusion module to combine the audio, visual and speaker embeddings, and a
self-attention module to predict the ASD scores. The lock represents the
speaker encoder is frozen in our framework.

##  2 TS-TalkNet: Target Active Speaker Detection

As depicted in Figure 1, our proposed TS-TalkNet framework takes the cropped
face track video, corresponding audio signal and the target speaker's enrolled
speech signal as inputs, and outputs the binary decision of the speech
activity for each video frame. TS-TalkNet consists of a feature representation
frontend and a speaker detection backend.

###  2.1 Feature representation frontend

The feature representation frontend includes an audio temporal encoder, a
visual temporal encoder and a speaker encoder. Following TalkNet, the audio
and visual temporal encoders are used to learn audio and visual embeddings
from the audio and face track inputs. The speaker encoder is proposed to
supplement the target speaker information.
Visual temporal encoder. The visual temporal encoder is designed to learn the
long-term representations of facial expression dynamics by encoding the visual
stream into a sequence of visual embeddings $F_{v}$ with consistent time
resolution. To achieve this, the visual frontend captures spatial information
within each video frame and encodes the video frame stream into a sequence of
frame-based embeddings. The visual temporal network then employs a video
temporal convolutional block, comprising of five residual connected ReLU,
batch normalization, and depth-wise separable convolutional layers, followed
by a Conv1D layer that reduces the embedding dimension. The purpose of this
encoder is to capture the temporal content within a long-term visual spatio-
temporal structure.
Audio temporal encoder. The audio temporal encoder is a ResNet34 network [31]
with a squeeze-and-excitation (SE) module [32]. This network extracts audio
content representation from the temporal dynamics of a sequence of audio
frames, which are initially represented by a vector of Mel-frequency cepstral
coefficients (MFCCs). The output of this network is a sequence of audio
embeddings, denoted as $F_{a}$. $F_{a}$ matches the time resolution of the
visual embeddings, $F_{v}$.
Speaker encoder. To begin, we construct the face-speaker library, which
contains multiple reference speeches for each target speaker. Further details
regarding the construction process can be found in Subsection 3.2.

We designate the individual of each face track input as the `target speaker'.
Then for each face track, we randomly select one pre-enrolled speech from all
the associated speeches in the face-speaker library. The pre-enrolled speech
is then used as the input of the speaker encoder. To incorporate target
speaker characteristics into our TS-TalkNet, we leverage a pre-trained speaker
recognition model to obtain the target speaker embedding. To guarantee robust
performance, the ECAPA-TDNN model [33] is used as the speaker encoder. It has
demonstrated reliable performance for speaker recognition task.

The ECAPA-TDNN model employs emphasized channel attention to selectively focus
on critical parts of the speech signal, propagating that information through
the network and aggregating it to make a final decision. From the variable
lengths of input utterances, the output speaker embedding has the fixed
dimension. This capability enables the model to handle speech signals in
various scenarios and generate robust speaker representations more
effectively.

From the enrolled speech input, we obtain the target speaker embedding $F_{s}$
and convey the target speaker information to the speaker detection backend. We
freeze the parameters of this module in our framework, since our purpose is to
obtain robust embedding for the target speaker.

###  2.2 Speaker detection backend

The speaker detection backend comprises a cross-attention module to achieve
audio-visual synchronization, a fusion module to combine the speaker embedding
with audio-visual embeddings, in addition to a self-attention module to
capture speaking activities from the temporal context at the utterance level.
We aims to combine three embeddings generated from the feature representation
frontend: $F_{a}$ for audio modality, $F_{v}$ for visual modality, and $F_{s}$
for speaker characteristic, to provide a comprehensive representation of the
speaking activities for ASD prediction.
Cross-attention module. Firstly, we use a cross-attention structure along the
temporal dimension to achieve audio-visual synchronization and interaction.
The inputs are the vectors of query ($Q_{a}$, $Q_{v}$), key ($K_{a}$,
$K_{v}$), and value ($V_{a}$, $V_{v}$) from audio and visual embeddings,
$F_{a}$ and $F_{v}$ respectively, projected by a linear layer. As formulated
in Equation 1 and Equation 2, the outputs are the audio attention embedding
$F_{a\rightarrow v}$ and visual attention embedding $F_{v\rightarrow a}$,
where $d$ denotes the dimension of $Q$, $K$and $V$. The outputs are
concatenated together along the temporal direction as audio-visual embedding
$F_{av}$.

$F_{a\rightarrow v}=softmax(\frac{Q_{v}K_{a}^{\mathrm{T}}}{\sqrt{d}})V_{a}$
(1) $F_{v\rightarrow
a}=softmax(\frac{Q_{a}K_{v}^{\mathrm{T}}}{\sqrt{d}})V_{v}$ (2)

Figure 2: The different fusion approaches for audio, visual and speaker
embeddings. We denote the structure (a) and (b) as `Fus1' and `Fus2',
respectively. $\otimes$ represents embedding concatenation along the time
dimension.

Fusion module. The fusion module is used to combine the speaker embedding with
the audio-visual embedding to obtain the overall audio-visual-speaker
representation $F_{avs}$.

Noted that $F_{s}$ is not a temporal embedding, while $F_{av}$,
$F_{v\rightarrow a}$ and $F_{a\rightarrow v}$ are all temporal embedding
sequences. For alignment, we replicate $F_{s}$ along the time dimension to
$\hat{F}_{s}$ to implement the fusion module.

We have investigated three different fusion structures to demonstrate that our
idea of leveraging voice characteristics isn't restricted to a specific
network architecture. As shown in Fig 2 (a) and (b), `Fus1' structure uses
$F_{a\rightarrow v}$ and $\hat{F}_{s}$ as inputs for cross-attention. `Fus2'
structure applies $F_{av}$ to achieve the cross-attention process with
$\hat{F}_{s}$. The third structure is to concatenate the $\hat{F}_{s}$ and
$F_{av}$ along the temporal dimension, which is denoted as `Concat'.
Self-attention module. After obtaining the audio-visual-speaker representation
$F_{avs}$, we follow the same strategy as TalkNet [21] and apply a self-
attention structure to model $F_{avs}$ temporal information to distinguish
between speaking and non-speaking frames. This structure is similar with the
cross-attention structure, except that the query, key and value in the
attention layer all come from the joint embedding $F_{avs}$.

###  2.3 Loss function

Treating ASD as a frame-level classification task, we project the output of
the self-attention structure to an ASD label sequence using a fully connected
layer and the softmax operation. We then compare the predicted label sequence
to the ground truth label sequence using cross-entropy loss, which is given in
Equation 3. Here, $\hat{y}_{i}$ and $y_{i}$ represent the predicted and ground
truth ASD labels for the $i^{th}$ video frame, respectively, where
$i\in[1,T]$. The value of $T$ represents the number of video frames.

$\mathcal{L}=-\frac{1}{T}\sum_{i=1}^{T}(y_{i}\cdot
log\hat{y}_{i})+(1-y_{i})\cdot log(1-\hat{y}_{i})$ (3)

##  3 Experiments

In this section, we describe the datasets and experimental details to evaluate
the proposed TS-TalkNet framework.

###  3.1 Dataset

AVA-ActiveSpeaker (AVA) dataset [1] is a large-scale audio-visual active
speaker detection dataset. It consists of 262 videos extracted from Hollywood
movies, with 120 for training, 33 for validation, and 109 for testing. This
dataset contains about 3.65 million human-labeled video frames or 38.5 hours
of face tracks, and the corresponding audio.
The Active Speakers in the Wild (ASW) dataset [34] consists of 212 videos
randomly selected from the VoxConverse dataset [35]. It contains 13.4 hours
videos in which 56.7% are active in the training set, 9.6 hours videos where
60.4% are active in the validation set, and 7.9 hours videos where 57.0% are
active in the test set.

###  3.2 Face-speaker library

We pre-process the dataset to build the face-speaker library by searching the
face tracks from the same person using a face recognition module. We utilize
the pre-trained ResNet50 model on the Glint360K dataset [36] as the face
recognition module and set the face similarity threshold at 0.7. The enrolled
speeches come from the corresponding audio of the found tracks, which are
located by extracting the active segments using ground-truth labels. Then for
each input, we can retrieve the pre-enrolled speeches of the target speaker
from the face-speaker library according to the input faces. If the reference
speeches exist, we randomly choose one as the target speaker enrolled speech
and feed it into TS-TalkNet. Otherwise, we set the enrolled speech to zero
vectors to process detection. In this way, our TS-TalkNet can handle different
situations, whether there is an enrolled speech record or not.

###  3.3 Implementation details

The proposed TS-TalkNet is implemented by PyTorch using the Adam optimizer. We
set the initial learning rate to $10^{-4}$ and decrease it by 5% for each
epoch. The MFCC dimension is set to 13, and all faces are resized into
$112\times 112$ pixels. Both audio and visual embeddings are set to 128
dimensions. For both cross-attention and self-attention structures, a single
transformer layer with 8 attention heads is used. We apply visual augmentation
such as random flipping, rotating, and cropping the original images to improve
framework performance.

###  3.4 Evaluation metric

For the AVA dataset, we use the official ActivityNet evaluation tool to
compute mean average precision (mAP) and evaluate on the validation (val) set.
For the ASW dataset, following [34, 21], we compute three metrics: mean
precision (AP), area under the receiver operating characteristic (AUC), and
equal error rate (EER) using sklearn package and evaluate on the ASW val and
test set.

Table 1: Comparison with the state of the arts on the AVA val set in terms of mAP. The bold rows represent the results of our proposed methods. Method |  mAP (%)$\uparrow$
---|---
Roth et al. [13, 1] |  79.2
Zhang et al. [14] |  84.0
MAAS-LAN [13] |  85.1
Alcazar et al. [37] |  87.1
Chung et al [19] |  87.8
MAAS-TAN [13] |  88.8
UniCon [38] |  92.0
TalkNet [21] |  92.3
ASDNet [39] |  93.5
Light-ASD [40] |  94.1
UniCon+ [41] |  94.5
SPELL+ [23] |  94.9
TS-TalkNet (Fus1) |  93.3
TS-TalkNet (Fus2) |  93.5
TS-TalkNet (Concat) |  93.9
Table 2: Comparison with the state of the arts on the ASW val and test set in terms of AP, AUC and EER. The bold rows represent the results of our proposed methods. Set |  Method |  AP (%)$\uparrow$ |  AUC (%)$\uparrow$ |  EER (%)$\downarrow$
---|---|---|---|---
Val |  TalkNet [21] |  96.4 |  98.2 |  6.0
|  TS-TalkNet (Fus1) |  97.3 |  98.6 |  5.4
|  TS-TalkNet (Fus2) |  97.5 |  98.7 |  5.1
|  TS-TalkNet (Concat) |  97.7 |  98.7 |  5.1
Test |  RothNet [1] |  89.7 |  - |  -
|  SyncNet [20] |  92.4 |  - |  -
|  ASW-SSL [34] |  90.5 |  - |  -
|  LoCoNet [42] |  93.4 |  95.1 |  9.8
|  ASW-BGRUs [34] |  96.6 |  97.2 |  6.2
|  TalkNet [21] |  97.7 |  98.6 |  5.1
|  TS-TalkNet (Fus1) |  98.3 |  98.9 |  4.8
|  TS-TalkNet (Fus2) |  98.3 |  98.9 |  4.4
|  TS-TalkNet (Concat) |  98.5 |  99.0 |  4.3

##  4 Results and Analysis

In this section, we present the performance comparison of our proposed TS-
TalkNet with state-of-the-art methods on the AVA and ASW datasets to show its
efficiency. We also explore the architectures of the fusion module and analyze
the results to find out the improvement under the different test conditions.

###  4.1 AVA results

We firstly report the performance of TS-TalkNet on the AVA val dataset in
Table 1. With the inclusion of enrolled speech, we observe that TS-TalkNet
(Concat) achieves 93.9% mAP and 1.6% mAP improvements over TalkNet on the AVA
val set. Moreover, for TS-TalkNet (Fus1) and TS-TalkNet (Fus2) frameworks, we
obtain the improvements by 1.0% and 1.2% mAP, respectively.

###  4.2 ASW results

The results of TS-TalkNet on the ASW val and test dataset are reported in
Table 2. Specially, the proposed TS-TalkNet (Concat) framework outperforms
TalkNet by 1.3%, 0.6% and 0.9% in terms of AP, AUC and EER, respectively, on
the ASW val set; and 0.8%, 0.4% and 0.8% in terms of AP, AUC and EER,
respectively, on the ASW test set. Similarly, both TS-TalkNet (Fus1) and TS-
TalkNet (Fus2) frameworks outperform TalkNet by a large margin.

We visualize the ASD results in Figure 3 for two videos from the ASW dataset.
Our results verify the benefits of speaker embedding on ASD task and
demonstrate the superiority of TS-TalkNet.

Table 3: Results comparison of different percentages of the frames with active labels on the ASW val set. For example, `0-20' represents the tracks with less than 20% active labels. Active Frame (%) | Method |  AP (%)$\uparrow$ |  AUC (%)$\uparrow$ |  EER (%)$\downarrow$
---|---|---|---|---
`0 - 20' | TalkNet | 8.29 | 95.92 | 9.45
TS-TalkNet | 11.54 | 96.65 | 7.63
`20 - 40' | TalkNet | 83.47 | 94.85 | 11.22
TS-TalkNet | 89.37 | 96.06 | 9.41
`40 - 60' | TalkNet | 92.54 | 93.33 | 13.95
TS-TalkNet | 93.39 | 93.94 | 13.35
`60 - 80' | TalkNet | 96.95 | 93.16 | 14.37
TS-TalkNet | 97.30 | 93.99 | 12.97
`80 - 100' | TalkNet | 99.71 | 93.44 | 13.89
TS-TalkNet | 99.75 | 94.01 | 13.28

###  4.3 Qualitative studies

Fusion structure. We implement three fusion approaches to combine the audio-
visual embeddings and speaker embedding, as mentioned in Subsection 2.2. As
shown in Table 2, for TS-TalkNet (Fus1), we obtain the improvements by 0.9%,
0.4% and 0.6% in terms of AP, AUC and EER, respectively, over the TalkNet on
the ASW val set; and 0.6%, 0.3% and 0.3% in terms of AP, AUC and EER,
respectively, on the ASW test set. For TS-TalkNet (Fus2), we achieve the
improvements by 1.1%, 0.5% and 0.9% in terms of AP, AUC and EER, respectively,
over TalkNet on the ASW val set; and 0.6%, 0.3% and 0.7% in terms of AP, AUC
and EER respectively on the ASW test set. Similar observations can be found in
Table 1.

These results indicate that incorporating the speaker embedding can improve
the performance of TS-TalkNet, regardless of the fusion approaches. Therefore,
our idea of using speaker characteristics to assist ASD is not restricted to a
specific network architecture.
Analysis study. We aim to investigate the effect of speaker embedding
incorporation for face tracks with varying proportions of active frames. So we
analyze the performance improvement of TS-TalkNet on the ASW val set by
dividing the instances into five sections based on the percentage of the
frames with active labels of each track input. As shown in Table 3, TS-TalkNet
can improve the detection accuracy for each section, indicating that the
speaker-specific characteristics are valuable for detecting the target
speaker's active frames across different scenarios.

Figure 3: The results of TS-TalkNet for real-world videos with one person (a)
and multiple persons (b) on the screen. The green box denotes the active
speaker. The red box denotes the inactive speaker. As demonstrated in (a),
despite the occlusion of the lip in certain video frames, which compromises
the audio-visual synchronization cue, TS-TalkNet can still accurately detect
the speaker with the aid of speaker embedding complementation.

##  5 Conclusion

In this paper, we propose the Target Speaker TalkNet (TS-TalkNet) for active
speaker detection by incorporating target speaker embeddings. Our experiments
and studies demonstrate that speaker characteristic plays a crucial role in
solving the ASD problem. Our work provides a new perspective and potential
solution for improving the performance of ASD systems in complicated acoustic
environments. In future work, we can investigate the potential applications of
TS-TalkNet in other speech-related tasks.

##  6 Acknowledgements

This work is funded by 1) Huawei Noah’s Ark Lab; 2) National Natural Science
Foundation of China (Grant No. 62271432); 3) Agency for Science, Technology
and Research (A*STAR) under its AME Programmatic Funding Scheme (Project No.
A18A2b0046)

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