# Disordered arcs and Harer stability

Oscar Harr, Max Vistrup, and Nathalie Wahl

# Disordered arcs and Harer stability

Oscar Harr, Max Vistrup, and Nathalie Wahl

###### Abstract.

We give a new proof of homological stability with the best known isomorphism
range for mapping class groups of surfaces with respect to genus. The proof
uses the framework of Randal-Williams–Wahl and Krannich applied to disk
stabilization in the category of bidecorated surfaces, using the Euler
characteristic instead of the genus as a grading. The monoidal category of
bidecorated surfaces does not admit a braiding, distinguishing it from
previously known settings for homological stability. Nevertheless, we find
that it admits a suitable Yang–Baxter element, which we show is sufficient
structure for homological stability arguments.

##  1\. Introduction

Let $S_{g,r}^{s}$ be a surface of genus $g$ with $r$ boundary components and
$s$ punctures. The mapping class group
$\Gamma(S_{g,r}^{s}):=\pi_{0}{\operatorname{Homeo}}(S_{g,r}^{s}\textrm{ rel
}\partial S)$ of $S$ satisfies homological stability: the homology group
$H_{i}(\Gamma(S_{g,r}^{s});\mathbb{Z})$ is independent of $g$ and $r$ when $g$
is large relative to $i$. This stability result was originally proved by Harer
in [Har85], and later improved by Ivanov, Boldsen and Randal-Williams [Iva89,
Bol12, RW16], see also [Har93, Wah13, HV17, GKRW19]. We recast the result here
as a stability theorem in the category of bidecorated surfaces, and give a new
proof of the best know stability range using the most straightforward
inductive argument originally designed by Quillen, and formalized in [RWW17,
Kra19]. Our proof at the same time illustrates how little is needed to run the
stability machines of these two papers.

Our main stability result is the following, recovering precisely the ranges of
[Bol12, Thm 1] and [RW16, Thm 7.1 (i),(ii)]:

######  Theorem A.

Let $S_{g,b}^{s}$ be a surface of genus $g\geq 0$, with $r\geq 1$ marked
boundary components and $s\geq 0$ punctures, and let
$\Gamma(S_{g,r}^{s})=\pi_{0}{\operatorname{Homeo}}(S_{g,r}^{s}\textrm{ rel
}\partial S)$ denote its mapping class group. The map

$H_{i}(\Gamma(S_{g,r}^{s});\mathbb{Z})\mathrel{\smash{\meno}}\mathrel{\mkern-3.0mu}\rightarrow
H_{i}(\Gamma(S^{s}_{g,r+1});\mathbb{Z})$

induced by gluing a pair or pants along one boundary component is always
injective, and an isomorphism when $i\leq\frac{2g}{3}$, and the map

$H_{i}(\Gamma(S_{g,r+1}^{s});\mathbb{Z})\mathrel{\smash{\meno}}\mathrel{\mkern-3.0mu}\rightarrow
H_{i}(\Gamma(S^{s}_{g+1,r});\mathbb{Z})$

induced by gluing a pair of pants along two boundary components is an
epimorphism when $i\leq\frac{2g+1}{3}$ and an isomorphism when
$i\leq\frac{2g-2}{3}$.

Combining the two maps in the theorem gives a genus stabilization that is
known to be close to optimal by a computation of Morita [Mor03] and low
dimensional computations, see Remarks 2.5 and 4.11. While we do not know
whether the two ranges in the above statement can be individually improved, it
is remarkable that three rather different proofs (those of Boldsen [Bol12],
Randal-Williams [RW16], and ours) end up with the exact same ranges.

A particular feature of our proof is that the two maps occurring in the
theorem will be for us “the same map”, namely a disk stabilization in the
category ${{\mathbf{M}}_{2}}$ of bidecorated surfaces. A bidecorated surface
is a surface $S$ with two marked intervals $I_{0},I_{1}$ in its boundary. The
two intervals may lie on the same or on different boundary components.
Morphisms in ${{\mathbf{M}}_{2}}$ are mapping classes, i.e. isotopy classes of
homeomorphisms, and ${{\mathbf{M}}_{2}}$ admits a monoidal structure
$\mathbin{\text{{\\#}}}$ defined by identifying the marked intervals in pairs.

Our main example of a bidecorated surface will be the bidecorated disk $D$. As
shown in Lemma 3.1, taking sums of the disk with itself in
${{\mathbf{M}}_{2}}$ produces surfaces of any genus:
$D^{\mathbin{\text{{\\#}}}2g+1}$ is a surface $S_{g,1}$ of genus $g$ with a
single boundary component, while $D^{\mathbin{\text{{\\#}}}2g+2}$ is a surface
$S_{g,2}$ of genus $g$ with two boundary components, each containing a marked
interval. To obtain any surface $S_{g,r}^{s}$ with $r\geq 1$, we will consider
the object $S\mathbin{\text{{\\#}}}D^{\mathbin{\text{{\\#}}}2g}$ in
${{\mathbf{M}}_{2}}$, for $S=S_{0,r}^{s}$ a genus 0 surface with $r$ boundary
components and $s$ punctures. Now the maps in Theorem A are precisely the disk
stabilization maps in ${{\mathbf{M}}_{2}}$:

${\operatorname{Aut}}_{{\mathbf{M}}_{2}}(S\mathbin{\text{{\\#}}}D^{\mathbin{\text{{\\#}}}2g})\xrightarrow{\mathbin{\text{{\\#}}}D}{\operatorname{Aut}}_{{\mathbf{M}}_{2}}(S\mathbin{\text{{\\#}}}D^{\mathbin{\text{{\\#}}}2g+1})\xrightarrow{\mathbin{\text{{\\#}}}D}{\operatorname{Aut}}_{{\mathbf{M}}_{2}}(S\mathbin{\text{{\\#}}}D^{\mathbin{\text{{\\#}}}2g+2})$

for these particular choices of surfaces.

Theorem A is thus the statement that disk stabilization
$\mathbin{\text{{\\#}}}D$ in ${{\mathbf{M}}_{2}}$ induces isomorphisms on the
homology of these automorphism groups in a range. We show in the present paper
that this result can be obtained as a direct application of the main result of
[Kra19], from which an additional stability statement with twisted
coefficients automatically follows. We start by stating this additional
result.

### Twisted coefficients

Fix $r\geq 1$ and $s\geq 0$. In our setting, a coefficient system $F$ for the
mapping class group $\Gamma(S_{g,r}^{s})$ is a collection of
$\mathbb{Z}[\Gamma(S_{g,r}^{s})]$-modules $F_{2g}$ and
$\mathbb{Z}[\Gamma(S_{g,r+1}^{s})]$-modules $F_{2g+1}$ for each $g\geq 0$,
together with maps

$F_{n}\mathrel{\smash{\meno}}\mathrel{\mkern-3.0mu}\rightarrow F_{n+1}$

equivariant with respect to the disk stabilization and satisfying that a
certain Dehn twist acts trivially on the image of $F_{n}$ in $F_{n+2}$ under
double stabilization (see Definition 4.6). Given a coefficient system, one can
define a notion of degree; a constant coefficient systems has degree 0 and for
example the coefficient system
$F_{2g+i}=H_{1}(S_{g,r+i}^{s};\mathbb{Z})^{\otimes k}$, $i\in\\{0,1\\}$, has
degree $k$ (see Example 4.7).

We obtain the following twisted stability result:

######  Theorem B.

Let $\Gamma(S_{g,r}^{s})$ be as in Theorem A, and $F$ be a coefficient system
of degree $k$. The stabilization map

$H_{i}(\Gamma(S_{g,r}^{s});F_{2g})\mathrel{\smash{\meno}}\mathrel{\mkern-3.0mu}\rightarrow
H_{i}(\Gamma(S^{s}_{g,r+1});F_{2g+1})$

is an epimorphism for $i\leq\frac{2g-3k-2}{3}$ and an isomorphism for
$i\leq\frac{2g-3k-5}{3}$, and the map

$H_{i}(\Gamma(S_{g,r+1}^{s});F_{2g+1})\mathrel{\smash{\meno}}\mathrel{\mkern-3.0mu}\rightarrow
H_{i}(\Gamma(S^{s}_{g+1,r});F_{2g+2})$

is an epimorphism for $i\leq\frac{2g-3k-1}{3}$ and an isomorphism for
$i\leq\frac{2g-3k-4}{3}$. In these bounds, $3k$ can be replaced by $k$ if $F$
is in addition split in the sense of Definition 4.6.

Stability theorems for mapping class groups with twisted coefficients can be
found in the work of Ivanov, Boldsen, Randal-Williams–Wahl, and
Galatius–Kupers–Randal-Williams [Bol12, Iva93, RWW17, GKRW19]. The results are
not easy to compare as the types of coefficient system that are permitted
depend on the paper, but some classical examples such as the one described
above fit all frameworks (see Remarks 4.8 and 4.11 for more details).

### Braided action and Yang–Baxter operators

We want to obtain Theorems A and B as consequences of Theorems A and C of
[Kra19]. For this, we first have to show that disk stabilization in the
monoidal category $({{\mathbf{M}}_{2}},\mathbin{\text{{\\#}}})$ comes from an
action of a braided monoidal groupoid.

Let $\mathbf{B}$ denote the groupoid of braid groups, with object the natural
numbers and the braid group $B_{n}$ as automorphisms of $n$. We will construct
an action of $\mathbf{B}$ on ${{\mathbf{M}}_{2}}$ using an appropriate
Yang–Baxter operator in ${{\mathbf{M}}_{2}}$: The sum of bidecorated disks
$D\mathbin{\text{{\\#}}}D$ in ${{\mathbf{M}}_{2}}$ is a cylinder, whose
mapping class group is an infinite cyclic group generated by the Dehn twist
$T$ along the core circle of the cylinder. It turns out that this morphism
$T\in{\operatorname{Aut}}_{{{\mathbf{M}}_{2}}}(D\mathbin{\text{{\\#}}}D)$ is a
Yang–Baxter operator in ${{\mathbf{M}}_{2}}$, in the sense that it satisfies
the equation

$(T\mathbin{\text{{\\#}}}1)(1\mathbin{\text{{\\#}}}T)(T\mathbin{\text{{\\#}}}1)=(1\mathbin{\text{{\\#}}}T)(T\mathbin{\text{{\\#}}}1)(1\mathbin{\text{{\\#}}}T)$

in ${\operatorname{Aut}}_{{{\mathbf{M}}_{2}}}(D^{\mathbin{\text{{\\#}}}3})$.
The same holds for the inverse twist $T^{-1}$, that will turn out more
convenient for us. As explained in Section 5.1, we get an associated strong
monoidal functor
$\mathbf{B}\mathrel{\smash{\meno}}\mathrel{\mkern-3.0mu}\rightarrow{{\mathbf{M}}_{2}}$
taking the object $n$ to $D^{\mathbin{\text{{\\#}}}n}$. The corresponding
homomorphism
$B_{n}\mathrel{\smash{\meno}}\mathrel{\mkern-3.0mu}\rightarrow\text{Aut}_{{\mathbf{M}}_{2}}(D^{\mathbin{\text{{\\#}}}n})$
can be identified with the geometric embedding in the sense of [Waj99],
associated to the chain of curves $a_{1},\dots,a_{n-1}$ in

$D^{\mathbin{\text{{\\#}}}n}=D\mathbin{\text{{\\#}}}D\mathbin{\text{{\\#}}}\dots\mathbin{\text{{\\#}}}D,$

where the $i$th curve $a_{i}$ is the core circle in the $i$th cylinder
$D\mathbin{\text{{\\#}}}D$ in the above sum, see Lemma 3.5 and Example 5.3.

The strong monoidal functor
$\mathbf{B}\mathrel{\smash{\meno}}\mathrel{\mkern-3.0mu}\rightarrow{{\mathbf{M}}_{2}}$
from above endows ${{\mathbf{M}}_{2}}$ with the structure of an $E_{1}$-module
over the braid groupoid $\mathbf{B}$, and since the latter is braided
monoidal, we can apply the results of [Kra19] to study disk stabilization in
${{\mathbf{M}}_{2}}$.

######  Remark 1.1.

Homological stability frameworks such as [RWW17, Kra19, GKRW19] require an
$E_{2}$-algebra, or the weaker structure of $E_{1}$-module over an
$E_{2}$-algebra, as input. This is a priori a lot of data, and it may be that
the most natural choice in a given context simply does not admit an
$E_{2}$-structure. This turns out to be the case for the monoidal category of
bidecorated surfaces ${{\mathbf{M}}_{2}}$: In the context of categories,
$E_{2}$-structures are given by braided monoidal structures and we show in
Section 5.3 that even the full monoidal subcategory of ${{\mathbf{M}}_{2}}$
generated by our stabilizing object, the disk $D$, does not admit a braiding.
This distinguishes our situation from most previous examples of homological
stability.

On the other hand, it does not take much to equip a given monoidal category
$\mathscr{X}$ with the structure of an $E_{1}$-module over a braided monoidal
category. In fact, as shown in Section 5.1, any Yang–Baxter operator in
$\mathscr{X}$ determines a strong monoidal functor
$\mathbf{B}\mathrel{\smash{\meno}}\mathrel{\mkern-3.0mu}\rightarrow\mathscr{X}$
from the braid groupoid $\mathbf{B}$, and thus endows $\mathscr{X}$ with the
structure of an $E_{1}$-module over $\mathbf{B}$. This perspective also makes
sense if $\mathscr{X}$ itself acts on a category $\mathscr{M}$, and one is
interested in the stabilization

$\mathscr{M}\xrightarrow{\oplus X}\mathscr{M}\xrightarrow{\oplus X}\cdots$

induced by acting with an object $X$ of $\mathscr{X}$ admitting a Yang-Baxter
operator $\tau\in\text{Aut}_{\mathscr{X}}(X\oplus X)$. The category
$\mathscr{M}$ becomes this way likewise a module over $\mathbf{B}$, where the
object $n$ of $\mathbf{B}$ acts on $A\in\mathscr{M}$ via $A\oplus n=A\oplus
X^{\oplus n}$.

### Disordered arcs

Given a category $\mathscr{M}$ as above, with the structure of an
$E_{1}$-module over a monoidal category $\mathscr{X}$ with a distinguished
Yang–Baxter operator $(X,\tau)$, such that acting by $X$ satisfies a certain
injectivity property (see Proposition 3.4), the main result of [Kra19] implies
that homological stability for stabilization with $X$ is controlled by the
connectivity of certain complexes of destabilizations. In the category of
bidecorated surfaces ${{\mathbf{M}}_{2}}$, stabilizing with the bidecorated
disk $D$ corresponds homotopically to attaching an arc, and we show in
Proposition 4.4 that the relevant complex of destabilizations for stabilizing
a surface $S$ with a disk $n$ times identifies with the “disordered arc
complex”111We called those disordered arcs because it is the opposite ordering
convention than the one used in the “ordered arc complex” of [RW16].
associated to the surface
$S\mathbin{\text{{\\#}}}D^{\mathbin{\text{{\\#}}}n}$. This is a simplicial
complex whose vertices are isotopy classes of non-separating arcs in the
surface with endpoints $b_{0}=I_{0}({}^{1}{\mskip-5.0mu/\mskip-3.0mu}_{2})$
and $b_{1}=I_{1}({}^{1}{\mskip-5.0mu/\mskip-3.0mu}_{2})$, and where a
collection of isotopy classes forms a simplex if the classes can be
represented by arcs that are disjoint away from the endpoints, are jointly
non-separating, and such that the arcs have the same ordering at $I_{0}$ and
$I_{1}$.

Writing $\mathscr{D}^{\nu}(S_{g,r},b_{0},b_{1})$ for the disordered arc
complex of a surface $S_{g,r}$ with marked points $b_{0}$ and $b_{1}$ in
$\nu=1$ or $\nu=2$ boundary components, the main ingredient of our proof of
homological stability is the following connectivity result:

######  Theorem C.

(Theorem 2.4) The disordered arc complex
$\mathscr{D}^{\nu}(S_{g,r},b_{0},b_{1})$ is
$\left(\frac{2g+\nu-5}{3}\right)$-connected.

######  Remark 1.2.

It is conjectured in [RWW17, Conj C] that the complex of destabilizations is
highly connected if and only if stability holds with all appropriate twisted
coefficients. The slope $2/3$ bounds in Theorems A and B is precisely dictated
by the same slope $2/3$ in Theorem C in the connectivity of the arc complex,
which is the complex of destabilizations in that case. This connectivity bound
is best possible among linear bounds as a better bound would prove an
incorrect stability statement, see Remark 2.5.

### Organization of the paper.

In Section 2 we prove the high connectivity of the disordered arc complex. In
Section 3 we define the monoidal category of bidecorated surfaces
$({{\mathbf{M}}_{2}},\mathbin{\text{{\\#}}})$, as well as the action of the
braid groupoid $\mathbf{B}$ on this category. In Section 4, we show Theorems A
and B by showing that the disordered arc complex agrees with the complex of
destabilizations, and applying the main result of [Kra19]. Finally, in Section
5 we explain the relationship between homological stability and Yang–Baxter
operators, and show the non-braidedness of the category of bidecorated
surfaces.

### Acknowledgements

The first and third authors were partially supported by the Danish National
Research Foundation through the Copenhagen Centre for Geometry and Topology
(DRNF151), and the third author by the European Research Council (ERC) under
the European Union’s Horizon 2020 research and innovation programme (grant
agreement No. 772960).

##  2\. High connectivity of the disordered arc complex

In this section, we prove that the disordered arc complex is highly connected.
It will be defined as a subcomplex of the following simplicial complex of non-
separating arcs:

######  Definition 2.1.

Let $S$ be an orientable surface222 By surface we mean a topological
$2$-manifold $S$ which is compact except for a finite number of punctures,
i.e. there is a compact topological $2$-manifold $\overline{S}$ and an
embedding $i\colon
S\,\lhook\mathrel{\mkern-3.0mu}\mathrel{\smash{\meno}}\mathrel{\mkern-3.0mu}\rightarrow\overline{S}$
so that $\overline{S}\setminus i(S)$ is a (possibly empty) finite union of
points.  with nonempty boundary, and let $b_{0},b_{1}$ be distinct points in
$\partial S$. The _complex of non-separating arcs_
$\mathscr{B}(S,b_{0},b_{1})$ is the simplicial complex whose $p$-simplices are
collections of $p+1$ distinct isotopy classes of arcs between $b_{0},b_{1}$
that admit representatives $a_{0},\ldots,a_{p}$ such that

  1. (a)

$a_{i}\cap a_{j}=\\{b_{0},b_{1}\\}$ for each $i\neq j$ and

  2. (b)

$S-(a_{0}\cup\cdots\cup a_{p})$ is connected.

For convenience, we will add a superscript $\mathscr{B}^{\nu}(S,b_{0},b_{1})$
to the notation of the complex, with $\nu=1$ indicating that $b_{0},b_{1}$ lie
on the same boundary component and $\nu=2$ indicating that they do not.

Note that the orientation of the surface defines orderings of the arcs
$a_{0},\dots,a_{p}$ representing a simplex at both $b_{0}$ and $b_{1}$.

######  Definition 2.2.

Let $(S,b_{0},b_{1})$ be as before. The _disordered arc complex_ is the
subcomplex
$\mathscr{D}^{\nu}(S_{g,r},b_{0},b_{1})\subseteq\mathscr{B}^{\nu}(S,b_{0},b_{1})$
consisting of those simplices $\sigma$ that admit arc representatives
$a_{0},\ldots,a_{p}$, again subject to Item (a), Item (b), satisfying in
addition

  * (c)

the ordering of the arcs at $b_{0}$ agrees with the ordering of the arcs at
$b_{1}$.

The name “disordered” was chosen to contrast with the pre-existing ordered arc
complex used by Ivanov [Iva89] in the case $\nu=1$ and Randall-Williams [RW16]
in their proofs of homological stability for the mapping class group of
surfaces; the “ordered” version is also a subcomplex of the
$\mathscr{B}^{\nu}(S,b_{0},b_{1})$, but with the requirement that the order of
the arcs at $b_{1}$ is reversed compared to the order at $b_{0}$. Fixing an
ordering condition has the effect that the action of the mapping class group
is transitive on the set of $p$-simplices for every $p$, see [Har85, Lem 3.2].
The ordered and disordered arc complexes represent the two extremes of how
fast the genus of the surface decreases when cutting along larger and larger
simplices: for the ordered arc complex, the genus goes down as fast as
possible, essentially every time one removes an arc, while for the disordered
arc complex, the genus goes down as slow as possible, only every other time:

######  Proposition 2.3.

For a $p$–simplex $\sigma=\langle
a_{0},\dots,a_{p}\rangle\in\mathscr{D}^{\nu}(S_{g,r},b_{0},b_{1})$, the
surface $S_{g,r}\setminus\sigma$ obtained by removing a tubular neighborhood
of $a_{i}$ for each $i$ has genus $g^{\prime}$ with $r^{\prime}$ boundary
components for

$g^{\prime}=g-\left\lfloor{\frac{p+3-\nu}{2}}\right\rfloor\ \ \ \textrm{and}\
\ \ r^{\prime}=\begin{cases}r+(-1)^{\nu},&\text{if }p\text{ is even,}\\\
r&\text{else}.\end{cases}$

###### Proof.

The number of boundary components $r^{\prime}$ can be obtained by a direct
inductive computation, with the genus $g^{\prime}$ then deduced using the
Euler characteristic. The computation is a special case of [Bol12, Prop 2.11],
applied to the case where the permutation $\alpha$ is the inversion
$[p\,(p-1)\dots 0]$, once one computes that the genus $S(\alpha)$ of a
neighborhood of the arcs is $\lfloor\frac{p+2-\nu}{2}\rfloor$, e.g. using
Corollary 2.15 of the same paper. $\square$

The complex $\mathscr{B}^{\nu}(S,b_{0},b_{1})$ is known to be
$(2g+\nu-3)$-connected. (This was first stated in [Har85]; see [Wah08, Thm
3.2] or [Wah13, Thm 4.8] for a complete proof.) We will here use this fact to
deduce that $\mathscr{D}^{\nu}(S_{g,r},b_{0},b_{1})$ is also highly-connected.
While the ordered arc complex is $(g-2)$-connected [RW16, Thm A.1], the
following result shows that the disordered arc complex is only slope
$\frac{2}{3}$ connected with respect to the genus, despite being $\sim
2g$-dimensional.

######  Theorem 2.4.

The disordered arc complex $\mathscr{D}^{\nu}(S_{g,r},b_{0},b_{1})$ is
$\left(\frac{2g+\nu-5}{3}\right)$-connected.

To prove the result, we use essentially the same argument as the one given in
[RW16] in the ordered case.

###### Proof.

Let $S=S_{g,r}$. In the case $g=0$, the statement for $\mathscr{D}^{1}(S)$ is
vacuous, and for $\mathscr{D}^{2}(S)$ it states that the complex is
$(-1)$-connected, i.e. nonempty, which holds as any arc in the surface
connecting $b_{0}$ and $b_{1}$ defines a vertex in $\mathscr{D}^{2}(S)$. We
prove the remaining cases by induction on $g$.

Let $g>0$. Suppose we are given $f:\partial
D^{k+1}\mathrel{\smash{\meno}}\mathrel{\mkern-3.0mu}\rightarrow\mathscr{D}^{\nu}(S,b_{0},b_{1})$
for some $k\leq(2g+\nu-5)/3$. We wish to exhibit a nullhomotopy of this map.
Since $(2g+\nu-5)/3\leq 2g+\nu-3$, Theorem 3.2 in [Wah08] enables us to choose
a map $\hat{f}$ such that the outer diagram

(2.1) ${\partial
D^{k+1}}$${\mathscr{D}^{\nu}(S,b_{0},b_{1})}$${D^{k+1}}$${\mathscr{B}^{\nu}(S,b_{0},b_{1}),}$$\scriptstyle{f}$$\scriptstyle{\hat{f}}$

commutes. Using PL-approximation, we may assume that $\hat{f}$ and $f$ are
simplicial with respect to some PL-triangulation of $D^{k+1}$. We will
repeatedly replace $\hat{f}$ until the dotted arrow exists, thereby giving the
desired nullhomotopy.

Write $<_{0}$ and $<_{1}$ for the anti-clockwise orderings at $b_{0}$ and
$b_{1}$. We call a $p$-simplex $\sigma$ in $D^{k+1}$ _regular bad_ if
$\hat{f}(\sigma)=\langle a_{0},\cdots,a_{p^{\prime}}\rangle$, indexed in such
a way that $a_{0}<_{0}\dots<_{0}a_{p^{\prime}}$ are anticlockwise at $b_{0}$,
and there is $j>0$ with $a_{j}<_{1}a_{0}$ at $b_{1}$. Here $p^{\prime}\leq p$
is the dimension of the image simplex $\hat{f}(\sigma)$, and $p^{\prime}\geq
1$ if $\sigma$ is regular bad. This condition is “dense” in the sense that any
simplex $\sigma$ in $D^{k+1}$ with image not included in
$\mathscr{D}^{\nu}(S,b_{0},b_{1})$ must contain a regular bad simplex as a
face. Thus it suffices to give a procedure for exchanging $\hat{f}$ with a map
having strictly fewer regular bad simplices, while maintaining commutativity
of the outer diagram Eq. 2.1.

Let $\sigma$ be a regular bad simplex of $D^{k+1}$ of maximal dimension $p$
and consider its link $\operatorname{Lk}\sigma\subset D^{k+1}$. Maximality
implies that $\hat{f}|_{\operatorname{Lk}\sigma}$ factors as

$\hat{f}|_{\operatorname{Lk}\sigma}\colon\operatorname{Lk}\sigma\mathrel{\smash{\meno}}\mathrel{\mkern-3.0mu}\rightarrow\mathscr{D}^{\mu}(S\setminus\hat{f}(\sigma),b_{0}^{\prime},b_{1}^{\prime})\mathrel{\smash{\meno}}\mathrel{\mkern-3.0mu}\rightarrow\mathscr{D}^{\nu}(S,b_{0},b_{1})\subseteq\mathscr{B}^{\nu}(S,b_{0},b_{1}),$

where $S\setminus\hat{f}(\sigma)$ is the closure of the surface obtained from
$S$ by cutting out the collection of arcs $\hat{f}(\sigma)$, and
$b_{0}^{\prime}$ and $b_{1}^{\prime}$ in $S\setminus\hat{f}(\sigma)$ are the
first copies of $b_{0}$ and $b_{1}$ in the cut surfaces as depicted in Figure
1.

$a_{0}^{\prime}$
---
$\dots$
---
$a_{q}^{\prime}$
---
$a_{0}$
---
$a_{1}$
---
$\dots$
---
$a_{p}$
---
$a_{0}^{\prime}$
---
$\dots$
---
$a_{q}^{\prime}$
---
$a_{j}$
---
$\dots$
---
$a_{0}$
---
$\dots$
---
$b_{0}$
---
$b_{1}$
---
Figure 1. Maximal regular bad simplex $\\{a_{0},\dots,a_{p}\\}$ and simplex
$\\{a^{\prime}_{0},\dots,a^{\prime}_{q}\\}$ in its link.

Indeed, suppose that $\tau\in\operatorname{Lk}\sigma$ and write
$\hat{f}(\tau)=\langle a_{0}^{\prime},\cdots,a_{q}^{\prime}\rangle.$ If
$a_{0}\leq_{0}a_{i}^{\prime}$ at $b_{0}$ for any $a_{i}^{\prime}$, then the
simplex $\sigma*\langle a_{i}^{\prime}\rangle$ is regular bad of a larger
dimension, contradicting maximality. So we must have
$a_{i}^{\prime}<_{0}a_{0}$ for each $i$, i.e. the arcs of $\tau$ are at
$b_{0}^{\prime}$ in the cut surface. Now we must also have that each
$a_{i}^{\prime}<_{1}a_{0}$ as otherwise $\sigma*\langle a_{i}^{\prime}\rangle$
would again be regular bad. Finally, maximality of $\sigma$ would also be
contradicted if the orderings of the arcs
$a_{0}^{\prime},\cdots,a_{q}^{\prime}$ does not agree at $b_{0}$ and $b_{1}$
as, if $a_{i}^{\prime}<_{0}a_{j}^{\prime}$ with
$a_{j}^{\prime}<_{1}a_{i}^{\prime}$ for some $i,j$, then $\sigma*\langle
a^{\prime}_{i},a^{\prime}_{j}\rangle$ would again be regular bad, of larger
dimension. Thus $\hat{f}(\tau)$ must be disordered and, after cutting the
surface at the arcs of $\sigma$, can be viewed as a simplex of
$\mathscr{D}^{\nu}(S\setminus\hat{f}(\sigma),b_{0}^{\prime},b_{1}^{\prime}).$

The link $\operatorname{Lk}(\sigma)$ is a simplicial sphere $S^{k-p}\subset
D^{k+1}$. We want to show that the map $\hat{f}|_{\operatorname{Lk}(\sigma)}$
extends to a simplicial map

(2.2)
$F:D^{k-p+1}\mathrel{\smash{\meno}}\mathrel{\mkern-3.0mu}\rightarrow\mathscr{D}^{\mu}(S-\hat{f}(\sigma),b_{0}^{\prime},b_{1}^{\prime})\mathrel{\smash{\meno}}\mathrel{\mkern-3.0mu}\rightarrow\mathscr{D}^{\nu}(S,b_{0},b_{1})\subseteq\mathscr{B}(S,b_{0},b_{1})$

for $D^{k-p+1}$ a disk with some PL-structure extending that of
$\operatorname{Lk}(\sigma)$. This will follow if we can show that the complex
$\mathscr{D}^{\mu}(S\setminus\hat{f}(\sigma),b_{0}^{\prime},b_{1}^{\prime})$
is at least $(k-p)$-connected. Note that necessarily have
$g(S\setminus\hat{f}(\sigma))<g$ as $f(\sigma)$ is a non-separating
$p^{\prime}$-simplex with $p^{\prime}\geq 1$. Hence we can use our induction
hypothesis. We consider the cases $\nu=1$ and $\nu=2$ separately.

#### Case 1: $\nu=1$

We have that $g(S\setminus\hat{f}(\sigma))\geq g-p^{\prime}-1\geq g-p-1,$ as
removing $p^{\prime}+1$ arcs reduces the genus by at most $p^{\prime}+1\leq
p+1$. Hence by induction we have that
$\mathscr{D}^{\mu}(S\setminus\hat{f}(\sigma),b_{0}^{\prime},b_{1}^{\prime})$
is at least $(\frac{2(g-p-1)-4}{3})$-connected, using also that $\mu\geq 1$.
If $p\geq 2$, we have

$k-p\leq\frac{2g-4}{3}-p=\frac{2g-3p-4}{3}\leq\frac{2(g-p-1)-4}{3}.$

For $p=p^{\prime}=1$, note that $b_{0}^{\prime},b_{1}^{\prime}$ necessarily
lie in different boundary components, so that $\mu=2$ in that case. (See
Figure 2.) Hence in that case
$\mathscr{D}^{\mu}(S\setminus\hat{f}(\sigma),b_{0}^{\prime},b_{1}^{\prime})$
is $(\frac{2(g-2)-3}{3})$-connected, and

$k-1\leq\frac{2g-4}{3}-1=\frac{2g-7}{3}=\frac{2(g-2)-3}{3}.$

so we get the desired extension in both subcases.

$\dots$
---
$b_{0}^{\prime}$
---
$b_{0}$
---
$b_{1}$
---
$b_{1}^{\prime}$
---
Figure 2. Regular bad 1-simplex with $\nu=1$.

#### Case 2: $\nu=2$

The fact that $b_{0},b_{1}$ lie in different components implies that

$g(S\setminus\hat{f}(\sigma))\geq g-p^{\prime}\geq g-p$

as cutting along the first arc has no effect on the genus. Hence induction
here gives that
$\mathscr{D}^{\mu}(S\setminus\hat{f}(\sigma),b_{0}^{\prime},b_{1}^{\prime})$
is at least $(\frac{2(g-p)-4}{3})$-connected. Now for all $p\geq 1$,

$k-p\leq\frac{2g-4}{3}-p=\frac{2g-3p-4}{3}\leq\frac{2(g-p)-4}{3}$

yielding the desired connectivity.

We will use the map $F$ of (2.2) to modify $\hat{f}$ in the star
${\operatorname{St}}(\sigma)$. For this purpose, note that as simplicial
subcomplexes of $D^{k+1}$,

$\displaystyle{\operatorname{St}}(\sigma)$
$\displaystyle=\sigma*{\operatorname{Lk}}(\sigma),\quad$
$\displaystyle\partial{\operatorname{St}}(\sigma)$
$\displaystyle=\partial\sigma*{\operatorname{Lk}}(\sigma).$

In particular, we get an identification
$\partial(\partial\sigma*D^{k-p+1})\cong\partial{\operatorname{St}}(\sigma)$
for $D^{k-p+1}$ the simplicial disk that is the source of the map $F$ above.

We replace $\hat{f}\,|\,_{{\operatorname{St}}(\sigma)}$ by the unique
simplicial map

$\hat{f}*F:\partial\sigma*D^{k-p+1}\mathrel{\smash{\meno}}\mathrel{\mkern-3.0mu}\rightarrow\mathscr{B}^{\nu}(S,b_{0},b_{1}).$

It remains to show that this has improved the situation. Indeed, suppose that
$\tau=\tau_{0}*\tau_{1}$ is a regular bad simplex in
$\partial\sigma*D^{k-p+1}$. By construction, $\tau_{1}$ has image in
$\mathscr{D}^{\mu}(S\setminus\hat{f}(\sigma),b_{0}^{\prime},b_{1}^{\prime})\subset\mathscr{D}^{\nu}(S,b_{0},b_{1})$,
so the ordering of the arcs of $\tau$ at $b_{0}$ and $b_{1}$ starts with the
arcs of $\tau_{1}$, all in anti-clockwise order. Hence, if $\tau$ is regular
bad, we must have $\tau=\tau_{0}$ is a strict face of $\sigma$. In particular,
no new regular bad simplices have been added. As the simplex $\sigma$ has been
removed, we have thus reduced the total number of regular bad simplices in the
disk. Repeating this procedure, we will after finitely many stages remove
every regular bad simplex, thus making the dashed arrow exist, which proves
the result. $\square$

######  Remark 2.5.

The connectivity estimate above can be shown to be optimal in certain low-
genus examples, corresponding to known computations of the unstable homology
of mapping class groups. Indeed, $\mathscr{D}^{2}(S_{1,r})$ is disconnected.
To see this, consider the spectral sequence associated to the action of the
mapping class group $\Gamma(S_{1,r})$ on the simplicial complex
$\mathscr{D}^{2}(S_{1,r})$. This is the spectral sequence arising from the
vertical filtration of the double complex
$\mathbb{Z}\mathscr{D}^{2}(S_{1,r})_{\bullet}\otimes_{\Gamma(S_{1,r})}F_{\bullet}$,
where
$F_{\bullet}\mathrel{\smash{\meno}}\mathrel{\mkern-3.0mu}\rightarrow\mathbb{Z}$
is a free resolution of the trivial $\Gamma(S_{1,r})$-module. By a standard
argument using Shapiro’s lemma (see e.g. [HW10, Thm 5.1] or [HV17, Sec 1]),
one finds that the first page of this spectral sequence is given by

$E^{1}_{p,q}\cong\begin{cases}\widetilde{H}_{q}(\Gamma(S_{1,r}))&\text{if
}p=-1,\\\ \widetilde{H}_{q}(\Gamma(S_{1,r-1}))&\text{if }p=0,\\\
\widetilde{H}_{q}(\Gamma(S_{0,r}))&\text{if }p=1,\\\
\widetilde{H}_{q}(\Gamma(S_{0,r-1}))&\text{if }p=2,\\\
0&\text{otherwise.}\end{cases}$

Assume for contradiction that $\mathscr{D}^{2}(S_{1,r})$ is connected. Then an
analysis of the horizontal filtration of the double complex
$\mathbb{Z}\mathscr{D}^{2}(S_{1,r})_{\bullet}\otimes_{\Gamma(S_{1,r})}F_{\bullet}$
shows that $E^{\infty}_{p,q}=0$ for $p+q\leq 0$, so the differential
$d^{1}\colon
H_{1}(\Gamma(S_{1,r-1}))\mathrel{\smash{\meno}}\mathrel{\mkern-3.0mu}\rightarrow
H_{1}(\Gamma(S_{1,r}))$ must be surjective. This contradicts the fact that
$H_{1}(\Gamma(S_{1,s}))\cong\mathbb{Z}^{s}$ for $s\geq 1$ (see [Kor02, Thm
5.1]). Hence it is not true that $\mathscr{D}^{\nu}$ is
$\left(\frac{2g+\nu-4}{3}\right)$-connected when $\nu=2$.

Similarly, one finds that $H_{1}(\mathscr{D}^{1}(S_{3,r}))\neq 0$ by
considering the spectral sequence associated to the action of
$\Gamma(S_{3,r})$ on $\mathscr{D}^{1}(S_{2,r+1})$ and noting that the
differential $d^{1}\colon
H_{1}(\Gamma(S_{2,r+1}))\mathrel{\smash{\meno}}\mathrel{\mkern-3.0mu}\rightarrow
H_{1}(\Gamma(S_{3,r}))$ cannot be injective since the source identifies with
$\mathbb{Z}/10\mathbb{Z}$ and the target is zero (see [Kor02, Thm 5.1]). Thus
$\mathscr{D}^{\nu}$ fails to be $\left(\frac{2g+\nu-4}{3}\right)$-connected
when $\nu=1$ also.

Note that these low dimensional computations also show that the first and last
ranges in Theorem A cannot be improved by a constant.

##  3\. The monoidal category of bidecorated surfaces

In this section, we describe a monoidal groupoid
$({{\mathbf{M}}_{2}},\mathbin{\text{{\\#}}})$ of surfaces decorated by two
intervals in their boundary, where the monoidal structure glues the intervals
in pairs. We show that this groupoid is a module over the braided monoidal
groupoid $\mathbf{B}$ of braid groups, giving, on classifying spaces, the
structure of an $E_{1}$-module over an $E_{2}$-algebra in the sense of
[Kra19].

###  3.1. Bidecorated surfaces and the monoidal structure

The groupoid ${{\mathbf{M}}_{2}}$ has objects bidecorated surfaces, that are,
informally, surfaces with two intervals marked in their boundary. To give a
precise definition of the objects that is convenient for the monoidal
structure, we start by constructing a special sequence of bidecorated surfaces
$X_{n}$, built out of disks, and defined inductively.

Let $X_{1}=D^{2}\subset\mathbb{C}$ denote the unit disk in the complex plane,
and define the embeddings $\iota_{1}^{0},\iota_{1}^{1}\colon
I\mathrel{\smash{\meno}}\mathrel{\mkern-3.0mu}\rightarrow X_{1}$ by

$\iota_{1}^{0}(t)=e^{i(\pi/4+t\pi/2)}\quad\text{and}\quad\iota_{1}^{1}(t)=e^{i(5\pi/4+t\pi/2)}.$

We denote by $\overline{\iota_{1}^{i}}\colon
I\mathrel{\smash{\meno}}\mathrel{\mkern-3.0mu}\rightarrow X_{1}$ the reversed
map $t\longmapsto\iota_{1}^{i}(1-t)$ for $i=0,1$.

Recursively, suppose we have defined $(X_{m},\iota^{0}_{m},\iota_{m}^{1})$ for
some $m\geq 1$. We construct $X_{m+1}$ from $X_{m}$ by gluing an additional
disk along two half intervals, with new markings
$\iota_{m+1}^{0},\iota_{m+1}^{1}$ coming from the first half of the markings
of $X_{m}$ and the second half of the markings of the attached disk:

$X_{m+1}\vcentcolon=\frac{X_{m}\sqcup
X_{1}}{\begin{subarray}{c}\iota^{i}_{m}(t)\sim\overline{\iota^{i}_{1}}(t),\
t\in{[{}^{1}{\mskip-5.0mu/\mskip-3.0mu}_{2},1]},\end{subarray}}\ \ \
\textrm{with}\ \ \ \iota_{m+1}^{i}(t)=\begin{cases}\iota_{m}^{i}(t),&\text{if
}t\leq 1/2,\\\ \iota_{1}^{i}(t),&\text{else.}\end{cases}$

for $i=0,1$. Note that the marked intervals in the boundary of $X_{m}$ might
live in different boundary components (in fact this will happen every other
time). Figure 3 shows what happens when a disk is glued to a surface in the
above described manner, in each of these two possible cases.

$S$
---
$D^{2}$
---
$S$
---
$D^{2}$
---
Figure 3. Gluing a disk $X_{1}=D^{2}$ to a bidecorated surface $S$

######  Lemma 3.1.

Let $m\geq 1$. Then $X_{m}\cong S_{g,r}$ is a surface of genus $g$ with $r$
boundary components, where

$(g,r)=\begin{cases}(\frac{m}{2}-1,2),&\text{if $m$ is even},\\\
(\frac{m-1}{2},1),&\text{if $m$ is odd}.\end{cases}$

###### Proof.

Note first that $X_{m}$ is a connected surface for each $m$, since $X_{1}$ is
a disk and $X_{m}$ is obtained from $X_{1}$ by successively adding disks (or
strips), attached along two disjoint intervals in the boundary. For the same
reason, we get that the Euler characteristic of $X_{m}$ is

$\chi(X_{m+1})=\chi(X_{m})-1=\dots=1-m.$

By the classification of surfaces, we are left to compute the number of
boundary components of $X_{m}$. For this, observing Figure 3, we notice that
if we glue a disk along two intervals of $S$ that lie in the same boundary
component, the new marked intervals given by the above procedure will give new
intervals in different boundary components and vice versa, and no boundary
component without marked intervals are ever created. It follows that the
number of boundary components of $X_{m}$ alternates between $1$ and $2$. The
result follows. $\square$

We are now ready to define the objects of the groupoid ${{\mathbf{M}}_{2}}$.
We will use the boundary of the above defined surfaces $X_{m}$ to parametrize
the boundary components of the surfaces that contain the marked intervals, to
allow us to work with parametrized boundary components instead of parametrized
arcs, in order to simplify some definitions.

######  Definition 3.2.

A _bidecorated surface_ is a tuple $(S,m,\varphi)$ where $S$ is a surface,
$m\geq 1$ is an integer, and

$\varphi\colon\partial X_{m}\sqcup(\sqcup_{k}S^{1})\xrightarrow{\sim}\partial
S$

is a homeomorphism, giving a parametrization of the boundary of $S$. We think
of $(S,m,\varphi)$ as a surface with two parametrized arcs

$I_{0}\vcentcolon=\varphi\circ\iota_{m}^{0}\quad\text{and}\quad
I_{1}\vcentcolon=\varphi\circ\iota_{m}^{1}$

in its boundary, and $k$ additional parametrized boundaries. The surface $S$
may also have punctures.

The monoidal groupoid $({{\mathbf{M}}_{2}},\mathbin{\text{{\\#}}},U)$ has
objects the bidecorated surfaces together with a formal unit $U$. There are no
morphisms between two bidecorated surfaces $(S,m,\varphi)$ and
$(S^{\prime},m^{\prime},\varphi^{\prime})$ unless $S$ and $S^{\prime}$ are
homeomorphic and $m=m^{\prime}$, in which case we define the set of morphisms
to be all the mapping classes of homeomorphisms that preserve the boundary
parametrizations

${\operatorname{Hom}}_{{\mathbf{M}}_{2}}((S,m,\varphi),(S^{\prime},m,\varphi^{\prime}))\vcentcolon=\pi_{0}{\operatorname{Homeo}}_{\partial}(S,S^{\prime})=\pi_{0}\\{f\in{\operatorname{Homeo}}(S,S^{\prime})\mid
f\circ\varphi=\varphi^{\prime}\\},$

where ${\operatorname{Homeo}}(S,S^{\prime})$ is endowed with the compact-open
topology, and ${\operatorname{Homeo}}_{\partial}(S,S^{\prime})$ with the
subspace topology. The only morphism involving the unit $U$ is the identity
$\text{id}_{U}$.

######  Remark 3.3.

Our definition of the morphisms in the category ${{\mathbf{M}}_{2}}$ is such
that punctures in a surfaces $S$ can be permuted by automorphisms of $S$ in
${{\mathbf{M}}_{2}}$. Our argument works just as well with labeled punctures,
that are not permutable by homeomorphims, or both labeled and unlabeled
punctures, just like we could also have additional boundary components that
are only marked up to a permutation. The only changes this would cause to the
argument would be that it would make the notations and conventions more
cumbersome.

The monoidal structure $\mathbin{\text{{\\#}}}$ is defined as follows. The
object $U$ is by definition a unit for $\mathbin{\text{{\\#}}}$. For the
remaining objects, the monoidal product $\mathbin{\text{{\\#}}}$ is defined by

$\displaystyle(S,m,\varphi)\mathbin{\text{{\\#}}}(S^{\prime},m^{\prime},\varphi^{\prime})\vcentcolon=\left(\frac{S\sqcup
S^{\prime}}{\begin{subarray}{c}I_{i}(t)\sim\overline{I_{i}^{\prime}}(t),t\in{[{}^{1}{\mskip-5.0mu/\mskip-3.0mu}_{2},1]},\end{subarray}},m+m^{\prime},\varphi\mathbin{\text{{\\#}}}\varphi^{\prime}\right),$

where $i=0,1$, and where

$\varphi\mathbin{\text{{\\#}}}\varphi^{\prime}\colon\partial
X_{m+m^{\prime}}\sqcup(\sqcup_{k+k^{\prime}}S^{1})\ \hookrightarrow\
\partial(S\mathbin{\text{{\\#}}}S^{\prime}),$

is obtained using the canonical identification $\partial
X_{m+m^{\prime}}\cong(\partial
X_{n}\backslash\iota_{m}(\frac{1}{2},1))\cup(\partial
X_{m^{\prime}}\backslash\iota_{m^{\prime}}(0,\frac{1}{2}))$. On morphisms, the
monoidal product is given by juxtaposition.

The monoidal category ${{\mathbf{M}}_{2}}$ has the following injectivity
property with respect to gluing a disk, that will be useful in the proof of
our stability result.

######  Proposition 3.4.

For any object $S=(S,m,\varphi)$ of ${{\mathbf{M}}_{2}}$, and any $p\geq 0$,
the map

${\operatorname{Aut}}_{{\mathbf{M}}_{2}}(S)\xrightarrow{\
{\mathbin{\text{{\\#}}}D^{\mathbin{\text{{\\#}}}{p+1}}}\
}{\operatorname{Aut}}_{{\mathbf{M}}_{2}}(S\mathbin{\text{{\\#}}}D^{\mathbin{\text{{\\#}}}{p+1}})$

is injective, where $D=(X_{1},1,\operatorname{id})$ is our chosen disk.

###### Proof.

Recall that the underlying surface of $D^{\mathbin{\text{{\\#}}}p+1}$ is the
surface $X_{p+1}$ defined above. Picking a smooth representative of the
underlying surface of $S\mathbin{\text{{\\#}}}X_{p+1}$, with $S$ a smooth
subsurface in its interior, we can model the map in the statement using the
description of the mapping class group of surfaces in terms of isotopy classes
of diffeomorphisms rather than homeomorphisms. (See e.g., [Bol09, Thm 1.2] for
a detailed account of the classical isomorphism
$\pi_{0}{\operatorname{Homeo}}_{\partial}(S)\cong\pi_{0}{\operatorname{Diff}}_{\partial}(S)$
when $S$ is compact.) Now the result follows by essentially the same argument
as the case of attaching of surface along a single arc instead of two, as
treated in [RWW17, Prop 5.18], using the fibration

$\displaystyle{\operatorname{Diff}}(S\mathbin{\text{{\\#}}}X_{p+1}\
\textrm{rel}\ \partial S\cup X_{p+1})$
$\displaystyle\mathrel{\smash{\meno}}\mathrel{\mkern-3.0mu}\rightarrow{\operatorname{Diff}}(S\mathbin{\text{{\\#}}}X_{p+1}\
\textrm{rel}\ \partial(S\mathbin{\text{{\\#}}}X_{p+1}))$
$\displaystyle\mathrel{\smash{\meno}}\mathrel{\mkern-3.0mu}\rightarrow\text{Emb}((X_{p+1},I_{0}|_{[\frac{1}{2},1]}\cup
I_{1}|_{[\frac{1}{2},1]}),(S\mathbin{\text{{\\#}}}X_{p+1},I_{0}|_{[\frac{1}{2},1]}\cup
I_{1}|_{[\frac{1}{2},1]}))$

where the fiber identifies with ${\operatorname{Diff}}(S\ \textrm{rel}\
\partial_{0}S)$ and where we note that $I_{0}|_{[\frac{1}{2},1]}\cup
I_{1}|_{[\frac{1}{2},1]}=\partial
X_{p+1}\cap\partial(S\mathbin{\text{{\\#}}}X_{p+1})$. Injectivity of the first
map on $\pi_{0}$ follows if we can show that the base is simply-connected. In
fact the base can be shown inductively to have contractible components, using
that $X_{p+1}$ is built inductively by attaching disks along two intervals, or
homotopically attaching arcs, and using the contractibility of the components
of embeddings of arcs in a surface, as proved in [Gra73, Thm 5]. $\square$

###  3.2. Braided action

We want to apply the homological stability machine of [Kra19] to stabilization
in ${{\mathbf{M}}_{2}}$ with the bidecorated disk

$D\vcentcolon=(X_{1},1,\text{id}).$

For this, we need that the classifying space of ${{\mathbf{M}}_{2}}$ is an
$E_{1}$–module over an $E_{2}$–algebra. This will follow if we can show on the
categorical level that ${{\mathbf{M}}_{2}}$ admits an appropriate action of a
braided monoidal groupoid. We will build such an action in this section, using
as braided monoidal groupoid the groupoid of braid groups. In constrast with
most classical examples of homological stability, we will show in Section 5.3
that this action of the braid groupoid does not come from a braided structure
on ${{\mathbf{M}}_{2}}$, or the full monoidal subcategory generated by $D$. It
is instead constructed using a Yang–Baxter element in ${{\mathbf{M}}_{2}}$,
associated to a braid subgroup of the mapping class group of $X_{m}$, that we
will describe now.

Write

$D^{\mathbin{\text{{\\#}}}m}=D_{1}\mathbin{\text{{\\#}}}\ldots\mathbin{\text{{\\#}}}(D_{i}\mathbin{\text{{\\#}}}D_{i+1})\mathbin{\text{{\\#}}}\ldots\mathbin{\text{{\\#}}}D_{m},$

where we use subscript to enumerate the disks, and where the underlying
surface is $X_{m}$. We let $a_{i}$ denote the isotopy class of a curve in the
interior $D_{i}\mathbin{\text{{\\#}}}D_{i+1}\cong S^{1}\times I$ that is
parallel to its boundary components, as shown in Figure 4.

$b_{1}$
---
$b_{0}$
---
$a_{i}$
---
Figure 4. The curve $a_{i}$ in $D_{i}\mathbin{\text{{\\#}}}D_{i+1}$

######  Lemma 3.5.

The curves $a_{1},\dots,a_{m-1}$ form a chain in
$D^{\mathbin{\text{{\\#}}}m}$, i.e. $a_{i}$ and $a_{i+1}$ have intersection
number 1 for each $i$, and $a_{i}\cap a_{j}=\emptyset$ if $|i-j|>1$.

###### Proof.

The curve $a_{i}$ lives in the disks $D_{i}$ and $D_{i+1}$, so it can only
intersect $a_{i-1}$ and $a_{i+1}$ non-trivially, and hence it suffices to
consider the subsurface of $D^{\mathbin{\text{{\\#}}}m}$ corresponding to
$D_{i}\mathbin{\text{{\\#}}}D_{i+1}\mathbin{\text{{\\#}}}D_{i+2}$. Here the
claim can be checked by hand, see Figure 5. $\square$

$D_{i}$
---
$D_{i+1}$
---
$D_{i+2}$
---
Figure 5. Intersection of $a_{i}$ (blue) and $a_{i+1}$ (green) in the
underlying surface of
$D_{i}\mathbin{\text{{\\#}}}D_{i+1}\mathbin{\text{{\\#}}}D_{i+2}$.

Let $T_{i}\in\text{Aut}_{{{\mathbf{M}}_{2}}}(D^{\mathbin{\text{{\\#}}}m})$
denote the Dehn twist along the curve $a_{i}$ in
$D^{\mathbin{\text{{\\#}}}m}$. A classical fact states that the Dehn twists
along a chain of embedded curves satisfy the braid relations (see e.g. [FM11,
3.9 and 3.11]):

(3.1) $\displaystyle T_{i}T_{i+1}T_{i}$ $\displaystyle=T_{i+1}T_{i}T_{i+1}$
$\displaystyle\quad\text{for all }i,$ $\displaystyle T_{i}T_{j}$
$\displaystyle=T_{j}T_{i}$ $\displaystyle\quad\text{if }|i-j|>1,$

Note that the same relations are satisfied by the inverse twists $T_{i}^{-1}$,
that will turn out to be more convenient for us. Also, adding a disk to the
right or left of $D^{\mathbin{\text{{\\#}}}m}$ gives the relations

$T_{i}\mathbin{\text{{\\#}}}\text{id}_{D}=T_{i}\quad\text{and}\quad\text{id}_{D}\mathbin{\text{{\\#}}}T_{i}=T_{i+1}$

in ${\operatorname{Aut}}_{{{\mathbf{M}}_{2}}}(D^{\mathbin{\text{{\\#}}}m+1})$.
In particular (3.1) includes the relation

$(T_{1}^{-1}\mathbin{\text{{\\#}}}\text{id}_{D})(\text{id}_{D}\mathbin{\text{{\\#}}}T_{1}^{-1})(T_{1}^{-1}\mathbin{\text{{\\#}}}\text{id}_{D})=(\text{id}_{D}\mathbin{\text{{\\#}}}T_{1}^{-1})(T_{1}^{-1}\mathbin{\text{{\\#}}}\text{id}_{D})(\text{id}_{D}\mathbin{\text{{\\#}}}T_{1}^{-1})$

in ${\operatorname{Aut}}_{{\mathbf{M}}_{2}}(D^{\mathbin{\text{{\\#}}}3})$, so
in other words, the inverse Dehn twist
$T_{1}^{-1}\in{\operatorname{Aut}}_{{\mathbf{M}}_{2}}(D{\mathbin{\text{{\\#}}}}D)$
is a Yang–Baxter operator in the sense of Section 5.1.

Recall from the introduction that $\mathbf{B}$ denotes the groupoid of braid
groups, with objects the natural numbers $\\{0,1,2,\dots\\}$, automorphisms of
$n$ the braid group $B_{n}$, and no other non-trivial morphisms. In Section
5.1 we show that, being a Yang-Baxter operator, $T_{1}^{-1}$ yields a strong
monoidal functor

$\Phi=\Phi_{D,T_{1}^{-1}}\colon(\mathbf{B},\oplus)\mathrel{\smash{\meno}}\mathrel{\mkern-3.0mu}\rightarrow({{\mathbf{M}}_{2}},\mathbin{\text{{\\#}}}),$

uniquely determined up to monoidal natural isomorphism by the fact that
$\Phi(1)=D$ and, for the standard generator $\sigma_{1}\in
B_{2}=\text{Aut}_{\mathbf{B}}(1)$, $\Phi(\sigma_{1})=T_{1}^{-1}$.

Such a functor $\Phi$ endows ${{\mathbf{M}}_{2}}$ with the structure of an
$E_{1}$-module over $\mathbf{B}$ via the associated functor

$\displaystyle\alpha=(-\mathbin{\text{{\\#}}}\Phi(-))\colon\
{{\mathbf{M}}_{2}}\times\mathbf{B}$
$\displaystyle\mathrel{\smash{\meno}}\mathrel{\mkern-3.0mu}\rightarrow{{\mathbf{M}}_{2}},$

given on objects by
$\alpha(S,n)=S\mathbin{\text{{\\#}}}\Phi(n)=S\mathbin{\text{{\\#}}}D^{\mathbin{\text{{\\#}}}n}$,
and likewise for morphisms. On classifying spaces, this yields exactly the
kind of input needed in Krannich’s homological stability framework, see
[Kra19, Lem 7.2].

######  Remark 3.6.

For each $m$, the restriction of the functor
$\Phi:\mathbf{B}\mathrel{\smash{\meno}}\mathrel{\mkern-3.0mu}\rightarrow{{\mathbf{M}}_{2}}$
to $B_{m}=\text{Aut}_{\mathbf{B}}(m)$ maps the standard generator $\sigma_{i}$
to the inverse Dehn twist
$T_{i}^{-1}\in{\operatorname{Aut}}_{{\mathbf{M}}_{2}}(D^{\mathbin{\text{{\\#}}}m})=\pi_{0}\textrm{Homeo}_{\partial}(X_{m})$.
By Birman–Hilden theory [BH72, BH73] the homomorphisms $\Phi|_{B_{m}}\colon
B_{m}\mathrel{\smash{\meno}}\mathrel{\mkern-3.0mu}\rightarrow\text{Aut}_{{\mathbf{M}}_{2}}(D^{\mathbin{\text{{\\#}}}m})$
are actually injective.

##  4\. Homological stability

Generalizing the main result of [RWW17], Krannich associates to an
$E_{1}$-module $\mathscr{M}$ over an $E_{2}$-algebra $\mathscr{X}$ with a
chosen stabilizing object $X\in\mathscr{X}$, a space of destabilizations at
every $A\in\mathscr{M}$, whose high connectivity implies homological stability
at $A$ when stabilizing by $X$. We are interested in the case where
$\mathscr{M}=B{{\mathbf{M}}_{2}}$ is the classifying space of
${{\mathbf{M}}_{2}}$ and $\mathscr{X}=B\mathbf{B}$, acting on
$B{{\mathbf{M}}_{2}}$ via the map
$\alpha\colon{{\mathbf{M}}_{2}}\times\mathbf{B}\mathrel{\smash{\meno}}\mathrel{\mkern-3.0mu}\rightarrow{{\mathbf{M}}_{2}}$
defined in Section 3.2. We will pick $A=S\in{{\mathbf{M}}_{2}}$ to be some
surface, with $X=1\in\mathbf{B}$ modelling stabilization with the disk as
$\alpha(-,X)=-\mathbin{\text{{\\#}}}D$ is the sum with the bidecorated disk
$D=(X_{1},1,\operatorname{id})$ of Section 3.1.

Generally, the space of destabilizations is a semi-simplicial space, but in
settings such as ours, it is actually levelwise homotopy discrete. Indeed, by
[Kra19, Lem 7.6]), when the structure of $E_{1}$-module over an
$E_{2}$-algebra is induced by an action of a braided monoidal category on a
groupoid, and under the injectivity condition given in Proposition 3.4, the
space of destabilizations is equivalent to the following semi-simplicial set,
defined just as in [RWW17] in the case of a braided monoidal groupoid acting
on itself.

######  Definition 4.1.

([Kra19, Def 7.5]) Let $(\mathscr{M},\oplus)$ be a right module over a braided
monoidal groupoid $(\mathscr{X},\oplus,b)$, where we denote also by $\oplus$
the module action. Let $A$ and $X$ be objects of $\mathscr{M}$ and
$\mathscr{X}$ respectively. The space of destabilizations
$W_{n}(A,X)_{\bullet}$ is the semi-simplicial set with set of $p$-simplices

$\displaystyle W_{n}(A,X)_{p}=$ $\displaystyle\ \\{(B,f)\ |\
B\in\textrm{Ob}(\mathscr{M})\ \textrm{and}\ f\colon B\oplus X^{\oplus
p+1}\mathrel{\smash{\meno}}\mathrel{\mkern-3.0mu}\rightarrow A\oplus X^{\oplus
n}\ \textrm{in}\ \mathscr{M}\\}/_{\sim}$

where $(B,f)\sim(B^{\prime},f^{\prime})$ if there exists an isomorphism
$g\colon B\mathrel{\smash{\meno}}\mathrel{\mkern-3.0mu}\rightarrow B^{\prime}$
in $\mathscr{C}$ satisfying that
$f=f^{\prime}\circ(g\oplus\operatorname{id}_{X^{\oplus p+1}})$. The face map
$d_{i}\colon
W_{n}(A,X)_{p}\mathrel{\smash{\meno}}\mathrel{\mkern-3.0mu}\rightarrow
W_{n}(A,X)_{p-1}$ is defined by $d_{i}[B,f]=[B\oplus X,d_{i}f]$ for

$d_{i}f\colon B\oplus X\oplus X^{p}\xrightarrow{\operatorname{id}_{B}\oplus
b_{X^{\oplus i},X}^{-1}\oplus\operatorname{id}_{X^{\oplus p-i}}}B\oplus
X^{\oplus i}\oplus X\oplus X^{\oplus p-i}\xrightarrow{\ f\ }A\oplus X^{\oplus
n},$

for $b_{X^{\oplus i},X}^{-1}\colon X\oplus X^{\oplus
i}\mathrel{\smash{\meno}}\mathrel{\mkern-3.0mu}\rightarrow X^{\oplus i}\oplus
X$ coming from the braiding in $\mathscr{X}$.

###  4.1. Disk destabilizations and disordered arcs

Given a bidecorated orientable333The definition of the disordered arc complex
naturally extend to non-orientable bidecorated surfaces, ordering the arcs
according to the orientations of $I_{0}$ and $I_{1}$, but we will only
consider orientable surfaces here surface $S=(S,m,\varphi)$, with
$I_{0},I_{1}$ compatibly oriented, let
$\mathscr{D}(S)=\mathscr{D}^{\nu}(S,b_{0},b_{1})$ denote the disordered arc
complex of $S$ as in Section 2, where

$b_{0}=I_{0}(1/2)\quad\text{and}\quad b_{1}=I_{1}(1/2)$

are the midpoints of the marked intervals, and $\nu=1$ if $I_{0}$ and $I_{1}$
lie on the same boundary component and $\nu=2$ otherwise. The vertices of a
simplex in $\mathscr{D}(S)$ are canonically ordered by the anti-clockwise
ordering at $b_{0}$ (or equivalently at $b_{1}$). Hence we can associate to
this simplicial complex a semi-simplicial set that we denote
$\mathscr{D}(S)_{\bullet}$, with same set of $p$-simplices and whose $i$th
face map is given by forgetting the $(i+1)$st arc with respect to that
ordering. As $\mathscr{D}(S)$ and $\mathscr{D}(S)_{\bullet}$ have homeomorphic
realizations, they have the same connectivity.

Write $W_{n}(S,D)_{\bullet}$ for the space of destabilization of Definition
4.1 associated to the module $\mathscr{M}={{\mathbf{M}}_{2}}$ over the
$E_{2}$–algebra $\mathscr{X}=\mathbf{B}$ acting on ${{\mathbf{M}}_{2}}$ as
above, with $X=1\in\mathbf{B}$, and $A=S=(S_{g,r}^{s},m,\varphi)$ some
bidecorated orientable surface of small genus $g\geq 0$, with $r$ boundary
components and $s$ punctures. The space $W_{n}(S,D)_{\bullet}$ is then the
space of destabilizations of the stabilization map

${\operatorname{Aut}}_{{\mathbf{M}}_{2}}(S\mathbin{\text{{\\#}}}D^{\mathbin{\text{{\\#}}}n-1})\xrightarrow{\mathbin{\text{{\\#}}}D}{\operatorname{Aut}}_{{\mathbf{M}}_{2}}(S\mathbin{\text{{\\#}}}D^{\mathbin{\text{{\\#}}}n})$

that attaches an additional disk to the surface along the two marked
intervals.

We want to identify $W_{n}(S,D)_{\bullet}$ with
$\mathscr{D}(S\mathbin{\text{{\\#}}}D^{\mathbin{\text{{\\#}}}n})_{\bullet}$.
For this, we start by constructing a particular disordered collection of arcs
in $D^{\mathbin{\text{{\\#}}}n}$. Write again

$D^{\mathbin{\text{{\\#}}}n}=D_{1}\mathbin{\text{{\\#}}}\dots\mathbin{\text{{\\#}}}D_{i}\mathbin{\text{{\\#}}}\dots\mathbin{\text{{\\#}}}D_{n},$

and let $\rho_{i}$ denote the unique isotopy class of arc in the $i$th disk
$D_{i}$ going from $b_{0}=I_{0}(1/2)$ to $b_{1}=I_{1}(1/2)$.

######  Lemma 4.2.

The arcs $\rho_{1},\dots,\rho_{m}$ are ordered anti-clockwise at both $b_{0}$
and $b_{1}$.

###### Proof.

It suffices to show that $\rho_{i}$ and $\rho_{i+1}$ are ordered anti-
clockwise at $b_{0}$ and $b_{1}$ for each $i$. Thus we need only consider what
happens in the subsurface $D_{i}\mathbin{\text{{\\#}}}D_{i+1}$. The gluing
being defined in exactly the same way at $I_{0}$ and $I_{1}$, the arcs are
ordered in the same way at both endpoints, and the particular choice of gluing
gives the anti-clockwise ordering, see Figure 6. $\square$

$b_{1}$
---
$b_{0}$
---
$\rho_{i}$
---
$\rho_{i+1}$
---
Figure 6. Ordering of the arcs $\rho_{i}$ at their endpoints

Recall from Section 3.2 the Dehn twist $T_{i}$ along the curve $a_{i}$ in
$D_{i}\mathbin{\text{{\\#}}}D_{i+1}$. The union of the arcs $\rho_{i}$ in
$D^{\mathbin{\text{{\\#}}}m}$ define a deformation retract of the surface, as
each disk $D_{i}$ retracts onto the corresponding arc $\rho_{i}$, and we can
understand the action of the twists $T_{i}$ on the surface by considering
their action on the arcs $\rho_{i}$. The action is given by the following
result, that will be needed to compare the face maps in the semi-simplicial
sets $W_{n}(S,D)_{\bullet}$ with
$\mathscr{D}(S\mathbin{\text{{\\#}}}D^{\mathbin{\text{{\\#}}}n})_{\bullet}$.

######  Lemma 4.3.

The action of the Dehn twist $T_{i}$ along the curve $a_{i}$ on the homotopy
classes of the arcs $\rho_{i}$, relative to their endpoints, is

$T_{i}(\rho_{j})=\begin{cases}\rho_{i}\overline{\rho_{i+1}}\rho_{i}&\text{if
}j=i,\\\ \rho_{i}&\text{if }j=i+1,\\\ \rho_{j}&\text{else.}\end{cases}$

Equivalently,

$T_{i}^{-1}(\rho_{i})=\rho_{i+1}\ \ \ \textrm{and}\ \ \
T_{i}^{-1}(\rho_{i+1})=\rho_{i+1}\overline{\rho_{i}}\rho_{i+1}$

and $T_{i}^{-1}$ leaves the other $\rho_{j}$ invariant.

###### Proof.

The Dehn twist $T_{i}$ can only affect $\rho_{i}$ and $\rho_{i+1}$ as the
curve $a_{i}$ only intersects these two arcs, from which the last case in the
statement follows. The computation for the arcs $\rho_{i}$ and $\rho_{i+1}$ is
local to $D_{i}\mathbin{\text{{\\#}}}D_{i+1}$, where, as shown in Figure 7, we
have $T_{i}(\rho_{i})\simeq\rho_{i}\overline{\rho_{i+1}}\rho_{i}$, giving the
first case in the statement, and $T_{i}(\rho_{i+1})\simeq\rho_{i}$, giving the
second case. $\square$

$T_{i}$
---
$T_{i}$
---
$\simeq$
---
$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\rho_{i}$
---
$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\rho_{i+1}$
---
Figure 7. The action of the Dehn twist $T_{i}$ on the arcs $\rho_{i}$ (top)
and $\rho_{i+1}$ (bottom)

######  Proposition 4.4.

Let $S=(S,m,\varphi)$ be an object of ${{\mathbf{M}}_{2}}$. There is an
isomorphism of semi-simplicial sets

$W_{n}(S,D)_{\bullet}\cong\mathscr{D}^{\nu}(S\mathbin{\text{{\\#}}}D^{\mathbin{\text{{\\#}}}n})_{\bullet}$

where the marked points $b_{0}$ and $b_{1}$ are the midpoints of the intervals
$I_{0}$ and $I_{1}$ in $S\mathbin{\text{{\\#}}}D^{\mathbin{\text{{\\#}}}n}$
and with $\nu=\operatorname{parity}(m+n)$, that is $\nu=1$ if $I_{0}$ and
$I_{1}$ lie in the same boundary component of
$S\mathbin{\text{{\\#}}}D^{\mathbin{\text{{\\#}}}n}$ and $\nu=2$ otherwise.

###### Proof.

We first show that both $W_{n}(S,D)_{p}$ and
$\mathscr{D}^{\nu}(S\mathbin{\text{{\\#}}}D^{\mathbin{\text{{\\#}}}n})_{p}$
are isomorphic to
${\operatorname{Aut}}_{{\mathbf{M}}_{2}}(S\mathbin{\text{{\\#}}}D^{\mathbin{\text{{\\#}}}n})/{\operatorname{Aut}}_{{\mathbf{M}}_{2}}(S\mathbin{\text{{\\#}}}D^{\mathbin{\text{{\\#}}}n-p-1})$
for every $p\geq 0$. This holds by definition for the first semi-simplicial
set. For
$\mathscr{D}^{\nu}(S\mathbin{\text{{\\#}}}D^{\mathbin{\text{{\\#}}}n})_{p}$,
it will follow from two facts: (1) the natural action of

${\operatorname{Aut}}_{{\mathbf{M}}_{2}}(S\mathbin{\text{{\\#}}}D^{\mathbin{\text{{\\#}}}n})=\pi_{0}{\operatorname{Homeo}}_{\partial}(S\mathbin{\text{{\\#}}}D^{\mathbin{\text{{\\#}}}n})$

on this set of $p$-simplices is transitive, and (2) the stabilizer of a
$p$-simplex is isomorphic to
${\operatorname{Aut}}_{{\mathbf{M}}_{2}}(S\mathbin{\text{{\\#}}}D^{\mathbin{\text{{\\#}}}n-p-1})$.
The first fact follows because the homeomorphism type of the complement
$S\setminus\sigma$ of a collection of non-separating arcs $\sigma=\langle
a_{0},\dots,a_{p}\rangle$ is determined by the orderings of the arcs at the
endpoints as this determines the number of boundary components of the
complement (see [Har85, Lem 3.2]), and the second from the fact that this
complement is precisely diffeomorphic to
$S\mathbin{\text{{\\#}}}D^{\mathbin{\text{{\\#}}}n-p-1}$ for any $p$-simplex
in the disordered arc complex. Indeed, this diffeomorphism type does not
depend on the simplex by transitivity of the action, so it is enough to check
the claim for any chosen simplex. Let

$\sigma_{p}=\langle\rho_{n-p},\dots,\rho_{n}\rangle$

be the collection of arcs in
$S\mathbin{\text{{\\#}}}D^{\mathbin{\text{{\\#}}}n}$ consisting of the cores
$\rho_{i}$ of the last $p+1$ disks. Recall from Lemma 4.2 that this is a
disordered simplex, once we note additionally that the arcs are also non-
separating. Now Figure 8 shows that the operation of cutting along the core
$\rho$ of a disk exactly undoes the gluing operation, which proves the claim
in that case.

$D$
---
$\rho$
---
$S$
---
$S$
---
Figure 8. Cutting along the core of a disk

Note that the actions on both sets of simplices are given by post-composition
with mapping classes, where we think here of an arc as an isotopy class of
embedding. There is then a unique equivariant isomorphism $\varphi_{p}\colon
W_{n}(S,D)_{p}\xrightarrow{\cong}\mathscr{D}^{\nu}(S\mathbin{\text{{\\#}}}D^{\mathbin{\text{{\\#}}}n})_{p}$
taking the $p$-simplex

$f_{p}=(S\mathbin{\text{{\\#}}}D^{\mathbin{\text{{\\#}}}n-p-1},\,\operatorname{id}_{S\mathbin{\text{{\\#}}}D^{\mathbin{\text{{\\#}}}n}})$

of $W_{n}(S,D)$ to the $p$-simplex
$\sigma_{p}=\langle\rho_{n-p},\dots,\rho_{n}\rangle$ of the target already
considered above.

We are left to check that the face maps $d_{i}$ correspond to each other under
the isomorphisms $\varphi_{p}$. Because the face maps are equivariant with
respect to the
${\operatorname{Aut}}_{{\mathbf{M}}_{2}}(S\mathbin{\text{{\\#}}}D^{\mathbin{\text{{\\#}}}n})$-action
in both cases, and the actions are transitive, it is enough to check that the
face maps agree for the simplices $f_{p}$ and $\sigma_{p}=\varphi_{p}(f_{p})$.
By definition,

$d_{i}f_{p}=((S\mathbin{\text{{\\#}}}D^{\mathbin{\text{{\\#}}}n-p-1})\mathbin{\text{{\\#}}}D,\,\operatorname{id}_{S\mathbin{\text{{\\#}}}D^{\mathbin{\text{{\\#}}}n-p-1}}\mathbin{\text{{\\#}}}b_{D^{\mathbin{\text{{\\#}}}i},D}^{-1}\mathbin{\text{{\\#}}}\operatorname{id}_{D^{\mathbin{\text{{\\#}}}p-i}})$

while

$d_{i}\sigma_{p}=\langle\rho_{n-p},\dots,\widehat{\rho_{n-p+i}},\dots,\rho_{n}\rangle$

is the simplex obtained by forgetting the $(i+1)$st arc. In particular, we
immediately have that $d_{0}(f_{p})=f_{p-1}$ and
$d_{0}(\sigma_{p})=\sigma_{p-1}=\varphi_{p-1}(f_{p-1})$ giving that the face
maps agree in that case.

For the remaining face maps, note that

$\operatorname{id}_{S\mathbin{\text{{\\#}}}D^{\mathbin{\text{{\\#}}}n-p-1}}\mathbin{\text{{\\#}}}b_{D^{\mathbin{\text{{\\#}}}i},D}^{-1}\oplus\operatorname{id}_{D^{\mathbin{\text{{\\#}}}p-i}}=T_{n-p+i-1}\circ\dots\circ
T_{n-p}\colon
S\mathbin{\text{{\\#}}}D^{\mathbin{\text{{\\#}}}n}\mathrel{\smash{\meno}}\mathrel{\mkern-3.0mu}\rightarrow
S\mathbin{\text{{\\#}}}D^{\mathbin{\text{{\\#}}}n}$

as composition of Dehn twists $T_{i}$ of Section 3.2. We need to compute the
image of $\rho_{n-p+1},\dots,\rho_{n}$ under this map. By Lemma 4.3, we have
that for $1\leq j\leq i$,

$\displaystyle T_{n-p+i-1}\circ\dots\circ T_{n-p}(\rho_{n-p+j})$
$\displaystyle=T_{n-p+i-1}\circ\dots\circ T_{n-p+j-1}(\rho_{n-p+j})$
$\displaystyle=T_{n-p+i-1}\circ\dots\circ T_{n-p+j}(\rho_{n-p+j-1})$
$\displaystyle=\rho_{n-p+j-1}$

while for $i+1\leq j\leq p$,

$T_{n-p+i-1}\circ\dots\circ T_{n-p}(\rho_{n-p+j})=\rho_{n-p+j}.$

Hence $d_{i}(f_{p})$ takes the arcs $\rho_{n-p+1},\dots,\rho_{n}$ to the arcs

$\rho_{n-p},\dots,\rho_{n-p+i-1},\rho_{n-p+i+1},\dots,\rho_{n},$

i.e. precisely to the arcs of $d_{i}(\sigma_{p})$. So we indeed have that
$\varphi_{p-1}(d_{i}(f_{p}))=d_{i}(\varphi_{p-1}(f_{p}))$, which finishes the
proof. $\square$

###  4.2. Coefficient systems

Having identified the space of destabilizations with the semi-simplicial set
of disordered arcs in Proposition 4.4, we can now input the connectivity
computation of the disordered arc complex of Section 2 into the general
stability theorem of [Kra19]. To state the resulting stability theorem in full
generality, we need to introduce the notions of (split) finite degree
coefficient systems. We follow [Kra19, Sec 4], which generalizes [RWW17,
4.1-4] that unify the earlier definitions of Dwyer for the general linear
groups [Dwy80] and Ivanov for the mapping class groups [Iva93]. (The papers
[Kra19, RWW17] consider in addition abelian coefficient systems, but these are
not relevant here, because the abelianization of the mapping class group of
surfaces of large enough genus is trivial by a theorem of
Mumford–Birman–Powell, see Lemma 1.1 in [Har83].)

Fix a bidecorated surface $S=(S,m,\varphi)$, and let $D$ be the bidecorated
disk as above. Definition 4.1 of [Kra19] becomes in our case:

######  Definition 4.5.

A coefficient system for the groups
${\operatorname{Aut}}_{{\mathbf{M}}_{2}}(S\mathbin{\text{{\\#}}}D^{\mathbin{\text{{\\#}}}n})$
with respect to the stabilization by $D$ is a collection of
$\mathbb{Z}[{\operatorname{Aut}}(S\mathbin{\text{{\\#}}}D^{\mathbin{\text{{\\#}}}n})]$-modules
$M_{n}$ for $n\geq 0$, together with maps $s_{n}\colon
M_{n}\mathrel{\smash{\meno}}\mathrel{\mkern-3.0mu}\rightarrow M_{n+1}$ that
are equivariant with respect to the stabilization map
${\operatorname{Aut}}(S\mathbin{\text{{\\#}}}D^{\mathbin{\text{{\\#}}}n})\xrightarrow{\mathbin{\text{{\\#}}}D}{\operatorname{Aut}}(S\mathbin{\text{{\\#}}}D^{\mathbin{\text{{\\#}}}n+1})$,
satisfying the following condition:

(4.1)
$T_{n+1}\in{\operatorname{Aut}}(S\mathbin{\text{{\\#}}}D^{\mathbin{\text{{\\#}}}n+2})\
\textrm{acts trivially on the image of }M_{n}\xrightarrow{s_{n+1}\circ
s_{n}}M_{n+2}$

for $T_{n+1}$ the Dehn twist of Section 3.2 with support the last two disks in
$S\mathbin{\text{{\\#}}}D^{\mathbin{\text{{\\#}}}n+2}$.

We will encode the data of a coefficient system as a pair $(F,\sigma^{F})$
with

$F\colon{{\mathbf{M}}_{2}}|_{S,D}\mathrel{\smash{\meno}}\mathrel{\mkern-3.0mu}\rightarrow\operatorname{Mod}_{\mathbb{Z}}$

a functor from the full subcategory of ${{\mathbf{M}}_{2}}$ on the objects
$S\mathbin{\text{{\\#}}}D^{\mathbin{\text{{\\#}}}n}$ for $n\geq 0$ to abelian
groups, where $M_{n}=F(S\mathbin{\text{{\\#}}}D^{\mathbin{\text{{\\#}}}n})$
with its
${\operatorname{Aut}}(S\mathbin{\text{{\\#}}}D^{\mathbin{\text{{\\#}}}n})$-action
induced by $F$, and

$\sigma^{F}\colon F(-)\mathrel{\smash{\meno}}\mathrel{\mkern-3.0mu}\rightarrow
F(-\mathbin{\text{{\\#}}}D)$

is a natural transformation encoding the suspension maps $s_{n}$, where we
assume that $F(\operatorname{id}\mathbin{\text{{\\#}}}T)$ acts trivially on
the image of $(\sigma^{F})^{2}\colon
F(-)\mathrel{\smash{\meno}}\mathrel{\mkern-3.0mu}\rightarrow
F(-\mathbin{\text{{\\#}}}D^{\mathbin{\text{{\\#}}}2})$ for $T$ the Dehn twist
supported on the added disks $D^{\mathbin{\text{{\\#}}}2}$.

Given a coefficient system $F$, we define its suspension $\Sigma
F\colon{{\mathbf{M}}_{2}}|_{S,D}\mathrel{\smash{\meno}}\mathrel{\mkern-3.0mu}\rightarrow\operatorname{Mod}_{\mathbb{Z}}$
by $\Sigma F(-)=F(-\mathbin{\text{{\\#}}}D)$ with

$\sigma^{\Sigma F}\colon\Sigma
F(-)=F(-\mathbin{\text{{\\#}}}D)\xrightarrow{\sigma^{F}}F(-\mathbin{\text{{\\#}}}D^{\mathbin{\text{{\\#}}}2})\xrightarrow{\operatorname{id}\mathbin{\text{{\\#}}}T}F(-\mathbin{\text{{\\#}}}D^{\mathbin{\text{{\\#}}}2})=\Sigma
F(-\mathbin{\text{{\\#}}}D),$

where one checks that the triviality condition 4.1 is satisfied with this
choice of structure map $\sigma^{\Sigma F}$. (See [Kra19, Def 4.4].)

The structure map $\sigma^{F}$ induces a natural transformation
$F\mathrel{\smash{\meno}}\mathrel{\mkern-3.0mu}\rightarrow\Sigma F$, called
the suspension map. We define the kernel $\ker F$ and cokernel
${\operatorname{coker}}F$ to be the kernel and cokernel functors of that
natural transformation. We call $F$ split if the suspension map is split
injective in the category of coefficient systems.

######  Definition 4.6.

[RWW17, Def 4.10] A coefficient system $F$ is

  1. (a)

of (split) degree $-1$ at $N$ if
$F(S\mathbin{\text{{\\#}}}(D^{\mathbin{\text{{\\#}}}n}))=0$ for all $n\geq N$;

  2. (b)

of degree $k\geq 0$ at $N$ if $\ker(F)$ has degree $-1$ at $N$ and
${\operatorname{coker}}(F)$ has degree $(k-1)$ at $(N-1)$;

  3. (c)

of split degree $k\geq 0$ at $N$ if $F$ is split and
${\operatorname{coker}}(F)$ is of split degree $(k-1)$ at $(N-1)$.

######  Example 4.7.

  1. (a)

A coefficient system $F$ is of degree $0$ at $0$ if and only if $\sigma^{F}$
is a natural isomorphism. This is in particular the case for constant
coefficient systems.

  2. (b)

The functor
$F_{k}\colon{{\mathbf{M}}_{2}}\mathrel{\smash{\meno}}\mathrel{\mkern-3.0mu}\rightarrow\operatorname{Mod}_{\mathbb{Z}}$
defined by

$F_{k}(S)=H_{1}(S,\mathbb{Z})^{\otimes k}$

is a split coefficient system of degree $k$ at $0$. (This is essentially a
result of Ivanov [Iva93, Sec 2.8], who considers a version of the composite
stabilization $\mathbin{\text{{\\#}}}D^{\mathbin{\text{{\\#}}}2}$. See also
[Bol12, Ex 4.3] for the case $k=1$, and [Sou20, Lem 2.9] that proves this in a
very general set-up, though in the case of a braided groupoid acting over
itself only.)

  3. (c)

Given a $k$-connected space $X$, the coefficient system
$F_{n}^{k}\colon{{\mathbf{M}}_{2}}\mathrel{\smash{\meno}}\mathrel{\mkern-3.0mu}\rightarrow\text{Mod}_{\mathbb{Z}}$
defined by

$F_{n}^{k}(S)=H_{n}(\operatorname{Map}(S/\partial S),X),$

which appears in the work of Cohen–Madsen [CM09], is a coefficient system of
degree $\left\lfloor{n/k}\right\rfloor$ (see [Bol12, Ex 4.3]).

######  Remark 4.8.

Although the above examples all makes sense in the different set-ups
considered in the literature, one should keep in mind that there are
variations in what precisely a finite degree coefficient system for the
mapping class groups of surfaces means in e.g. the papers [Iva93, CM09, Bol12,
RWW17] and [Kra19]. This is due to two facts: first, the definition of the
coefficient system depends on the category of surfaces considered and on the
stabilization map(s) one works with, and second, the triviality condition
(4.1) arising from Krannich’s framework is actually weaker than the one used
in earlier frameworks, see e.g. [Kra19, Rem 7.9].

In addition, the paper [GKRW19] uses a homological condition instead of a
finite degree condition (see 5.5.1 in that paper). The relationship between
that condition and finite degree conditions is discussed in [GKRW18, Rem
19.11].

###  4.3. The stability theorem

We are now ready to state our main theorem:

######  Theorem 4.9.

Let $S=(S,m,\varphi)$ be an object of ${{\mathbf{M}}_{2}}$ with $m$ odd, i.e.
such that $I_{0},I_{1}$ are in the same boundary component. Let
$F\colon{{\mathbf{M}}_{2}}|_{S,D}\mathrel{\smash{\meno}}\mathrel{\mkern-3.0mu}\rightarrow\operatorname{Mod}_{\mathbb{Z}}$
be a coefficient system and write
$F_{n}=F(S\mathbin{\text{{\\#}}}D^{\mathbin{\text{{\\#}}}n})$. The map

$H_{i}({\operatorname{Aut}}_{{\mathbf{M}}_{2}}(S\mathbin{\text{{\\#}}}D^{\mathbin{\text{{\\#}}}n});F_{n})\mathrel{\smash{\meno}}\mathrel{\mkern-3.0mu}\rightarrow
H_{i}({\operatorname{Aut}}_{{\mathbf{M}}_{2}}(S\mathbin{\text{{\\#}}}D^{\mathbin{\text{{\\#}}}n+1});F_{n+1})$

is

  1. (a)

an epimorphism for $i\leq\frac{n}{3}$ and an isomorphism for
$i\leq\frac{n-3}{3}$ if $F$ is constant.

  2. (b)

an epimorphism for $i\leq\frac{n-3k-2}{3}$ and an isomorphism for
$i\leq\frac{n-3k-5}{3}$ if $F$ has degree $k$ at $N\geq 0$ and $n>N$.

  3. (c)

an epimorphism for $i\leq\frac{n-k-2}{3}$ and an isomorphism for
$i\leq\frac{n-k-5}{3}$ if $F$ has split degree $k$ at $N\geq 0$ and $n>N$.

######  Remark 4.10.

We have stated the theorem in the case of an initial surface $S$ with $I_{0}$
and $I_{1}$ in the same boundary component for simplicity. The case of a
surface $S^{\prime}$ where the two intervals lie in different components is
actually also included in the statement, by writing
$S^{\prime}=S\mathbin{\text{{\\#}}}D$ for $S$ of the previous type, or
considering $S^{\prime}\mathbin{\text{{\\#}}}D$ if $S^{\prime}$ does not admit
such a decomposition. Indeed, as we have already seen in Section 3 (see Figure
3), gluing in a disk exactly changes whether $I_{0}$ and $I_{1}$ are in the
same boundary or not.

We will first show that the above results implies the two main theorems stated
in the introduction.

###### Proof of Theorems A and B from Theorem 4.9.

Let $S_{0,r}^{s}$ be a surface of genus 0 with $r\geq 1$ boundary components
and $s$ punctures, and consider the associated object
$S=(S_{0,r}^{s},1,\varphi)$ of ${{\mathbf{M}}_{2}}$, with two marked intervals
in the first boundary component. Then
$S\mathbin{\text{{\\#}}}D^{\mathbin{\text{{\\#}}}2g}$ has the form
$(S_{g,r}^{s},1+2g,\varphi)$ while
$S\mathbin{\text{{\\#}}}D^{\mathbin{\text{{\\#}}}2g+1}$ has the form
$(S_{g,r+1}^{s},2+2g,\varphi)$. Moreover, the maps

$S\mathbin{\text{{\\#}}}D^{\mathbin{\text{{\\#}}}2g}\xrightarrow{\mathbin{\text{{\\#}}}D}S\mathbin{\text{{\\#}}}D^{\mathbin{\text{{\\#}}}2g+1}\xrightarrow{\mathbin{\text{{\\#}}}D}S\mathbin{\text{{\\#}}}D^{\mathbin{\text{{\\#}}}2g+2}$

precisely induce on automorphism groups in ${{\mathbf{M}}_{2}}$ the two maps
appearing in Theorems A and B.

The fact that the first map is always injective in homology follows from the
fact that postcomposing the map
$S_{g,r}^{s}\mathrel{\smash{\meno}}\mathrel{\mkern-3.0mu}\rightarrow
S_{g,r+1}^{s}$, defined by the sum $\mathbin{\text{{\\#}}}D$, with the map
$S_{g,r+1}^{s}\mathrel{\smash{\meno}}\mathrel{\mkern-3.0mu}\rightarrow
S_{g,r+1}^{s}\cup_{S^{1}}D^{2}\simeq S_{g,r}^{s}$ filling in one of the newly
created boundary component, is homotopic to the identity. Now Theorem 4.9(a)
gives that the map

$H_{i}({\operatorname{Aut}}_{{{\mathbf{M}}_{2}}}(S\mathbin{\text{{\\#}}}D^{\mathbin{\text{{\\#}}}2g}))\xrightarrow{\mathbin{\text{{\\#}}}D}H_{i}({\operatorname{Aut}}_{{{\mathbf{M}}_{2}}}(S\mathbin{\text{{\\#}}}D^{\mathbin{\text{{\\#}}}2g+1}))$

is surjective for $i\leq\frac{2g}{3}$ in homology with constant coefficients.
Given that the map is always injective, we get an isomorphism in that same
range, proving the first part of Theorem A. Applying (b) and (c) instead gives
Theorem B for the first map.

For the second map, we now apply Theorem 4.9 in the case $n=2g+1$, but in that
case, there is no additional argument for injectivity, so the bounds translate
directly to surjectivity and isomorphism bounds. $\square$

###### Proof of Theorem 4.9.

Proposition 4.4 together with Theorem 2.4 give that $W_{n}(S,D)_{\bullet}$ is
$\left(\frac{2g+\nu-5}{3}\right)$-connected, for $g$ the genus of
$S\mathbin{\text{{\\#}}}D^{\mathbin{\text{{\\#}}}n}$ and $\nu=1$ if $I_{0}$
and $I_{1}$ are in the same boundary component of
$S\mathbin{\text{{\\#}}}D^{\mathbin{\text{{\\#}}}n}$, which is the case
precisely when $n$ is even, and $\nu=2$ otherwise. The surface
$S\mathbin{\text{{\\#}}}D^{\mathbin{\text{{\\#}}}n}$ has genus greater than or
equal to the genus of $D\mathbin{\text{{\\#}}}D^{\mathbin{\text{{\\#}}}n}$,
that is $\frac{n}{2}$ if $n$ is even and $\frac{n-1}{2}$ if $n$ is odd (see
Lemma 3.1). Hence $2g+\nu\geq n+1$ is both cases, and $W_{n}(S,D)_{\bullet}$
is at least $\left(\frac{n-4}{3}\right)$-connected.

Now $W_{n}(S,D)_{\bullet}$ is the semi-simplicial set denoted
$W^{\text{RW}}(S\mathbin{\text{{\\#}}}D^{\mathbin{\text{{\\#}}}n})_{\bullet}$
in [Kra19] (see Definition 7.5 in that paper). By Lemma 7.6 in the same paper,
using Proposition 3.4, this semi-simplicial set has the same connectivity as
the semi-simplicial space
$W(S\mathbin{\text{{\\#}}}D^{\mathbin{\text{{\\#}}}n})_{\bullet}$ of [Kra19],
which by Remark 2.7 of that paper determines the connectivity assumption of
Theorem A in that paper: the canonical resolution of the assumption of the
theorem is $m$-connected, if and only if the space
$W(S\mathbin{\text{{\\#}}}D^{\mathbin{\text{{\\#}}}n})_{\bullet}$ is
$(m-1)$-connected. Given that
$W(S\mathbin{\text{{\\#}}}D^{\mathbin{\text{{\\#}}}n})_{\bullet}$ is
$\left(\frac{n-4}{3}\right)$–connected, we have that the canonical resolution
of is $\left(\frac{n-4+3}{3}\right)$–connected. Hence we can apply [Kra19, Thm
A] with $k=3$ and grading
$g_{{{\mathbf{M}}_{2}}}:{{\mathbf{M}}_{2}}|_{S,D}\mathrel{\smash{\meno}}\mathrel{\mkern-3.0mu}\rightarrow\mathbb{N}$
given by
$g_{{{\mathbf{M}}_{2}}}(S\mathbin{\text{{\\#}}}D^{\mathbin{\text{{\\#}}}n})=n-2$;
see also [Kra19, Rem 2.24], where we can take $m=4$. The theorem, with the
improvement given by (i) in the remark, then gives that

$H_{i}(\text{Aut}_{{{\mathbf{M}}_{2}}}(S\mathbin{\text{{\\#}}}D^{\mathbin{\text{{\\#}}}n});\mathbb{Z})\mathrel{\smash{\meno}}\mathrel{\mkern-3.0mu}\rightarrow
H_{i}(\text{Aut}_{{{\mathbf{M}}_{2}}}(S\mathbin{\text{{\\#}}}D^{\mathbin{\text{{\\#}}}n+1});\mathbb{Z})$

is an isomorphism for $i\leq\frac{n-3}{3}$ and an epimorphism for
$i\leq\frac{n}{3}$, giving the stated result in the case of constant
coefficients. For a coefficient system $F$ of degree $k$ at $N$, [Kra19, Thm
C] gives that

$H_{i}(\text{Aut}_{{{\mathbf{M}}_{2}}}(S\mathbin{\text{{\\#}}}D^{\mathbin{\text{{\\#}}}n});F_{n})\mathrel{\smash{\meno}}\mathrel{\mkern-3.0mu}\rightarrow
H_{i}(\text{Aut}_{{{\mathbf{M}}_{2}}}(S\mathbin{\text{{\\#}}}D^{\mathbin{\text{{\\#}}}n+1});F_{n+1})$

is an isomorphism for $i\leq\frac{n-3k-5}{3}$ and an epimorphism for
$i\leq\frac{n-3k-2}{3}$ for $n>N$, improved to an isomorphism for
$i\leq\frac{n-k-5}{3}$ and an epimorphism for $i\leq\frac{n-k-2}{3}$ if $F$ is
split. $\square$

######  Remark 4.11 (Optimality of the stability bounds).

Combining the two maps in Theorem A, we obtain that the genus stabilization

$H_{i}(\Gamma(S_{g,r}^{s});\mathbb{Z})\mathrel{\smash{\meno}}\mathrel{\mkern-3.0mu}\rightarrow
H_{i}(\Gamma(S^{s}_{g+1,r});\mathbb{Z})$

is an epimorphism when $i\leq\frac{2g}{3}$ and an isomorphism when
$i\leq\frac{2g-2}{3}$. The slope $\frac{2}{3}$ is known to be optimal by a
computation of Morita [Mor03], with optimal isomorphism range since for
instance
$H_{1}(\Gamma(S_{2,r});\mathbb{Z})\mathrel{\smash{\meno}}\mathrel{\mkern-3.0mu}\rightarrow
H_{1}(\Gamma(S_{3,r});\mathbb{Z})$ is not injective as the source is
isomorphic to $\mathbb{Z}/12$ and the target is trivial, see e.g. [Kor02,
Theorem 5.1]. Our combined genus epimorphism range, on the other hand, falls
short of the range $i\leq\frac{2g+1}{3}$, as given in [GKRW19], a range that
is optimal by Morita’s computation (see Theorem B (i) of [GKRW19]).

Our results for twisted coefficients are most easily compared with those of
Boldsen [Bol12, Thm 3], whose coefficient systems are coefficient systems of
finite split degree in our sense, though with a stricter triviality condition
upon double stabilization. For these coefficient systems, he obtains slightly
better ranges, with improvement $+{}^{2}{\mskip-5.0mu/\mskip-3.0mu}_{3}$ for
the first map and $+{}^{5}{\mskip-5.0mu/\mskip-3.0mu}_{3}$ for the second. The
papers [RWW17, GKRW19] only consider genus stability. In [RWW17], the
stability slope obtained is only $\frac{1}{2}$, while in [GKRW19, Sec 5.5.1],
the finite degree condition is replaced by a more general homological
condition that applies to some finite coefficient systems [GKRW18, Sec 19.2].
In the particular case of the $k$th tensor power of the first homology of the
surface, they do however only get the epimorphism range
$i\leq\frac{2g-2k+1}{3}$ and isomorphism range $i\leq\frac{2g-2k-2}{3}$, see
Example 5.22 in that paper.

##  5\. Braiding and homological stability for groups

In order to use the framework of Krannich [Kra19] to prove homological
stability for a sequence of groups, one needs the structure of an
“$E_{1}$-module over an $E_{2}$-algebra”. We give in Proposition 5.1 below a
simple way to construct such a module structure, in terms of Yang–Baxter
operators. Compared to earlier approaches to homological stability such as
[RWW17], which Krannich’s work generalizes, this has the advantage of being
very lightweight. Instead of having to provide the structure of a braiding on
the monoidal category whose automorphism groups one is interested in, it
suffices to provide a single morphism satisfying a simple equation.

Our main example of a Yang–Baxter operator is the inverse Dehn twist
$T_{1}^{-1}\in{\operatorname{Aut}}_{{{\mathbf{M}}_{2}}}(D\mathbin{\text{{\\#}}}D)$,
defined in Section 3.2 and used to prove our main result. In Section 5.3, we
show that this Yang–Baxter operator is not part of a braided monoidal
structure on the category ${{\mathbf{M}}_{2}}$, but gives instead a twisted
version of such a structure.

###  5.1. Yang–Baxter operators and braid groupoid actions

Let $\mathscr{X}=(\mathscr{X},\oplus,\mathbbm{1})$ be a monoidal category. A
Yang–Baxter operator in $\mathscr{X}$ is a pair $(X,\tau)$ consisting of an
object $X\in\mathscr{X}$ and a morphism
$\tau\in{\operatorname{Aut}}_{\mathscr{C}}(X\oplus X)$, satisfying the
Yang–Baxter equation

$(\tau\oplus 1)(1\oplus\tau)(\tau\oplus 1)=(1\oplus\tau)(\tau\oplus
1)(1\oplus\tau)\in{\operatorname{Aut}}_{\mathscr{C}}(X\oplus X\oplus X),$

where we suppress associators from the notation.

Yang–Baxter operators are closely related to the braid groupoid: Recall from
Section 3.2 the braid groupoid $\mathbf{B}$, with objects the natural numbers
and only non-trivial morphisms ${\operatorname{Aut}}_{\mathbf{B}}(n)=B_{n}$. A
variant of the coherence theorem for braided monoidal categories says that the
category of strong monoidal functors from the braid groupoid into
$\mathscr{X}$ is equivalent to a naturally defined category of Yang–Baxter
operators in $\mathscr{X}$ [JS93, Prop 2.2].444 In other words, the pair
consisting of the braid groupoid $\mathbf{B}$ and the Yang–Baxter operator
$\sigma_{1}\in\text{Aut}_{\mathbf{B}}(2)$, is the initial monoidal category
with a distinguished Yang–Baxter element. To a Yang–Baxter operator $(X,\tau)$
in $\mathscr{X}$, this equivalence associates the strong monoidal functor
$\Phi_{X,\tau}\colon\mathbf{B}\mathrel{\smash{\meno}}\mathrel{\mkern-3.0mu}\rightarrow\mathscr{X}$
given by $\Phi_{X,\tau}(n)=X^{\oplus n}$ on objects, and on morphisms by
letting

$\Phi_{X,\tau}\colon
B_{n}\mathrel{\smash{\meno}}\mathrel{\mkern-3.0mu}\rightarrow{\operatorname{Aut}}_{\mathscr{X}}(X^{\oplus
n})$

send the $i$th standard generator $\sigma_{i}$ to $\text{id}_{X^{\oplus
i-1}}\oplus\tau\oplus\text{id}_{X^{\oplus n-i-1}}$, where the required maps
$\Phi_{X,\tau}(m)\oplus\Phi_{X,\tau}(n)\mathrel{\smash{\meno}}\mathrel{\mkern-3.0mu}\rightarrow\Phi_{X,\tau}(m+n)$
are given by the monoidal structure of $\mathscr{X}$.

Suppose now that the monoidal category $\mathscr{X}$ acts on a category
$\mathscr{M}$ via a functor
$\mathscr{M}\times\mathscr{X}\mathrel{\smash{\meno}}\mathrel{\mkern-3.0mu}\rightarrow\mathscr{M}$,
which we also denote by $\oplus$, compatible with the monoidal sum in
$\mathscr{X}$. The following result shows that the choice of a Yang–Baxter
operator defines an action of the braid groupoid $\mathbf{B}$ on
$\mathscr{M}$, and hence is appropriate data to apply the stability framework
of [Kra19]:

######  Proposition 5.1.

Let $(\mathscr{X},\oplus,\mathbbm{1})$ be a monoidal category with
$\tau\in{\operatorname{Aut}}_{\mathscr{X}}(X\oplus X)$ a Yang–Baxter operator
in $\mathscr{X}$. Suppose $\mathscr{X}$ acts on a category $\mathscr{M}$. Then
there is an action of the braid groupoid

$\alpha_{\tau}\colon\mathscr{M}\times\mathbf{B}\mathrel{\smash{\meno}}\mathrel{\mkern-3.0mu}\rightarrow\mathscr{M}$

given on objects by $\alpha_{\tau}(A,n)=A\oplus X^{\oplus n}$ and determined
on morphisms by

$\alpha_{\tau}(f,\sigma_{i})=f\oplus\text{id}_{X^{\oplus
i-1}}\oplus\tau\oplus\text{id}_{X^{\oplus n-i-1}},$

for $\sigma_{i}$ the $i$th elementary braid in $B_{n}$. Furthermore, taking
classifying spaces this endows $B\mathscr{M}$ with the structure of an
$E_{1}$-module over the $E_{2}$-algebra $B\mathbf{B}$.

Note that if we are interested in homological stability for stabilization by
$X$ for the automorphism groups
$G_{n}:={\operatorname{Aut}}_{\mathscr{M}}(A\oplus X^{\oplus n})$ for some
object $A$ of $\mathscr{M}$, only the full subcategory
$\mathscr{M}_{A,X}\subseteq\mathscr{M}$ spanned by objects of the form
$A\oplus X^{\oplus n}$, is relevant. So for stability purposes, it is enough
to consider the subfunctor

$\alpha_{\tau}\colon\mathscr{M}_{A,X}\times\mathbf{B}\mathrel{\smash{\meno}}\mathrel{\mkern-3.0mu}\rightarrow\mathscr{M}_{A,X}.$

In fact, to make sure that the structure of $E_{1}$–module over the
$E_{2}$–algebra $B\mathbf{B}$ is graded, one can even replace the category
$\mathscr{M}_{A,X}$ by a category with objects the natural numbers and setting
$\text{Aut}(n)=\text{Aut}_{\mathscr{M}}(A\oplus X^{\oplus n})$, avoiding any
potential issue coming from unwanted equalities $A\oplus X^{\oplus n}=A\oplus
X^{\oplus m}$ for $m\neq n$.

###### Proof.

The functor
$\alpha_{\tau}\colon\mathscr{M}\times\mathbf{B}\mathrel{\smash{\meno}}\mathrel{\mkern-3.0mu}\rightarrow\mathscr{M}$
is defined as the composite functor

$\alpha(-,-)=(-)\oplus\Phi_{X,\tau}(-),$

for
$\Phi_{X,\tau}\colon\mathbf{B}\mathrel{\smash{\meno}}\mathrel{\mkern-3.0mu}\rightarrow\mathscr{X}$
as above. The result follows from [Kra19, Lem 7.2] because $\alpha$ makes
$\mathscr{M}$ into a module over $\mathbf{B}$ and $\mathbf{B}$ is braided
monoidal. $\square$

######  Example 5.2.

If $\mathscr{X}=(\mathscr{X},\oplus,\mathbbm{1})$ admits a braiding $b$, then
$\tau=b_{X,X}\in{\operatorname{Aut}}_{\mathscr{X}}(X\oplus X)$ is a
Yang–Baxter operator for any object $X$. For $\mathscr{X}$ a groupoid acting
on itself or $\mathscr{X}$ acting on a category $\mathscr{M}$, this recovers
the basic set-up for homological stability of the paper [RWW17], or Section 7
of [Kra19].

######  Example 5.3 (Mapping class groups of surfaces).

As explained above, a Yang–Baxter operator
$\tau\in{\operatorname{Aut}}_{\mathscr{X}}(X\oplus X)$ gives in particular a
collection of homomorphisms $\Phi_{X,\tau}\colon
B_{n}\mathrel{\smash{\meno}}\mathrel{\mkern-3.0mu}\rightarrow{\operatorname{Aut}}_{\mathscr{X}}(X^{\oplus
n})$ from the braid groups to the automorphism group of $n$ copies of $X$.
There are two standard ways to embed braid groups in mapping class groups of
surfaces, and we explain here how they both come from Yang–Baxter elements in
appropriate categories of surfaces.

  1. (a)

Let ${{\mathbf{M}}_{2}}$ be the category of bidecorated surfaces of Section 3.
As explained in Section 3.2, the Dehn twist
$T\in{\operatorname{Aut}}_{{{\mathbf{M}}_{2}}}(D\mathbin{\text{{\\#}}}D)\cong\pi_{0}\operatorname{Homeo}_{\partial}(S^{1}\times
I)\cong\mathbb{Z}$, or its inverse $T^{-1}$, is a Yang–Baxter operator. The
associated map $\Phi_{D,T}\colon
B_{n}\mathrel{\smash{\meno}}\mathrel{\mkern-3.0mu}\rightarrow{\operatorname{Aut}}_{{{\mathbf{M}}_{2}}}(D^{\mathbin{\text{{\\#}}}n})$
is the embedding of braid group in the mapping class groups of $S_{g,1}$ (when
$n=2g+1$) and of $S_{g,2}$ (when $n=2g+2$) associated to Dehn twists along the
chain of embedded curves in the surfaces described in Lemma 3.5. This
embedding goes back at least to the work of Birman and Hilden [BH72, BH73].

  2. (b)

Let ${\mathbf{M}}_{1}$ denote instead the category of surfaces decorated by a
single interval, with monoidal structure $\oplus$ defined just as in the case
of $\mathbf{M}_{1}$ but gluing only along one interval. Then $\mathbf{M}_{1}$
in braided monoidal, see [RWW17, Sec 5.6.1]. Hence by Example 5.2, for any
object $X$ of $\mathbf{M}_{1}$, we have a Yang–Baxter element
$\tau_{X}\in{\operatorname{Aut}}_{\mathbf{M}_{1}}(X\oplus X)$. For
$X=S_{1,2}$, this can be used to prove genus stabilization (albeit with the
suboptimal slope ${}^{1}{\mskip-5.0mu/\mskip-3.0mu}_{2}$), and in the case
$X=S^{1}\times I$ marked by an interval in one of its boundary components, we
have that $X^{\oplus n}$ has underlying surface an $n$-legged pair of pants
$D^{2}\backslash(\sqcup_{n}\mathring{D}^{2})$ and the associated morphism

$\Phi_{X,\tau_{X}}\colon
B_{n}\mathrel{\smash{\meno}}\mathrel{\mkern-3.0mu}\rightarrow{\operatorname{Aut}}_{\mathbf{M}_{1}}(X^{\oplus
n})=\pi_{0}\operatorname{Homeo}_{\partial}(D^{2}\backslash(\sqcup_{n}\mathring{D}^{2}))$

is the standard embedding of the braid group as the subgroup of the mapping
class group of the multi-legged pants that does not twist the legs, see e.g.
[RWW17, Sec 5.6.1].

We will show in Proposition 5.7 below that the Yang–Baxter operator $T$ of the
first example, in the category ${{\mathbf{M}}_{2}}$, does not come from a
braiding in ${{\mathbf{M}}_{2}}$.

###  5.2. Homological stability from Yang–Baxter elements

Suppose we are given the data of a monoidal category
$(\mathscr{X},\oplus,\mathbbm{1})$ acting on a category $\mathscr{M}$, along
with a choice of stabilizing object $X\in\mathscr{X}$ and Yang–Baxter operator
$\tau\in\text{Aut}_{\mathscr{X}}(X\oplus X)$. Proposition 5.1 above allows to
apply [Kra19, Thm A], which in this case says that for any $A\in\mathscr{M}$,
there is a sequence of simplicial spaces $W_{n}(X,A)_{\bullet}$, for $n\geq
0$, so that if $W_{n}(X,A)$ is highly-connected for large $n$, then the
sequence

$\text{Aut}_{\mathscr{M}}(A)\xrightarrow{{-}\oplus
X}\text{Aut}_{\mathscr{M}}(A\oplus X)\xrightarrow{{-}\oplus
X}\text{Aut}_{\mathscr{M}}(A\oplus X\oplus X)\xrightarrow{{-}\oplus X}\cdots$

satisfies homological stability. Theorem B of the same paper gives in addition
a stability statement with twisted coefficients. Under an injectivity
assumption of the form of Proposition 3.4, this simplicial space is homotopy
discrete, and modeled by the space of destabilizations as described in
Definition 4.1.

######  Remark 5.4.

The fact that $(X,\tau)$ is a Yang–Baxter operator is precisely what is needed
for the collection of sets $W_{n}(A,X)_{p}$ and maps $d_{i}\colon
W_{n}(A,X)_{p}\mathrel{\smash{\meno}}\mathrel{\mkern-3.0mu}\rightarrow
W_{n}(A,X)_{p-1}$, defined as in Definition 4.1, to assemble into a semi-
simplicial set; indeed, the Yang–Baxter equation implies the necessary
simplicial identities.

For a fixed monoidal category $\mathscr{X}=(\mathscr{X},\oplus,\mathbbm{1})$
acting on a category $\mathscr{M}$, and a stabilizing object
$X\in\mathscr{X}$, the choice of Yang–Baxter element will not affect the
stabilizing map, but it will affect the spaces $W_{n}(X,A)_{\bullet}$. The
identity map $1\in\text{Aut}_{\mathscr{X}}(X\oplus X)$ is a trivial choice of
Yang–Baxter operator. But, as is to be expected, this trivial twist is not
useful for proving stability:

######  Proposition 5.5.

Let $\mathscr{X},\mathscr{M}$, $A$ and $X$ be as above. If we choose the
Yang–Baxter operator $\tau\in{\operatorname{Aut}}_{\mathscr{X}}(X\oplus X)$ to
be the identity element, then the semi-simplicial set $W_{n}(A,X)_{\bullet}$
is connected if and only if the map

$G_{n-1}=\text{Aut}_{\mathscr{M}}(A\oplus X^{\oplus
n-1})\xrightarrow{{-}\oplus X}\text{Aut}_{\mathscr{M}}(A\oplus X^{\oplus
n})=G_{n}$

is an isomorphism.

###### Proof.

If $\tau$ is the identity element, all face maps $d_{i}$ are equal to the
canonical map
$G_{n}/G_{n-p-1}\mathrel{\smash{\meno}}\mathrel{\mkern-3.0mu}\rightarrow
G_{n}/G_{n-p}$. In particular, the vertices of any $p$-simplex are all equal,
so the semi-simplicial set $W_{n}(A,X)_{\bullet}$ is isomorphic to a disjoint
union of semi-simplicial sets, one for each $0$-simplex. The result follows
from the fact that the set of $0$-simplices is precisely the quotient
$G_{n}/G_{n-1}$. $\square$

In fact, Barucco proved in his master thesis a result that translates to the
following stronger statement (stated in the thesis in the context of a
groupoid acting on itself, i.e. $\mathscr{M}=\mathscr{X}$):

######  Lemma 5.6.

[Bar17, Lem 3.1] The space $W_{n}(A,X)$ is connected if and only if $1^{\oplus
n-2}\oplus\tau$ and $G_{n-1}\oplus 1$ together generate
$G_{n}={\operatorname{Aut}}(A\oplus X^{\oplus n})$.

The connectivity of the semi-simplicial set $W_{n}(A,X)$ (or of the associated
simplical complex defined in [RWW17, Def 2.8]) can be thought of as a measure
a form of higher generation of the group $G_{n}$ by the cosets of the
subgroups $G_{n-p}$ for $p\geq 1$ and braid subgroups generated by the chosen
Yang–Baxter element $t$, in a way similar to the notion of higher generation
for a family of subgroups of a group defined in [AH93, 2.1].

###  5.3. Braidings and bidecorated surfaces

We show in this section that the Yang–Baxter operator $T$ on the bidecorated
disk $D$ in the groupoid ${{\mathbf{M}}_{2}}$ does not come from a braiding on
the subcategory of ${{\mathbf{M}}_{2}}$ generated by the disk. In fact, we
will show that this subcategory does not admit a braiding.

Let $D=(D^{2},1,\operatorname{id})$ be the standard bidecorated disk of
Section 3, where we recall that $X_{1}=D^{2}$. We define a “rotated”
bidecorated disk $\overline{D}=(D^{2},1,r_{\pi})$, where $r_{\pi}$ is the
rotation of $\partial X_{1}=\partial D^{2}$ by $\pi$ radians, which has the
effect of interchanging the intervals $I_{0}$ and $I_{1}$. Rotating all of
$D^{2}$ by $\pi$ then induces a morphism $\iota\colon
D\mathrel{\smash{\meno}}\mathrel{\mkern-3.0mu}\rightarrow\overline{D}$ in
${{\mathbf{M}}_{2}}$, and likewise morphisms

$\iota^{\mathbin{\text{{\\#}}}m}\colon
D^{\mathbin{\text{{\\#}}}m}\mathrel{\smash{\meno}}\mathrel{\mkern-3.0mu}\rightarrow\overline{D}^{\mathbin{\text{{\\#}}}m}$

for every $m\geq 1$, each which we will by abuse of notation also denote by
$\iota$. The morphism $\iota$ can be identified with the hyperelliptic
involution of the underlying surface depicted in Figure 9 for the two cases
$m=2g$ and $m=2g+1$, where in the latter case the boundary components are
exchanged by $\iota$.

$\cdots$
---
$\cdots$
---
Figure 9. The hyperelliptic involutions $\iota$ of $S_{g,1}$ and $S_{g,2}$

The morphism $\iota$ induces an identification

$\displaystyle\text{Aut}_{{{\mathbf{M}}_{2}}}(D^{\mathbin{\text{{\\#}}}m})$
$\displaystyle\xrightarrow{\ {\cong}\
}\text{Aut}_{{{\mathbf{M}}_{2}}}(\overline{D}^{\mathbin{\text{{\\#}}}m})$
$\displaystyle f$ $\displaystyle\ext@arrow
0395\arrowfill@{\mapstochar\mathrel{\smash{\meno}}}\mathrel{\smash{\meno}}\rightarrow{}{\
{\phantom{\sim}}\ }\iota\circ f\circ\iota^{-1}$

In order to precisely state the failure of $T$ to extend to a braiding, we
will also need the identification

$\displaystyle
I\colon\text{Aut}_{{{\mathbf{M}}_{2}}}(D^{\mathbin{\text{{\\#}}}m})$
$\displaystyle\xrightarrow{\ {\cong}\
}\text{Aut}_{{{\mathbf{M}}_{2}}}(\overline{D}^{\mathbin{\text{{\\#}}}m})$
$\displaystyle f$ $\displaystyle\ext@arrow
0395\arrowfill@{\mapstochar\mathrel{\smash{\meno}}}\mathrel{\smash{\meno}}\rightarrow{}{\
{\phantom{\sim}}\ }f$

that comes from the fact that an element
$f\in\text{Aut}_{{{\mathbf{M}}_{2}}}(D^{\mathbin{\text{{\\#}}}m})$ is just a
mapping class for the underlying surface of $D^{\mathbin{\text{{\\#}}}m}$,
which is the same as the underlying surface of
$\overline{D}^{\mathbin{\text{{\\#}}}m}$, so $f$ can just as well be viewed as
an element of
$\text{Aut}_{{{\mathbf{M}}_{2}}}(\bar{D}^{\mathbin{\text{{\\#}}}m})$. In
contrast with the identification induced by $\iota$, the second identification
is “external”, in the sense that it does not come from a morphism in
${{\mathbf{M}}_{2}}$.

Viewing $\iota$ as a diffeomorphism of the underlying surface $X_{m}$ of
$D^{\mathbin{\text{{\\#}}}m}$ that does not fix the boundary, and specifically
exchanges the marked points
$b_{0}=I_{0}({}^{1}{\mskip-5.0mu/\mskip-3.0mu}_{2})$ and
$b_{1}=I_{1}({}^{1}{\mskip-5.0mu/\mskip-3.0mu}_{2})$, we see that it takes the
isotopy class of arc $\rho_{i}$ of Section 4.1 to the reversed arc
$\overline{\rho_{i}}$. We will use in the proof of the following result that
the homotopy classes $\rho_{i}$ generate the fundamental groupoid of $X_{m}$
based at the points $b_{0},b_{1}$.555 As a full subgroupoid of the ordinary
fundamental groupoid of $X_{m}$, this groupoid is the one spanned by the
objects corresponding to the points $b_{0},b_{1}\in X_{m}$.  The mapping class
$\iota$ is in fact completely determined by the fact that
$\iota(\rho_{i})=\overline{\rho_{i}}$.

######  Proposition 5.7.

Let $\mathbf{D}\subset{{\mathbf{M}}_{2}}$ denote the full monoidal subcategory
generated by $D$.

  1. (i)

The monoidal category $\mathbf{D}$ does not admit a braiding. In particular,
the monoidal functor

$\Phi\colon(\mathbf{B},\oplus)\mathrel{\smash{\meno}}\mathrel{\mkern-3.0mu}\rightarrow(\mathbf{D},\mathbin{\text{{\\#}}})\subset({{\mathbf{M}}_{2}},\mathbin{\text{{\\#}}})$

does not come from a braiding on $\mathbf{D}$.

  2. (ii)

Let $f\in{\operatorname{Aut}}_{{\mathbf{M}}_{2}}(D^{\mathbin{\text{{\\#}}}m})$
and
$g\in{\operatorname{Aut}}_{{\mathbf{M}}_{2}}(D^{\mathbin{\text{{\\#}}}n})$,
and put $\beta_{m,n}=\Phi(b_{m,n})$, where the block braid $b_{m,n}$ is the
braid which passes the last $n$ strands over the first $m$ strands. Then

$\beta_{m,n}\circ(f\mathbin{\text{{\\#}}}g)\circ\beta_{n,m}^{-1}=\begin{cases}g\mathbin{\text{{\\#}}}(\iota^{-1}\circ
f\circ\iota)&\text{if }n\text{ is odd},\\\
g\mathbin{\text{{\\#}}}f&\text{else,}\end{cases}$

for $\iota\colon
D^{\mathbin{\text{{\\#}}}m}\mathrel{\smash{\meno}}\mathrel{\mkern-3.0mu}\rightarrow\overline{D}^{\mathbin{\text{{\\#}}}m}$
the involution defined above, and where $f$ in the rightmost expression is the
map $f$ considered as an element of
${\operatorname{Aut}}_{{\mathbf{M}}_{2}}(\overline{D}^{\mathbin{\text{{\\#}}}m})$
via the isomorphism $I$ defined above.

###### Proof.

We start by proving (ii). It is enough to check the statement when $f$ and $g$
are Dehn twists, as those generate the mapping class groups. Note that if $c$
is a curve in the underlying surface $X_{m+n}$ of
$D^{\mathbin{\text{{\\#}}}m+n}$, and $T_{c}$ denotes the Dehn twist along $c$,
then conjugating $T_{c}$ by a diffeomorphism $\varphi$ of the surface gives

$\varphi\circ T_{c}\circ\varphi^{-1}=T_{\varphi(c)}.$

Recall further that the isotopy class of a Dehn twist $T_{c}$ depends only on
the free homotopy class of the curve $c$. We are therefore to compute the
images of curves in $D^{\mathbin{\text{{\\#}}}m}$ and
$D^{\mathbin{\text{{\\#}}}n}$ under the map $\beta_{m,n}$, as free homotopy
classes. A curve $c$ can be written, up to free homotopy, as a concatenation
of the arcs $\rho_{i}$ and their inverses $\overline{\rho}_{i}$, as the
homotopy classes of these arcs generate the fundamental groupoid of the
surface $X_{m+n}$ based at $b_{0},b_{1}$. In particular, write

(5.1)
$c\simeq\rho_{i_{1}}*\overline{\rho}_{i_{2}}*\rho_{i_{3}}\dots*\overline{\rho}_{i_{k}}.$

The mapping class $\beta_{m,n}$ can be written as the composition

$\beta_{m,n}=(T_{n}\circ\dots\circ
T_{m+n-1})\circ\dots\circ(T_{2}\circ\dots\circ
T_{m+1})\circ(T_{1}\circ\dots\circ T_{m})$

and hence we can compute the image of each $\rho_{i}$ using Lemma 4.3. For
$r>0$, denote by $T_{i,i+r}$ the composition of Dehn twists $T_{i}\circ
T_{i+1}\circ\dots\circ T_{i+r}$. Note first that

$T_{i,j}(\rho_{j+1})\simeq T_{i,j-1}(\rho_{j})\simeq\dots\simeq\rho_{i}.$

From this, it follows that for $i\geq 1$,

$\displaystyle\beta_{m,n}(\rho_{m+i})$
$\displaystyle\simeq(T_{n,m+n-1})\circ\dots\circ(T_{1,m})(\rho_{m+i})$
$\displaystyle\simeq(T_{n,m+n-1})\circ\dots\circ(T_{i,m+i-1})(\rho_{m+i})$
$\displaystyle\simeq(T_{n,m+n-1})\circ\dots\circ(T_{i+1,m+i})(\rho_{i})$
$\displaystyle\simeq\rho_{i}.$

On the other hand, for $i\leq k\leq j$, we have

$T_{i,j}(\rho_{k})\simeq T_{i,k}(\rho_{k})\simeq
T_{i,k-1}(\rho_{k}*\overline{\rho}_{k+1}*\rho_{k})\simeq\rho_{i}*\overline{\rho}_{k+1}*\rho_{i},$

from which we can deduce that for $i\leq m$,

$\displaystyle\beta_{m,n}(\rho_{i})$
$\displaystyle\simeq(T_{n,m+n-1})\circ\dots\circ(T_{1,m})(\rho_{i})$
$\displaystyle\simeq(T_{n,m+n-1})\circ\dots\circ(T_{2,m+1})(\rho_{1}*\overline{\rho}_{i+1}*\rho_{1})$
$\displaystyle\simeq(T_{n,m+n-1})\circ\dots\circ(T_{3,m+2})(\rho_{1}*\overline{\rho}_{2}*\rho_{i+2}*\overline{\rho}_{2}*\rho_{1})$
$\displaystyle\simeq\cdots$
$\displaystyle\simeq\rho_{1}*\iota(\rho_{2})*\dots*\iota^{n-1}(\rho_{n})*\iota^{n}(\rho_{i+n})*\iota^{n-1}(\rho_{n})*\dots*\iota(\rho_{2})*\rho_{1}$

since $\iota^{j}(\rho_{i})$ is $\rho_{i}$ when $j$ is even and
$\overline{\rho}_{i}$ when $j$ is odd.

If the curve $c$ lies in the last $n$ disks $D^{\mathbin{\text{{\\#}}}n}$
inside $D^{\mathbin{\text{{\\#}}}m+n}$, it can be written as a product (5.1)
with each $i_{j}>m$. Then the above computation gives that

$\beta_{m,n}(c)\simeq\rho_{i_{1}-m}*\overline{\rho}_{i_{2}-m}*\rho_{i_{3}-m}*\dots*\overline{\rho}_{i_{k}-m},$

that is, $c$ is mapped to the corresponding curve in the first $n$ disks
$D^{\mathbin{\text{{\\#}}}n}$ inside
$D^{\mathbin{\text{{\\#}}}n+m}=D^{\mathbin{\text{{\\#}}}m+n}$.

If the curve $c$ instead lies in the first $m$ disks
$D^{\mathbin{\text{{\\#}}}m}$ inside $D^{\mathbin{\text{{\\#}}}m+n}$, it can
be written as a product (5.1) with each $i_{j}\leq m$. Then the above
computation gives that

$\displaystyle\beta_{m,n}(c)$
$\displaystyle\simeq\rho_{1}*\iota(\rho_{2})*\dots*\iota^{n-1}(\rho_{n})*\iota^{n}(\rho_{i_{1}+n})*\iota^{n+1}(\rho_{i_{2}+n})*\dots*\iota^{n+1}(\rho_{i_{k}+n})$
$\displaystyle\hskip
189.49545pt*\iota^{n}(\rho_{n})*\dots*\iota^{2}(\rho_{2})*\iota(\rho_{1})$
$\displaystyle\simeq\iota^{n}(\rho_{i_{1}+n})*\iota^{n+1}(\rho_{i_{2}+n})*\iota^{n}(\rho_{i_{3}+n})\dots*\iota^{n+1}(\rho_{i_{k}+n})$
$\displaystyle\simeq\iota^{n}(\rho_{i_{1}+n}*\overline{\rho}_{i_{2}+n}*\rho_{i_{3}+n}\dots*\overline{\rho}_{i_{k}+n})$

Hence $c$ is mapped to the curve $\iota^{n}(c)$ in the last $m$ disks
$D^{\mathbin{\text{{\\#}}}m}$ inside
$D^{\mathbin{\text{{\\#}}}n+m}=D^{\mathbin{\text{{\\#}}}m+n}$, from which the
statement follows.

We are left to prove (i). To see that the images $\beta_{m,n}$ of block braids
under $\beta$ do not define a braiding in $\mathbf{D}$, using (ii) it is
enough to find a curve $c$ in $D^{\mathbin{\text{{\\#}}}m}$ for some $m$ so
that $\iota(c)\not\simeq c$, and such curves are plentiful. The same argument
shows that the inverses $\beta_{m,n}^{-1}$ likewise do not define a braiding.

Now suppose that $\widetilde{\beta}$ is a braiding on $\mathbf{D}$. The
braiding is determined by
$\widetilde{\beta}_{1,1}\in{\operatorname{Aut}}_{{\mathbf{M}}_{2}}(D^{\mathbin{\text{{\\#}}}2})\cong\mathbb{Z}$,
a group generated by the Dehn twist $T_{1}$. We have excluded the
possibilities $\widetilde{\beta}_{1,1}=T_{1}^{\pm 1}$, and
$\widetilde{\beta}_{1,1}=\operatorname{id}$ is similarly ruled out using now
the fact that curves are not moved at all by the identity. So assume that
$\widetilde{\beta}_{1,1}=T_{1}^{k}$, with $|k|>1$. Then
$\widetilde{\beta}_{2,1}=T_{1}^{k}T_{2}^{k}$ would have to satisfy
$T_{1}^{k}T_{2}^{k}(a_{1})=a_{2}$ in order for naturality to hold, where
$T_{i}$ is the Dehn twist along the curve $a_{i}$ as in Section 3.2. Applying
Proposition 3.2 in [FM11] twice, we get that the intersection number
$i(a_{2},T_{2}^{k}(a_{1}))=i(T_{1}^{k}(T_{2}^{k}(a_{1})),T_{2}^{k}(a_{1}))=|k|i(a_{1},T_{2}^{k}(a_{1})^{2}=|k|^{2}i(a_{1},a_{2})^{4}=|k|^{2}$.
On the other hand, using Proposition 3.4 in [FM11] we obtain

$|k|^{2}=i(a_{2},T_{2}^{k}(a_{1}))=|i(T_{2}^{k}(a_{1}),a_{2})-|k|i(a_{1},a_{1})i(a_{1},a_{2})|\leq
i(a_{1},a_{2})=1,$

where we have also used that $i(a_{1},a_{1})=0$. This contradicts our
assumption of $\widetilde{\beta}_{1,1}$. $\square$

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