boundary circles. Then $\mathcal{P}(W_{g}^{b},A)$ is flag. Moreover,
$\mathcal{P}(W_{g}^{b},A)$ is $wCM$ of dimension $m$ provided that
$m\leq\frac{g-1}{2}$ and $m\leq\frac{|A|+2d}{2d-1}$.

######  Remark 9.2.

As we will see below, the genus bound in the statement above is not needed
when $Y$ is a sphere.

Yet again, we proceed along the same lines as in Section 7. Once more, the
topological analysis becomes a bit more involved: in two dimensions, an arc
can be separating, and cutting a surface open along an arc affects the
fundamental group.

### The ball complex

In the surface case, we will not analyze the connectivity properties of the
ball complex $\operatorname{\mathbb{B}}(W_{g}^{b},A)$ by means of its
projection to the $d$-hypergraph complex. Instead we appeal to work by
Skipper–Wu [62, Theorem 3.12], who have calculated the connectivity of the
$d$-holed disk complex of a punctured surface. Capping each boundary component
with a once-punctured disk yields an isomorphism between the ball complex (of
the surface with boundary) and the $d$-holed disk complex (of the punctured
surface). Thus, we conclude:

######  Lemma 9.3 ([62]).

$\operatorname{\mathbb{B}}(W_{g}^{b},A)$ is $m$-connected for
$m\leq\frac{|A|+1}{2d-1}-2$.

Since links in the ball complex are again ball complexes, we can also deduce
the weak Cohen–Macaulay property.

######  Corollary 9.4.

$\operatorname{\mathbb{B}}(W_{g}^{b},A)$ is $wCM$ of dimension
$\lfloor\frac{|A|+1}{2d-1}-1\rfloor$.

###### Proof of Theorem 9.1 for $Y\cong{\mathbb{S}}^{2}$.

If $Y$ is a $2$-sphere, the geometric piece complex is isomorphic to the ball
complex, which is a flag complex. Thus, Corollary 9.4 establishes this case of
Theorem 9.1. ∎

From now on, we assume that $Y\cong{\mathbb{S}}^{1}\times{\mathbb{S}}^{1}$.

### The handle complex

As in the previous sections, the handle complex $\mathcal{H}(W_{g}^{b})$ has
as vertices isotopy classes of handles (i.e. embedded copies of
${\mathbb{S}}^{1}\times{\mathbb{S}}^{1}\smallsetminus\mathbb{B}^{2}$) and the
adjacency relation is given by disjointness. To analyze the handle complex in
the surface case, we will make use of the so-called _chain complex_ introduced
by Hatcher–Vogtmann [42].

A _chain_ is an unordered pair of simple closed curves intersecting
transversely at a single point; in particular, both curves are non-separating.
The chain complex $Ch({W_{g}^{b}})$ has simplices corresponding to isotopy
classes of systems of pairwise disjoint chains in $W_{g}^{b}$; observe that
every such system is coconnected, i.e. its complement has exactly one
connected component. The following was proved by Hatcher–Vogtmann [42,
Proposition 5.6]:

######  Theorem 9.5 ([42]).

The chain complex $Ch({W_{g}^{b}})$ is
$\lfloor\frac{g-3}{2}\rfloor$-connected.

Observe that every chain has a tubular neighborhood diffeomorphic to
${\mathbb{S}}^{1}\times{\mathbb{S}}^{1}\smallsetminus\mathbb{B}^{2}$. As a
consequence, the link of a $k$-simplex $\sigma$ in $Ch({W_{g}^{b}})$ is
isomorphic to $Ch({W_{g-k-1}^{b+k+1}})$, and so we deduce the following:

######  Corollary 9.6.

The chain complex $Ch({W_{g}^{b}})$ is $wCM$ of dimension
$\lfloor\frac{g-1}{2}\rfloor$.

Similarly, given a $k$-simplex $\\{ch_{0},\cdots,ch_{k}\\}$ in
$Ch({W_{g}^{b}})$, we get a $k$-simplex $H_{0},\ldots,H_{k}$ by considering
(sufficiently small) neighborhoods $H_{i}$. In other words, there is a
simplicial map

$\alpha_{ch}:Ch({W_{g}^{b}})\to\mathcal{H}(W_{g}^{b}).$

We observe:

######  Lemma 9.7.

The map $\alpha_{ch}$ is a complete join. Therefore, $\mathcal{H}(W_{g}^{b})$
is $wCM$ of dimension $\lfloor\frac{g-1}{2}\rfloor$.

###### Proof.

First, any two disjoint non-isotopic chains have non-isotopic neighborhoods,
and so $\alpha_{ch}$ injective on individual simplices. Next, given any
simplex $\sigma=\\{H_{0},\ldots,H_{k}\\}$ in $\mathcal{H}(W_{g}^{b})$, we can
choose a chain $ch_{i}$ in each $H_{i}$. Since these chains are pairwise
disjoint, they form a simplex $\\{ch_{0},\ldots,ch_{k}\\}$ in
$Ch({W_{g}^{b}})$ that maps to $\sigma$ under $\alpha_{ch}$, and thus
$\alpha_{ch}$ is surjective. Moreover, any element in
$\alpha_{ch}^{-1}(H_{i})$ can be realized as a chain in $H_{i}$. Hence
$\alpha_{ch}^{-1}(\sigma)=\operatorname{\Asterisk}_{i=1}^{k}\alpha_{ch}^{-1}(H_{i})$,
and thus $\alpha_{ch}$ is a complete join. The connectivity of
$\mathcal{H}(W_{g}^{b})$ now follows from Remark A.15 and Theorem 9.5. ∎

### The handle-tether-ball complex

The complex $\operatorname{HTB}(W_{g}^{b},A)$ is defined as in the previous
sections. The same argument as in the proof of Lemma 7.8 yields:

######  Lemma 9.8.

The map
$\pi:\operatorname{HTB}(W_{g}^{b},A)\rightarrow\mathcal{P}(W_{g}^{b},A)$ is a
complete join. In particular, if $\operatorname{HTB}(W_{g}^{b},A)$ is
$m$-connected, so is $\mathcal{P}(W_{g}^{b},A)$.

Also, the proof of Lemma 8.9 also applies in two dimensions:

######  Lemma 9.9.

Let $\sigma$ be a $k$-simplex in $\operatorname{HTB}(W_{g}^{b},A)$. The link
of $\sigma$ in $\operatorname{HTB}(W_{g}^{b},A)$ is isomorphic to a complex
$\operatorname{HTB}(W_{g-k-1}^{b-(k+1)(d-1)},A^{*})$, where $A^{*}$ is
obtained from $A$ by removing those $(k+1)d$ boundary circles used in
$\sigma$.

### The extended handle-tether-ball complex

As was the case in previous sections, allowing multiple handles to be tethered
to the same ball leads to the complex $\operatorname{HTB_{e}}(W_{g}^{b},A)$,
which contains $\operatorname{HTB}(W_{g}^{b},A)$ as a subcomplex. Both
complexes have the same vertex set, although
$\operatorname{HTB_{e}}(W_{g}^{b},A)$ may contain additional simplices. The
goal of this section is to prove the $m$-connectivity of
$\operatorname{HTB_{e}}(W_{g}^{b},A)$ if $m$ is bounded as an (explicit)
linear function of $g$ and $|A|$. As in the higher dimensional cases we will
use the projection

$\pi_{b}:\operatorname{HTB_{e}}(W_{g}^{b},A)\rightarrow\operatorname{\mathbb{B}}(W_{g}^{b},A).$

Recall that we know the connectivity of the total space
$\operatorname{\mathbb{B}}(W_{g}^{b},A)$ (see Corollary 9.4). Thus it remains
to determine the connectivity of the fibers $\pi_{b}^{-1}(\sigma)$ of
simplices $\sigma$; for this we use tethered handle complexes
$\operatorname{TH}(W_{g}^{b},Q)$.

Even if the outline of the proof is similar, the technical differences between
the surface case and the cases of higher-dimensional manifolds are most
pronounced in this step. Since there is no room to isotope arbitrary two arcs
off each other, we cannot prove an analogue of Lemma 8.10. Instead, we will
consider an inductive argument, which requires to cut $W_{g}^{b}$ along
tethered handles. This procedure may cut $W_{g}^{b}$ into components, as well
as cut boundary loops of $W_{g}^{b}$ into intervals. Therefore we start with a
more general description of tethered handle complexes: we will drop the
subscripts and use $W$ to denote a connected surface and $Q$ to denote a
collection of pairwise disjoint circles and open intervals (with pairwise
disjoint closures) contained in the boundary of $W$. A $k$-simplex of
$\operatorname{TH}(W,Q)$ is given by $k+1$ isotopy classes of tethered-handles
in $W$ with disjoint representatives.

######  Remark 9.10.

We emphasize that even if we allow the surface $W$ to be a subsurface of
$W_{g}^{b}$ and the collection $Q$ to contain intervals, we are still
interested in the following situation: let $\sigma=\\{B_{0},\ldots,B_{k}\\}$
be a $k$-simplex in $\operatorname{\mathbb{B}}(W_{g}^{b},A)$. We excise the
balls $B_{i}$ from $W_{g}^{b}$ to obtain a new surface $W=W_{g}^{c}$, (where
$c=b-(k+1)(d-1)$, although the exact value is of no importance) and let $Q$ be
the set of those boundary circles in $W_{g}^{c}$ that come from the essential
boundary spheres $\partial^{0}B_{i}$.

Figure 6. Tethered handles whose tethers necessarily cross

Another departure from the higher-dimensional case is that the projection
$\pi_{h}:\operatorname{TH}(W,Q)\rightarrow\mathcal{H}(W)$ is not a join
complex, in stark contrast to Lemma 8.11; see Figure 6. Instead, we use a bad
simplex argument to understand the connectivity properties of
$\operatorname{TH}(W,Q)$ as the subcomplex of $\operatorname{TH_{e}}(W,Q)$,
which is a complex with the same vertex set as $\operatorname{TH}(W,Q)$, but
whose simplices are tethered-handle systems where we allow multiple tethers at
each handle and multiple tethers at each boundary loop. In other words: we
only insist that tethers are pairwise disjoint; however, their heads and tails
(handles or boundary loops, respectively) are either pairwise disjoint or
equal.

######  Lemma 9.11.

The complex $\operatorname{TH_{e}}(W,Q)$ is
$\lfloor\frac{g(W)-3}{2}\rfloor$-connected. In particular,
$\operatorname{TH_{e}}(W_{g}^{b},Q)$ is
$\lfloor\frac{g-3}{2}\rfloor$-connected.

###### Proof.

We consider the projection
$\pi_{h}:\operatorname{TH_{e}}(W,Q)\to\mathcal{H}(W)$ forgetting tethers and
balls. Let $\sigma=\\{H_{0},\ldots,H_{k}\\}$ be a $k$-simplex in
$\mathcal{H}(W)$. We claim that the preimage $\pi_{h}^{-1}(\sigma^{\circ})$
above the barycenter $\sigma^{\circ}$ of $\sigma$ is contractible. To see
this, we identify $\pi_{h}^{-1}(\sigma^{\circ})$ with the complex
$A^{\circ}_{k}(S,P,P,Q)$ from Proposition A.11. Namely, we obtain the surface
$S$ from $W$ by first collapsing the handles $T_{i}$ to punctures in $P$, and
then collapsing the boundary circles and intervals in $Q$ to punctures. Now,
the complex $\mathcal{A}^{\circ}(S,P,P,Q)$ consists of those collections of
tethers that connect $P$ to $Q$ and make use of all punctures in $P$. It is
this last condition that identifies $\mathcal{A}^{\circ}(S,P,P,Q)$ as the
barycentric fiber $\pi_{h}^{-1}(\sigma^{\circ})$.

Now, Proposition A.11 tells us that the barycentric fibers are all
contractible, and Lemma A.19 implies that $\operatorname{TH_{e}}(W,Q)$
inherits the connectivity properties of the base $\mathcal{H}(W)$. The claim
then follows from Lemma 9.7. ∎

######  Lemma 9.12.

$\operatorname{TH}(W,Q)$ is
$\left\lfloor\frac{g(W)-3}{2}\right\rfloor$-connected. In particular,
$\operatorname{TH}(W_{g}^{b},Q)$ is
$\left\lfloor\frac{g-3}{2}\right\rfloor$-connected.

###### Proof.

We induct on $g$, using a bad simplex argument on the inclusion
$\operatorname{TH}(W,Q)\subseteq\operatorname{TH_{e}}(W,Q)$. If $g\geq 1$ and
$|Q|\geq 1$ the complex $\operatorname{TH}(W,Q)$ is non-empty. By induction,
we assume that the result is true for surfaces with $g^{\prime}<g$ and
$b^{\prime}\geq 1$. The bad simplex argument follows Example A.8. We use the
projection to the handle complex as our coloring map; i.e., we call a simplex
$\sigma$ in $\operatorname{TH_{e}}(W,Q)$ bad if each handle used in $\sigma$
hosts at least two tethers. In particular, vertices cannot be bad. A simplex
$\sigma$ in $\operatorname{TH_{e}}(W,Q)$ is called good if at most one tether
attaches to each handle. The good link $G_{\sigma}$ of a bad simplex $\sigma$
consists of those systems of tethered handles whose handles are pairwise
distinct and not used in $\sigma$.

We can give an explicit description of the good links. Let $\sigma$ be a bad
simplex and $W_{1},\ldots,W_{l}$ be the connected components of
$W\smallsetminus\sigma$, obtained from $W$ by cutting out a regular
neighborhood of $\sigma$. Further, $(W\smallsetminus\sigma)\cap Q$ splits into
a non-empty collection of circles and intervals $Q_{i}=W_{i}\cap Q$, see
Figure 7.

Figure 7. Removing the two tethered handles on the left yields the surface on
the right. Note how a boundary circle from $Q$ splits into two intervals.

The good link $G_{\sigma}$ is isomorphic to the join
$\operatorname{\Asterisk}\operatorname{TH}(W_{i},Q_{i})$. If $m\leq k$ is the
number of handles in $\sigma$, we have $g=m+\sum g(W_{i})\leq k+\sum
g(W_{i})$. In particular, the components $W_{i}$ have genus strictly less than
$g$ so we can apply the induction hypothesis to deduce that $G_{\sigma}$ is
$\left(\sum_{i=1}^{l}\left\lfloor\frac{g(W_{i})-3}{2}\right\rfloor+2l-2\right)$-connected.
Recall that vertices cannot be bad, and hence $k\geq 1$. A direct calculation
now yields:

$\displaystyle\sum_{i=1}^{l}\left\lfloor\frac{g(W_{i})-3}{2}\right\rfloor+2l-2$
$\displaystyle\geq\left\lfloor\sum_{i=1}^{l}\frac{g(W_{i})-3}{2}-\frac{l}{2}\right\rfloor+2l-2$
$\displaystyle=\left\lfloor\frac{\sum_{i=1}^{l}g(W_{i})-3}{2}-\frac{3}{2}(l-1)-\frac{l}{2}+2l-2\right\rfloor$
$\displaystyle\geq\left\lfloor\frac{g-k-3}{2}-\frac{1}{2}\right\rfloor$
$\displaystyle=\left\lfloor\frac{g-3}{2}-\frac{k+1}{2}\right\rfloor(\text{since
}k\geq 1)$ $\displaystyle\geq\left\lfloor\frac{g-3}{2}-k\right\rfloor$

Hence $G_{\sigma}$ is at least
$\left\lfloor\frac{g-3}{2}-k\right\rfloor$-connected, at which point Theorem
A.7 implies that $\operatorname{TH}(W,Q)$ is
$\left\lfloor\frac{g-3}{2}\right\rfloor$-connected. ∎

######  Lemma 9.13.

Assume that $m\leq\frac{g-3}{2}$ and $m\leq\frac{|A|+1}{2d-1}-2$. Then the
extended handle-tether-ball complex $\operatorname{HTB_{e}}(W_{g}^{b},A)$ is
$m$-connected.

###### Proof.

We consider the projection
$\pi_{b}:\operatorname{HTB_{e}}(W_{g}^{b},A)\to\operatorname{\mathbb{B}}(W_{g}^{b},A)$
forgetting handles and tethers. As previously, we are interested in the
connectivity of fibers above closed simplices. So let
$\tau=\\{B_{0},\ldots,B_{k}\\}$ be a $k$-simplex in
$\operatorname{\mathbb{B}}(W_{g}^{b},A)$. The final difference with the
situation in the previous two sections is that the fiber $\pi_{b}^{-1}(\tau)$
is not isomorphic to the tethered-handle complex
$\operatorname{TH}(W_{g}^{c},\\{\partial^{0}B_{0},\ldots,\partial^{0}B_{k}\\})$.
In other words, the statement of Lemma 8.10 fails in the surface case. The
reason is that a tether connecting a handle to, say, $B_{0}$ may run through
$B_{1}$; and while we were able to push the tether off the $d$-holed ball in
three dimensions, we may not be able to push it of the $d$-holed disk $B_{1}$
in two dimensions. The holes are obstacles to moving arcs.

Nonetheless, we can describe the fiber $\pi_{b}^{-1}(\tau)$ in terms of
tethered-handle complexes. For an inclusion chain
$\rho\subseteq\sigma\subseteq\tau$, denote by $S_{\sigma}$ the surface
obtained from $W_{g}^{b}$ by excising the balls $B$ in $\sigma$, and by
$Q_{\rho}$ the collection of $\partial^{0}B$ for $B\in\rho$. Note that
$Q_{\rho}$ can be interpreted in as a collection of boundary circles in
$S_{\sigma}$. Also, observe that all $S_{\sigma}$ has genus $g$. The tethered-
handle complex $\operatorname{TH}(S_{\sigma},Q_{\rho})$ consists of those
tethered handles that are tethered to a balls in $\rho$ and can be pushed off
all the balls $\sigma$.

With this notation, the fiber $\pi_{b}^{-1}(\tau)$ can be described as the
following union:

(9.14)
$\pi_{b}^{-1}(\tau)=\bigcup_{\varnothing\neq\sigma\subseteq\tau}\operatorname{TH}(S_{\sigma},Q_{\sigma}).$

By Lemma 9.12, all terms in the union are $m$-connected complexes. We want to
apply a nerve-cover argument (see Proposition A.17). We therefore have to
consider the intersection of the terms, as well. Let
$\sigma_{0},\ldots,\sigma_{j}$ be a collection of faces of $\tau$. We observe:

(9.15)
$\bigcap_{i=0}^{j}\operatorname{TH}(S_{\sigma_{i}},Q_{\sigma_{i}})=\operatorname{TH}(S_{\sigma_{0}\cup\cdots\cup\sigma_{j}},Q_{\sigma_{0}\cap\cdots\cap\sigma_{j}}).$

This just states that a tethered handle belongs to the intersection if and
only if it is tethered to a ball that lies in all the $\sigma_{i}$ and if it
can be pushed off all the balls that lie in any of the $\sigma_{i}$.

From the description 9.15, we see that intersections are again $m$-connected
unless they are empty. The latter happens if and only if the intersection
$\sigma_{0}\cap\cdots\cap\sigma_{j}$ is empty. Therefore, the nerve of the
cover 9.14 is isomorphic to the barycentric subdivision of a standard simplex
and hence contractible. From Proposition A.17, we deduce that the fiber
$\pi_{b}^{-1}(\tau)$ is $m$-connected for each simplex $\tau$ in
$\operatorname{\mathbb{B}}(W_{g}^{b},A)$. Finally, by Quillen’s Fiber Theorem
A.18, the total space $\operatorname{HTB_{e}}(W_{g}^{b},A)$ is $m$-connected.
∎

### Connectivity of the handle-tether-ball complex

Again, we make use of the bad simplex argument as stated in Proposition A.7 in
the flavor presented in Example A.8. We consider the projection
$\pi_{b}:\operatorname{HTB_{e}}(W_{g}^{b},A)\to\operatorname{\mathbb{B}}(W_{g}^{b},A)$
as a coloring of the vertices in $\operatorname{HTB_{e}}(W_{g}^{b},A)$. The
good simplices form the subcomplex $\operatorname{HTB}(W_{g}^{b},A)$ and as
before, the good link of a bad $k$-simplex is of the form
$\operatorname{HTB}(W_{g-k-1}^{c},A^{*})$. As in the previous sections, we
have:

######  Lemma 9.16.

Let $\sigma$ be a bad $k$-simplex of $\operatorname{HTB_{e}}(W_{g}^{b},A)$.
The good link $G_{\sigma}$ is isomorphic to a complex
$\operatorname{HTB}(W_{g-k-1}^{c},A^{*})$, for some (explicit, albeit
unimportant) $c\leq b$, and where $A^{*}$ is obtained from $A$ by removing
those up to $kd$ boundary spheres used in $\pi_{b}(\tau)$.

We use this to deduce the connectivity of the handle-tether-ball complex:

######  Proposition 9.17.

Assume $m\leq\frac{|A|+1}{2d-1}-2$ and $m\leq\frac{g-4}{2}$. Then the handle-
tether-ball complex $\operatorname{HTB}(W_{g}^{b},A)$ is $m$-connected.

###### Proof.

The induction will apply once we show that the good link $G_{\sigma}$ of a bad
$k$-simplex is $(m-k)$-connected. In view of the preceding lemma, we have to
verify $m-k\leq\frac{(g-k-1)-3}{2}$ and $m-k\leq\frac{|A^{*}|+1}{2d-1}-2$. As
vertices are good, we find $k\geq 1$. For the first inequality, we deduce from
$m\leq\frac{g-3}{2}$ and $2k\geq k+1$ that
$m-k\leq\frac{g-2k-3}{2}\leq\frac{(g-k-1)-3}{2}$ holds. For the second
estimate, we find:

$\frac{|A^{*}|+1}{2d-1}\geq\frac{|A|-kd+1}{2d-1}\geq\frac{|A|+1}{2d-1}-\frac{kd}{2d-1}\geq
m-k.$

Thus, the induction hypothesis applies and the claim follows. ∎

Using Lemma 9.9, we also obtain the following:

######  Corollary 9.18.

Assume $m\leq\frac{|A|+1}{2d-1}-1$ and $m\leq\frac{g-1}{2}$. Then the handle-
tether-ball complex $\operatorname{HTB}(W_{g}^{b},A)$ is $wCM$ of dimension
$m$. Consequently, $\mathcal{P}(W_{g}^{b},A)$ is also $wCM$ of dimension $m$.

### The flag property

Finally, we prove that the piece complex is flag.

######  Proposition 9.19.

$\mathcal{P}(W_{g}^{b},A)$ is a flag complex.

###### Proof.

Suppose $Z_{0},\ldots,Z_{k}$ are vertices of $\mathcal{P}(W_{g}^{b},A)$ that
define the 1-skeleton of a $k$-simplex in $\mathcal{P}(W_{g}^{b},A)$, and
$S_{i}$ the unique boundary component of $Z_{i}$ that is essential in
$W_{g}^{b}$. At this point, there is an isotopy $\phi_{t}:W_{g}^{b}\to
W_{g}^{b}$ such that the curves $S_{i}^{\prime}=\phi_{1}(S_{i})$ are all
simultaneously disjoint; this follows immediately from the fact that
$W_{g}^{b}$ supports a hyperbolic metric. But then the subsurfaces
$Z_{i}^{\prime}=\phi_{1}(Z_{i})$ are simultaneously disjoint also, and hence
define a $k$-simplex in $\mathcal{P}(W_{g}^{b},A)$, as desired. ∎

###### Proof of Theorem 9.1 for $Y={\mathbb{S}}^{1}\times{\mathbb{S}}^{1}$..

We have just seen that $\mathcal{P}(W_{g}^{b},A)$ is a flag complex. The claim
about being $wCM$ follows from Corollary 8.16. ∎

##  10\. Homology of asymptotic mapping class groups

We study the homology of asymptotic mapping class groups
$\mathcal{B}_{d,r}(O,Y)$ following the ideas in the proof of [26, Theorem
3.1]. We will focus solely on the case $d=2$ and $r=1$; for this reason we
will simply write $\mathcal{C}(O,Y)$ and $\mathcal{B}(O,Y)$. Our proof uses
crucially the fact that $V=V_{2,1}$ is simple [45] and acyclic [67]; we remark
that neither fact is true for all Higman–Thompson groups. Recall from
Definition 4.2 that

$\operatorname{Map}_{c}(\mathcal{C}(O,Y))=\bigcup_{k=1}^{\infty}\operatorname{Map}(O_{k}),$

where $O_{k}$ are the compact $n$-manifolds used in the construction of
$\mathcal{C}(O,Y)$. Recall that $O_{k+1}$ is constructed from $O_{k}$ by
gluing $2^{k}$ copies of $Y^{2}$ along the suited boundaries of $O_{k}$, where
$Y^{d}$ denotes the result of removing $d+1$ open balls from $Y$. It will be
interesting for our discussion to further divide this gluing map as follows:

  1. Step 1.

gluing $Y^{1}$ to a suited boundary of $O_{k}$;

  2. Step 2.

gluing $S^{3}$ to the remaining boundary of $Y^{1}$, where $S^{3}$ denotes the
manifold obtained from the standard sphere ${\mathbb{S}}^{n}$ by deleting
three disjoint open balls.

In the presence of the inclusion property, the inclusion map
$O_{k}\hookrightarrow O_{k+1}$ induces an injective homomorphism
$j:\operatorname{Map}(O_{k})\to\operatorname{Map}(O_{k+1})$. In turn, the
injective homomorphism $j$ induces an injection
$j_{\ast}:H_{i}(\operatorname{Map}(O_{k}))\to
H_{i}(\operatorname{Map}(O_{k+1}))$, which is well-studied in the literature
about homological stability of mapping class groups; in short, this asserts
that the map $j_{\ast}$ is in fact an isomorphism for certain values of $i$
(known as the stable range) in terms of $k$. The following result summarizes
the results on homological stability of mapping class groups that are relevant
to our purposes:

######  Theorem 10.1.

  1. (i)

Let $O$ be a compact surface, and
$Y\cong{\mathbb{S}}^{1}\times{\mathbb{S}}^{1}$. Then $j_{\ast}$ is an
isomorphism when $i\leq\lfloor\frac{2^{k+1}-4}{3}\rfloor$.

  2. (ii)

Let $O$ be any compact $3$-manifold, and
$Y\cong{\mathbb{S}}^{2}\times{\mathbb{S}}^{1}$. Then $j_{\ast}$ is an
isomorphism when $i\leq\lfloor\frac{2^{k}-3}{2}\rfloor$.

###### Proof.

Part (i) follows from [36]; here, note that the genus of $O_{k}$ is at least
$2^{k}-1$. Part (ii) follows from results in [44, Theorem 6.1] and [44,
Corollary 6.2]. We remark that the map in [44, Theorem 6.1] is defined in
terms boundary connected sum, although it applies just the same to the gluing
construction detailed above. Again, it is important the number of
${\mathbb{S}}^{2}\times{\mathbb{S}}^{1}$ summands is at least $2^{k}-1$ in the
calculation of the stable range. ∎

Since the compactly supported mapping class group is the direct limit of the
mapping class groups of the $O_{k}$, we immediate obtain:

######  Corollary 10.2.

Let $O$ and $Y$ be as in Theorem 10.1. Then
$H_{i}(\operatorname{Map}_{c}(\mathcal{C}(O,Y)))$ is the $i$-th stable
homology group of $\operatorname{Map}(W_{g})$.

Note also the following:

######  Theorem 10.3.

$H_{i}(\operatorname{Map}_{c}(\mathcal{C}(O,Y)),\mathbb{Z})$ is finitely
generated for all $i$, if

  1. (i)

$O$ is a compact surface and $Y$ is a torus.

  2. (ii)

$O\cong{\mathbb{S}}^{3}$, and $Y\cong{\mathbb{S}}^{2}\times{\mathbb{S}}^{1}$.

###### Proof.

By Theorem 10.1, it suffices to show that $H_{i}(O_{k},\mathbb{Z})$ is
finitely generated for all $i$. At this point, (i) follows from Lemma 2.4, and
(ii) from Corollary 2.7. ∎

######  Remark 10.4.

$H_{i}(\operatorname{Map}_{c}(\mathcal{C}(O,Y)),\mathbb{Z})$ need not be
necessarily finitely generated, for example when
$O\cong\mathbb{S}^{2n},Y\cong{\mathbb{S}}^{n}\times{\mathbb{S}}^{n}$ ($n\geq
3$). In fact by Lemma B.9 in Appendix B, the number of $\mathbb{Z}/2$ summands
generated by Dehn twists around boundaries in $\operatorname{Map}(O_{k})$ will
grow exponentially. A similar phenomenon happens in dimension $2$ when one
takes $O$ to be a disk and $Y\cong\mathbb{S}^{2}$.

We finally prove our desired result about the relation between stable homology
and the homology of asymptotic mapping class groups, using the same argument
as in [26, Proposition 4.3]:

######  Proposition 10.5.

Suppose $H_{i}(\operatorname{Map}_{c}(\mathcal{C}(O,Y)),\mathbb{Z})$ is
finitely generated for all $i$. Then $H_{i}(\mathcal{B}(O,Y))\cong
H_{i}(\operatorname{Map}_{c}(\mathcal{C}(O,Y),\mathbb{Z})$ for all $i$.

###### Proof.

After Proposition 4.3, we have

$1\to\operatorname{Map}_{c}(\mathcal{C}(O,Y)),\mathbb{Z})\rightarrow\mathcal{B}(O,Y)\rightarrow
V\to 1.$

Now apply the Lyndon-Hochschild-Serre spectral sequence, getting

$E^{2}_{p,q}=H_{p}(V,H_{q}(\operatorname{Map}_{c}(\mathcal{C}(O,Y),\mathbb{Z}))\to
H_{i}(\mathcal{B}(O,Y),\mathbb{Z})$

For convenience, write
$A_{q}=H_{q}(\operatorname{Map}_{c}(\mathcal{C}(O,Y)),\mathbb{Z})$. By
assumption, we have $A_{q}$ is a finitely generated abelian group for all $q$.
In particular $\mathrm{Aut(A_{q})}$ is residually finite [6]. Since $V$ is
simple, we have any homomorphism $V\to\mathrm{Aut}(A_{q})$ must be trivial.
Thus the action of $V$ on $A_{q}$ is trivial. Furthermore, since V is acyclic
[67], we have $H_{p}(V,A_{q})\cong 0$ if $p>0$ and $H_{0}(V,A_{q})\cong
A_{q}$. Therefore the spectral sequence collapses, and we have
$H_{i}(\mathcal{B}(O,Y),\mathbb{Z})\cong
H_{i}(\operatorname{Map}_{c}(\mathcal{C}(O,Y),\mathbb{Z})$ for all $i$, as
claimed. ∎

######  Remark 10.6.

One might try to use the same argument to calculate the homology of the
braided Thompson’s group $V_{br}$ using the short exact sequence $1\to
PB_{\infty}\to V_{br}\to V\to 1$. However, $H_{1}(PB_{\infty},\mathbb{Z})$ is
already not finitely generated which one can prove using winding numbers.

After all this, we arrive at the proof of Theorem 1.15:

###### Proof of Theorem 1.15.

The result follows at once from the combination of Corollary 10.2 and
Proposition 10.5. ∎

Finally, we record the following, which may be of independent interest:

######  Corollary 10.7.

$\mathcal{B}({\mathbb{S}}^{3},{\mathbb{S}}^{2}\times{\mathbb{S}}^{1})$ is
rationally acyclic.

###### Proof.

We only need to show that $\operatorname{Map}_{c}(\mathcal{C}(O,Y))$ is
rationally acyclic. In turn, by Theorem 10.1, it suffices to prove that the
stable homology of $\operatorname{Map}(O_{k})$ is rationally acyclic. Using
[43, Theorem 1.1] and the Lyndon-Hochschild-Serre spectral sequence, it is
enough to show the groups $A_{n,0}^{s}$ of [43] are stably rationally acyclic;
this is proved in [64, Theorem A.2] based on the fact that the automorphism
group of free group is rationally acyclic [30]. ∎

##  Appendix A Connectivity tools

The purpose of this section is to give some details on some connectivity tools
that are essential for calculating the connectivity of our spaces. A good
reference is [42, Section 2], although not all the tools we use can be found
there.

###  A.1. Combinatorial Morse theory

As will have become apparent by now, we make extensive use of combinatorial
Morse theory. For simplicial complexes, that is mostly a modern name for
techniques to establish connectivity that have been employed for a long time.
Let $X$ be a simplicial complex with a subcomplex $L$. Consider $X$ with $L$
coned off, i.e., consider the space $Y:=X\cup_{L}C(L)$ obtained by gluing the
cone $C(L)$ to the space $X$ along the common subspace $L$. Using the long
exact sequence of homotopy groups for the pair $(Y,X)$, one deduces the
following:

######  Observation A.1.

If $L$ is $m$-connected, then the inclusion $X\hookrightarrow Y=X\cup_{L}C(L)$
induces isomorphisms in homotopy groups $\pi_{d}$ for $d\leq m$ and an
epimorphism in $\pi_{m+1}$.

One may also cone off several subcomplexes $L_{i}$ at the same time:

######  Observation A.2.

Let $X$ be a simplicial complex, $\\{L_{i}\\}_{i\in I}$ a family of
$m$-connected subcomplexes, and $Y$ the space obtained by gluing each cone
$C(L_{i})$ to $X$ along $L_{i}$. Then the inclusion $X\hookrightarrow Y$
induces isomorphisms in homotopy groups $\pi_{d}$ for $d\leq m$ and an
epimorphism in $\pi_{m+1}$.

Now, consider a map $h:X^{(0)}\to T$ defined on the vertex set of a simplicial
complex $X$ with values in a totally ordered set $(T,<)$. We say that $h$ is a
_combinatorial Morse function_ , or a _height function_ , if adjacent vertices
are mapped to different values (the slogan is: $h$ is non-constant on edges).
For any $t\in T$, denote by $X^{<t}$ the subcomplex spanned by vertices $v$
with $h(v)<t$ and denote by $X^{\leq t}$ the subcomplex of those $v$ with
$h(v)\leq t$. The _descending link_ $\operatorname{lk}^{\downarrow}(v)$ of the
vertex $v$ is the subcomplex of the link $\operatorname{lk}(v)$ spanned by
those neighbors $w$ of $v$ _below_ $v$, i.e., satisfying $h(w)<h(v)$. From $h$
being non-constant on edges and Observation A.2, we deduce:

######  Corollary A.3.

$X_{\leq t}$ is obtained from $X_{<t}$ by coning off the descending links
$\operatorname{lk}^{\downarrow}(v)$ of all vertices of height $h(v)=t$. If all
these descending links are $m$-connected, the inclusion $X_{<t}\hookrightarrow
X_{\leq t}$ induces isomorphisms in homotopy groups $\pi_{d}$ for $d\leq m$
and an epimorphism in $\pi_{m+1}$.

We call the height function $h$ _discrete_ if for any pair $s,t\in T$ the
_interval of values_ $\\{\,h(v)\,\mid\,v\in X\text{\ and\ }s\leq h(v)\leq
t\,\\}$ is finite. In that case, we can compare the homotopy type of a
sublevel set to the whole complex.

######  Proposition A.4 (Morse Lemma for simplicial complexes).

If $h$ is a discrete height function on $X$ and all descending links above
level $s\in T$ are $m$-connected, then the inclusion of $X_{\leq
s}\hookrightarrow X$ induces isomorphisms in homotopy groups $\pi_{d}$ for
$d\leq m$ and an epimorphism in $\pi_{m+1}$.

###### Proof.

For any inclusion $X_{\leq s}\hookrightarrow X_{\leq t}$, the claim follows
from Corollary A.3 as the interval of values between $s$ and $t$ is finite.
The whole complex $X$ is the direct limit of the $X_{t}$, and the homotopy
group functor commutes with taking direct limits. ∎

Bestvina–Brady [7] applied this line of reasoning to piecewise Euclidean
complexes in such a way that its parallels with Morse theory become apparent.
Let $K$ be a piecewise Euclidean cell complex, and denote its vertex set by
$K^{(0)}$. A function $h:K\to\mathbb{R}$ is called a _discrete Morse function_
or a _height function_ provided that the image $h(K)$ is a discrete subset of
$\mathbb{R}$ and that $h$ is affine on closed cells and non-constant on edges.
Since all cells carry a Euclidean structure, we can think of links
geometrically as spaces of directions. If a vertex $v$ occurs as a face of a
cell $c$, the cell contributes a spherical piece to the link of $v$ consisting
precisely of those directions based at $v$ pointing into $c$. The gluing maps
for the complex $K$ induce gluings for those spherical pieces (each of which
is a spherical polyhedron with faces coming from cells between $v$ and $c$).
The _descending link_ of $v$ consists of the contributions of those proper
cofaces $c$ such that $h$ attains its maximum on $c$ at $v$. The _ascending
link_ of $v$ comes from those proper cofaces such that $h$ attains its minimum
on $c$ at $v$.

######  Proposition A.5 (Morse Lemma for piecewise Euclidean complexes).

Let $K$ be a piecewise Euclidean complex, and $h:K\to\mathbb{R}$ a discrete
Morse function. Assume that the descending links of all vertices $v$ with
$h(v)\in(s,t]$ are $m$-connected. Then the inclusion $K_{\leq
s}\hookrightarrow K_{\leq t}$ induces isomorphisms in homotopy groups
$\pi_{d}$ for $d\leq m$ and an epimorphism in $\pi_{m+1}$.

###  A.2. Morse theory on the barycentric subdivision

We will use the _bad simplex argument_ of [42, Section 2.1]. Here, we shall
provide a treatment in terms of combinatorial Morse theory. Let $X$ be a
simplicial complex. By $X^{\circ}$ we denote its barycentric subdivision, and
by $\sigma^{\circ}$ we denote the barycenter of the simplex $\sigma$. Note
that $\sigma^{\circ}$ is a _vertex_ in $X^{\circ}$. Simplices in $X^{\circ}$
correspond to chains of simplices in $X$. In particular if $\sigma^{\circ}$
and $\tau^{\circ}$ are adjacent in $X^{\circ}$, the dimensions of $\sigma$ and
$\tau$ differ. Thus, on the barycentric subdivision $X^{\circ}$, we can use
dimension as a secondary function to break ties. Instead of making this
generic, we shall illustrate the method by means of example.

Assume that for each simplex $\sigma$, we are given a subset
$\bar{\sigma}\subseteq\sigma$ of vertices such that the following two
conditions hold:

  1. (i)

For $\sigma\subseteq\tau$, we have $\bar{\sigma}\subseteq\bar{\tau}$.
(Monotonicity)

  2. (ii)

For any simplex, $\bar{\bar{\sigma}}=\bar{\sigma}$. (Idempotence)

The simplex $\sigma$ is _good_ if $\bar{\sigma}$ is empty and _bad_ if
$\bar{\sigma}=\sigma$. The monotonicity condition implies that faces of good
simplices are good. In particular, the good simplices form a subcomplex
$X^{\text{good}}$ of $X$. The _good link_ $G_{\sigma}$ of a simplex $\sigma$
is induced by its proper cofaces $\tau$ with $\bar{\sigma}=\bar{\tau}$. It
follows also from monotonicity that the good link is a subcomplex of the link,
i.e., it is closed with respect to taking faces.

In [42, Section 2.1], Hatcher–Vogtmann phrase the bad simplex argument in
terms of the collection of bad simplices. The equivalence to our approach is
seen as follows:

######  Remark A.6.

If two bad simplices $\sigma$ and $\tau$ span a simplex, then the span
$\sigma\cup\tau$ will also be bad by monotonicity. Conversely, consider a
collection $\mathcal{B}$ of simplices with the property that if
$\sigma\cup\tau$ is a simplex, $\sigma,\tau\in\mathcal{B}$ implies
$\sigma\cup\tau\in\mathcal{B}$. Then defining $\bar{\sigma}$ as the union of
all faces of $\sigma$ that belong to $\mathcal{B}$ satisfies monotonicity and
idempotence. With this definition, $\mathcal{B}$ coincides with the class of
bad simplices, i.e., $\sigma\in\mathcal{B}$ if and only if
$\sigma=\bar{\sigma}$.

######  Proposition A.7 (Bad simplex argument).

Suppose there is $m$ such that for all bad simplices $\sigma$, the good link
$G_{\sigma}$ is $(m-\dim(\sigma))$-connected. Then the inclusion
$X^{\text{good}}\hookrightarrow X$ induces an isomorphism in homotopy groups
$\pi_{d}$ for $d\leq m$ and an epimorphism in $\pi_{m+1}$.

###### Proof.

We want to use the cardinality $\\#\bar{\sigma}$ as a Morse function, noting
that it is integer-valued on the vertices $\sigma^{\circ}$ of the barycentric
subdivision $X^{\circ}$. However, we may have $\bar{\sigma}=\bar{\tau}$ for a
simplex $\tau$ and a proper face $\sigma$ of $\tau$. Thus, we use the
dimension to break ties. More precisely, on the barycentric subdivision
$X^{\circ}$, we consider the following height function:

$\displaystyle h:X^{\circ}$ $\displaystyle\to\mathbb{Z}\times\mathbb{Z}$
$\displaystyle\sigma^{\circ}$
$\displaystyle\mapsto\langle\\#\bar{\sigma},-\dim(\sigma)\rangle$

We use the lexicographic order on $\mathbb{Z}\times\mathbb{Z}$. Then, the
barycentric subdivision of the subcomplex $X^{\text{good}}$ of good simplices
is the $h$-sublevel complex $X^{\circ}_{<(1,0)}$.

If $X$ has infinite dimension, $h$ may not be discrete. However, by passing to
the $(m+1)$-skeleton, we may assume w.l.o.g. that $X$ is of finite dimension.

In the barycentric subdivision, the link of a vertex $\sigma^{\circ}$ is best
described by recalling that $\sigma$ is a simplex in $X$. Then, adjacent
vertices are faces and cofaces of $\sigma$; correspondingly, the link of
$\sigma^{\circ}$ in $X$ is the join
$\operatorname{lk}_{-}(\sigma)\operatorname{\ast}\operatorname{lk}_{+}(\sigma)$
of the _down-link_ $\operatorname{lk}_{-}(\sigma)$, spanned by the barycenters
of proper faces of $\sigma$; and the _up-link_
$\operatorname{lk}_{+}(\sigma)$, spanned by the barycenters of proper cofaces
of $\sigma$. The down-link is the barycentric subdivision of the boundary
sphere $\partial\sigma$ whereas the up-link is the geometric realization of
the poset of proper cofaces of $\sigma$, i.e., it can be thought of as
barycentric subdivision of the ordinary link of the simplex $\sigma$ in the
complex $X$. The join decomposition
$\operatorname{lk}(\sigma^{\circ})=\operatorname{lk}_{-}(\sigma)\operatorname{\ast}\operatorname{lk}_{+}(\sigma)$
is inherited by the descending link
$\operatorname{lk}^{\downarrow}(\sigma^{\circ})=\operatorname{lk}_{-}^{\downarrow}(\sigma)\operatorname{\ast}\operatorname{lk}_{+}^{\downarrow}(\sigma)$.

Let $\rho$ be a proper face of $\sigma$. The vertex $\rho^{\circ}$ lies in the
down-link of $\sigma^{\circ}$ and it is descending if and only if $\bar{\rho}$
is strictly smaller than $\bar{\sigma}$ since the second coordinate of $h$
increases. Note that $\bar{\rho}\subseteq\bar{\sigma}$ follows from
monotonicity. On the other hand, $\bar{\sigma}\subseteq\rho$ implies
$\bar{\sigma}=\bar{\bar{\sigma}}\subseteq\bar{\rho}$ whence
$\bar{\sigma}=\bar{\rho}$; and conversely $\bar{\sigma}=\bar{\rho}$ implies
$\bar{\sigma}\subseteq\rho$. Thus,
$\rho^{\circ}\not\in\operatorname{lk}_{-}^{\downarrow}(\sigma)$ if and only if
$\bar{\sigma}\subseteq\rho$. This translates as:

$\operatorname{lk}_{-}^{\downarrow}(\sigma)=\\{\,\rho^{\circ}\in\operatorname{lk}_{-}(\sigma)\,\mid\,\bar{\sigma}\not\subseteq\rho\,\\}.$

If $\sigma$ is good, $\operatorname{lk}_{-}^{\downarrow}(\sigma)=\varnothing$.
If $\sigma$ is bad, $\operatorname{lk}_{-}^{\downarrow}(\sigma)$ is the entire
boundary sphere $\partial\sigma$. If $\sigma$ is neither good nor bad,
$\operatorname{lk}_{-}^{\downarrow}(\sigma)$ contractible as it is the
boundary sphere $\partial\sigma$ with the star of $\bar{\sigma}$ removed.

The descending up-link $\operatorname{lk}_{+}^{\downarrow}(\sigma)$ is spanned
by those $\tau^{\circ}\in\operatorname{lk}_{+}(\sigma)$, for which the first
coordinate of $h$ does not increase (it cannot decrease by monotonicity). Thus
$\operatorname{lk}_{+}^{\downarrow}(\sigma)$ is the barycentric subdivision of
the good link $G_{\sigma}$. Hence $\operatorname{lk}_{+}^{\downarrow}(\sigma)$
is $(m-\dim(\sigma))$-connected by hypothesis, provided that $\sigma$ is bad.

Hence, for any vertex $\sigma^{\circ}$ outside the sublevel set, the
descending link
$\operatorname{lk}^{\downarrow}(\sigma^{\circ})=\operatorname{lk}_{-}^{\downarrow}(\sigma)\operatorname{\ast}\operatorname{lk}_{+}^{\downarrow}(\sigma)$
is at least $m$-connected: if $\sigma$ is bad, the descending down-link is the
$(\dim(\sigma)-2)$-connected sphere $\partial\sigma$ and the descending up-
link in $(m-\dim(\sigma))$-connected; if $\sigma$ is not bad, the descending
link is even contractible and the descending up-link does not matter. The
claim now follows from the Morse Lemma. ∎

In the main body, we make use of the bad simplex argument in two particular
incarnations.

######  Example A.8.

Assume that the vertices of $X$ are colored. We consider a simplex $\sigma$
bad if no vertex has a unique color among the vertices of $\sigma$. Then
$\bar{\sigma}$ is the set of vertices in $\sigma$ whose color features in
$\sigma$ at least twice. The conditions of monotonicity and idempotence are
clearly satisfied. The subcomplex $X^{\text{good}}$ then consists precisely of
those simplices wherein each vertex has a unique color. The good link
$G_{\sigma}$ of $\sigma$ is induced by those cofaces $\tau$ such that
$\tau\smallsetminus\sigma$ is a good simplex and uses no color that is already
used in $\sigma$.

For concreteness, let $\pi:X\to Y$ be a simplicial map and consider the
vertices of $Y$ as the colors. Then $X^{\text{good}}$ consists precisely of
those simplices $\sigma$ such that the restriction $\pi|_{\sigma}$ is
injective. The good link $G_{\sigma}$ of a bad simplex $\sigma$ consists of
those simplices $\tau\in\operatorname{lk}(\sigma)$ such that $\pi|_{\tau}$ is
injective and such that the images $\pi(\tau)$ and $\pi(\sigma)$ are disjoint.

######  Example A.9.

Partition the vertices of $X$ into good vertices and bad vertices. For a
simplex $\sigma$ let $\bar{\sigma}$ denote the set of bad vertices in
$\sigma$. Clearly, this definition satisfies monotonicity and idempotence.
Equivalently, a simplex is bad if all its vertices are bad. Then
$X^{\text{good}}$ is the full subcomplex spanned by the good vertices. The
good link of a simplex is also easily seen to be
$G_{\sigma}=\operatorname{lk}(\sigma)\cap X^{\text{good}}$.

###  A.3. A variation of the arc complex

Let $S$ be a surface of negative Euler characteristic with two disjoint sets
of marked points $P=\\{p_{0},\ldots,p_{m}\\}$ and
$Q=\\{q_{0},\ldots,q_{n}\\}$. We regard interior marked points as punctures.
An arc is a simple curve with one endpoint in $P$ the other endpoint in $Q$
and otherwise disjoint from $P\cup Q$. We consider arcs up to isotopy relative
to $P\cup Q\cup\partial S$. We endow $S$ with a complete hyperbolic metric, so
that interior marked points form the set of cusps. Then each arc can be
represented by a unique geodesic. Let $\mathcal{A}(S,P,Q)$ be a simplicial
complex whose $k$-simplices are collections
$\\{\alpha_{0},\ldots,\alpha_{k}\\}$ of arcs that are pairwise non-isotopic
and disjoint except possibly at the endpoints (thinking of arcs as geodesics
connecting cusps, this amounts to the geodesics being pairwise disjoint). One
might call $\mathcal{A}(S,P,Q)$ the _bipartite arc complex_ to distinguish it
from the arc complex of a surface.

######  Proposition A.10.

The complex $\mathcal{A}(S,P,Q)$ is contractible.

This is a variation on [36, Theorem 1.6]. The difference is that our punctures
$P$ are not located on the boundary. Instead of recycling Harer’s proof, we
give a proof using more modern technology, namely the surgery flow method
introduced by Hatcher [40].

###### Proof.

Hatcher–Vogtmann [42, Lemma 2.9] have turned Hatcher’s idea of surgery flow
into a fairly generic method. In establishing the hypotheses, we shall already
employ their notation.

Fix an arc $w$ from $P$ to $Q$. For any other arc $v$, we define the
_complexity_ $c(v)$ as the geometric intersection number of $v$ and $w$. The
geometric intersection number will be realized if the representatives $v$ and
$w$ are chosen to be geodesic with respect to a hyperbolic metric. For a
simplex $\sigma$ in the arc complex, we define $c(\sigma)$ as the sum of the
complexity over all vertices of $\sigma$. Note that the star $Y$ of $w$ in
$X:=\mathcal{A}(S,P,Q)$ consists precisely of the zero-complexity simplices.

For an arc $v$ that intersects $w$, we consider the point of intersection
closest along $w$ to the $Q$-end of $w$. We surger $v$ along the final segment
of $w$ and obtain two new arcs: one, which we call $\Delta v$ from the $P$-end
of $v$ to the $Q$-end of $w$; and another one, which we discard, connecting
$Q$ to $Q$. Note that $\Delta v$ lies in the link of $v$ and satisfies
$c(\Delta v)<c(v)$.

For a simplex $\sigma$ not in $Y$, there is at least one vertex (arc) that
intersects $w$. Let $v_{\sigma}$ be that intersecting arc whose point of
intersection with $w$ is closest to the $Q$-end. Note that $\sigma$ and
$\Delta v_{\sigma}$ span a simplex. Also note that $v_{\sigma}=v_{\tau}$ for
any face $\tau$ of $\sigma$ that contains $v_{\sigma}$.

In this situation [42, Lemma 2.9] applies and we conclude that the
(contractible) star $Y$ of $w$ is a deformation retract of
$X=\mathcal{A}(S,P,Q)$. Hence $\mathcal{A}(S,P,Q)$ is contractible. ∎

Now, we fix a subset $P^{\prime}\subseteq P$ and consider the subcomplex
$\mathcal{A}^{\circ}(S,P,P^{\prime},Q)$ of the barycentric subdivision
$\mathcal{A}^{\circ}(S,P,Q)$ spanned by those vertices $\sigma^{\circ}$ where
the $P$-endpoints of the arcs in $\sigma$ form a superset of $P^{\prime}$. We
make use of the following result in the analysis of the extended tethered-
handle complex $\operatorname{TH_{e}}(W,Q)$ in the surface setting.

######  Proposition A.11.

The complex $\mathcal{A}^{\circ}(S,P,P^{\prime},Q)$ is contractible.

###### Proof.

For a simplex $\sigma$ of $\mathcal{A}(S,P,Q)$, we denote by $P(\sigma)$ the
set of $P$-points issuing arcs in $\sigma$. For non-empty $A\subseteq
B\subseteq P$, let $X(A,B)$ denote the subcomplex of
$\mathcal{A}^{\circ}(S,P,Q)$ spanned by $\\{\ \sigma^{\circ}\,\mid\,A\subseteq
P(\sigma)\subseteq B\,\\}$. By $Y(A,B)$ we denote the complex spanned by $\\{\
\sigma^{\circ}\,\mid\,P(\sigma)\subseteq B\text{\ and\ }A\cap
P(\sigma)\neq\varnothing\,\\}$. Note that $X(A,B)$ is a subcomplex of
$Y(A,B)$.

For a subset $A\subseteq P$, we denote by $\sigma\mid_{A}$ the _restriction_
of $\sigma$ to $A$, i.e., the collection of those arcs in $\sigma$ issuing
from $A$. For $A\subseteq B\subseteq C$, restriction to $B$ defines simplicial
retractions $\mid_{B}:X(A,C)\to X(A,B)$ and $\mid_{B}:Y(B,C)\to
Y(B,B)\text{.}$ By [61, 1.3 Homotopy Property], these restriction maps are
deformation retractions, i.e., they are homotopic to the identity map on their
respective domains of definition. Note that $Y(B,B)$ can be identified with
the barycentric subdivision of an arc complex $\mathcal{A}(S^{\prime},B,Q)$
where $S^{\prime}$ is obtained by puncturing $S$ at $P\smallsetminus B$. In
particular, the complexes $Y(B,C)$ are all contractible.

We show by induction on the cardinality of $A$ that the complexes $X(A,B)$ are
contractible. If $A$ contains just a single element, we find $X(A,B)=Y(A,B)$
and $Y(A,B)$ is contractible.

To complete the induction step, we need to consider a puncture $b\in B$ with
$b\not\in A$ and put $A^{\prime}:=A\cup\\{b\\}$. We want to understand the
inclusion $X(A^{\prime},B)\hookrightarrow X(A,B)$ by means of a bad simplex
argument as in Example A.9. In particular, we call a vertex $\sigma^{\circ}$
bad if $b\not\in P(\sigma)$. Then $X(A^{\prime},B)$ is the good subcomplex of
$X(A,B)$. Let $\\{\tau_{0}^{\circ},\ldots,\tau_{l}^{\circ}\\}$ be a bad
$l$-simplex with $\tau_{0}\subset\tau_{1}\subset\cdots\subset\tau_{l}$. Note
that for a good vertex $\sigma^{\circ}$, we never have
$\sigma\subseteq\tau_{l}$ because $b$ is used in $\sigma$. Therefore, the good
link $G$ of $\\{\tau_{0}^{\circ},\ldots,\tau_{l}^{\circ}\\}$ consists of good
simplices $\\{\sigma_{0}^{\circ},\ldots,\sigma_{k}^{\circ}\\}$ with
$\tau_{l}\subset\sigma_{i}$. Restriction to $\\{b\\}$ defines a deformation
retraction of $G$ onto the subcomplex spanned by those $\sigma^{\circ}$ where
$\sigma$ contains $\tau_{l}$ and $P(\sigma\smallsetminus\tau_{l})=\\{b\\}$.
This complex can be identified with the contractible complex
$\mathcal{A}^{\circ}(S^{\prime},\\{b\\},Q)$ where $S^{\prime}$ is obtained
from $S$ by cutting along $\tau_{l}$.

The claim now follows as
$\mathcal{A}^{\circ}(S,P,P^{\prime},Q)=X(P^{\prime},P)$. ∎

###  A.4. Join complexes

The concept of a join complex, introduced by Hatcher–Wahl in [44, Section 3]
will be another useful tool for us in order to analyze connectivity
properties. We review the basics here, referring the reader to [44] for more
details.

######  Definition A.12 (Join complex).

A join complex over a simplicial complex $L$ is a simplicial complex $K$
together with a simplicial map $\pi:K\to L$ satisfying the following
properties:

  1. (1)

$\pi$ is surjective.

  2. (2)

$\pi$ is injective on individual simplices.

  3. (3)

For each $p$-simplex $\sigma=\langle x_{0},\cdots,x_{p}\rangle$ of $L$, the
subcomplex $K(\sigma)$ of $K$ consisting of all the $p$-simplices that project
to $\sigma$ is the join
$K_{x_{0}}(\sigma)\operatorname{\ast}\cdots\operatorname{\ast}K_{x_{p}}(\sigma)$
of the vertex sets of $K_{x_{i}}(\sigma):=K(\sigma)\cap\pi^{-1}(x_{i})$.

Note that $K(\sigma)$ need not be equal to $\pi^{-1}(\sigma)$. If all the
inclusions $K_{x_{i}}(\sigma)\subseteq\pi^{-1}(x_{i})$ are in fact equalities,
then we will call $K$ a complete join complex over $L$. A reformulation of (3)
which sometimes is helpful is the following condition; see [44] for a proof of
their equivalence:

  1. (3′)

A collection of vertices $(y_{0},\cdots,y_{p})$ of $K$ spans a $p$-simplex if
and only if for each $y_{i}$ there exists a $p$-simplex $\sigma_{i}$ of $K$
such that $y_{i}\in\sigma_{i}$ and
$\pi(\sigma_{i})=\langle\pi(y_{0}),\cdots,\pi(y_{p})\rangle$.

As mentioned above, it is possible to use the concept of join in order to
analyze the connectivity of a given complex, if this belongs to the following
class of complexes, introduced by Hatcher–Wahl [44]:

######  Definition A.13.

A simplicial complex is weakly Cohen–Macaulay ($wCM$ for short) of dimension
$n$ if it is $(n-1)$-connected and the link of each $p$-simplex is
$(n-p-2)$-connected.

Armed with this definition, one has the following result in the case of a join
complex:

######  Proposition A.14.

[44, Proposition 3.5] If $K$ is a complete join complex over a $wCM$ complex
$L$ of dimension $n$, then $K$ is also $wCM$ of dimension $n$.

######  Remark A.15.

If $\pi:K\to L$ is a complete join, then $L$ is a retract of $K$. In fact, we
can define a simplicial map $s:L\to K$ with $\pi\circ
s=\operatorname{id}_{L}$, by sending a vertex $v\in L$ to any vertex in
$\pi^{-1}(v)$ and then extending it over to all simplices, which can be done
since $\pi$ is a complete join. In particular, if $K$ is $n$-connected, so is
$L$.

However, we will encounter situations when we will only have a join complex
instead of a complete one. In these cases we will make use of the following
theorem, also due to Hatcher–Wahl [44]:

######  Theorem A.16.

[44, Theorem 3.6] Let $K$ be a join complex over a $wCM$ complex $L$ of
dimension $n$. Suppose that, for each $p$-simplex $\sigma\subset K$, one has
that $\pi(\operatorname{lk}_{K}(\sigma))$ is $wCM$ of dimension $n-p-2$. Then
$K$ is $(\frac{n}{2}-1)$-connected.

###  A.5. Nerve covers

The nerve of a family of sets $(K_{i})_{i\in I}$ is the simplicial complex
$\mathcal{N}(K_{i})$ defined on the vertex set $I$ which contains a simplex
$J\subseteq I$ if $\bigcap_{j\in J}K_{j}$ is non-empty. The following result
allows to identify the homotopy groups of a certain simplicial complex with
those of a nerve complex, up to a fixed dimension.

######  Proposition A.17.

[8, Theorem 6] Let $K$ be a connected simplicial complex and $(K_{i})_{i\in
I}$ a family of subcomplexes that covers $K$. Suppose that every non-empty
finite intersection $K_{i_{1}}\cap K_{i_{2}}\cap\ldots\cap K_{i_{l}}$ is
$(k-l+1)$-connected for $l\geq 1$. Then there is a map
$f:\|X\|\rightarrow\|\mathcal{N}(K_{i})\|$ which induces isomorphisms of
homotopy groups
$f_{j}^{\ast}:\pi_{j}(K)\rightarrow\pi_{j}(\mathcal{N}(K_{i}))$ for all $j\leq
k$.

###  A.6. Fiber connectivity

Given a simplicial map $f:X\to Y$, there are several results relating the
connectivity of $X$ and $Y$ provided that we have information about the
connectivity of the fibers. A useful instance is due to Quillen [61,
Proposition 7.6], who stated his result in the more general terms of posets
and their geometric realizations. Here, we state the proposition specialized
for simplicial maps.

######  Proposition A.18 (Quillen’s Fiber Theorem for simplicial complexes).

Let $f:X\to Y$ be a simplicial map and assume that the preimage
$f^{-1}(\sigma)$ of each closed simplex $\sigma$ in $Y$ is an $n$-connected
subcomplex of $X$. Then $X$ is $n$-connected if and only if $Y$ is
$n$-connected.

One can also make use of fibers above open simplices. Technically, it is
convenient to phrase the statement in terms of the barycentric subdivision.

######  Lemma A.19.

[42, Lemma 2.8] Let $f:L\rightarrow K$ be a simplicial map of simplicial
complexes. Suppose that $K$ is $n$-connected and the fibers
$f^{-1}(\sigma^{\circ})$ over the barycenters $\sigma^{\circ}$ of all
$k$-simplices in $K$ are $(n-k)$-connected. Then $L$ is $n$-connected.

##  Appendix B The inclusion and intersection properties

In this section we prove the inclusion and intersection properties for suited
submanifolds of the Cantor manifolds that appear in the statements of Theorems
1.6, 1.7, 1.9 and 1.10. None of the material here is new, and the arguments
will be well-known to the appropriate community of experts.

###  B.1. Dimension 2

The two-dimensional case is by far the easiest, so we deal with it first.
First, the inclusion property appears, for example, in [20, Section 3.6]. In
turn, the intersection property is a direct consequence of the Alexander
method, see [20, Section 2.3].

We now move onto the cases of Cantor manifolds of dimension bigger than two.
We commence by briefly recalling the definition and properties of a Dehn twist
along a sphere; see [53, 10]. Let $M$ be a connected, orientable smooth
manifold of dimension $n+1\geq 3$, $S$ a smoothly embedded $n$-sphere in $M$
and $U\cong S\times[0,1]$ a tubular neighborhood of $S$. Let $\gamma$ be a
generator of $\pi_{1}(SO(n+1))\cong\mathbb{Z}/2$. The Dehn twist
$T_{S}\in\operatorname{Map}(M)$ is the isotopy class of the diffeomorphism
which is the identity on $M\smallsetminus U$ and is given by
$(x,t)\to(\gamma(t)x,t)$ on $U$. The mapping class $T_{S}$ depends only on the
isotopy class of $S$; moreover, $T_{S}$ has order at most 2.

######  Lemma B.1.

Let $n\geq 2$ and $M=N\\#(\mathbb{S}^{1}\times\mathbb{S}^{n})$ be a smooth
closed stably parallelizable manifold. Let $T_{S}$ be the Dehn twist along the
$n$-sphere $S=0\times{\mathbb{S}}^{n}\subseteq
N\\#({\mathbb{S}}^{1}\times{\mathbb{S}}^{n})$. Then $T_{S}$ is a nontrivial
element of order $2$ in $\operatorname{Map}(M)$.

###### Proof.

When $n=2$, this is due to Laudenbach [56]. The proof we give here follows
closely to [10, Sections 4 & 5]. As discussed above, $T_{S}$ has order at most
2, so it suffices to show $T_{S}$ is nontrivial.

Any diffeomorphism $f\in\operatorname{Diff}(M)$ induces a bundle map $Df:TM\to
TM$ by taking its differential. Fix a parallelization of the stable tangent
bundle $TM\oplus\varepsilon^{k}$, where $\varepsilon$ is the trivial line
bundle. The differential $Df$ then induces a map $\bar{f}:M\to SO(n+k+1)$. If
$f$ is isotopic to the identity map, we would have that $\bar{f}$ is homotopic
to the constant map. In particular, $\bar{f}\mid_{S^{1}\times 0}$ is homotopic
to a constant path in $SO(n+k+1)$, contrary to the construction of the twist
$T_{S}$. ∎

Now following the same proof of [55, Lemma 2.2], we have the following.

######  Corollary B.2.

Let $N$ be a manifold with at least two sphere boundaries $S_{1}$ and $S_{2}$,
and suppose that the manifold $M$ obtained by gluing together $S_{1}$ and
$S_{2}$ is stably parallelizable. Then $T_{S_{i}}$ is nontrivial in
$\operatorname{Map}(N)$.

###### Proof.

First, note that $M$ is the connected sum of some manifold with
$\mathbb{S}^{1}\times\mathbb{S}^{n}$. Now, there is an induced map
$\iota:\operatorname{Map}(N)\to\operatorname{Map}(M)$. By Lemma B.1,
$\iota(T_{S_{i}})$ is non-trivial and hence the same holds for $T_{S_{i}}$. ∎

###  B.2. Dimension 3

The inclusion property for 3-manifolds appears as [44, Proposition 2.3], and
so we focus on the intersection property. We start with the following well-
known result, which gives information about the subgroup of
$\operatorname{Map}(M)$ generated by Dehn twists about boundary spheres.
Recall that a manifold has spherical boundary if every boundary component is
diffeomorphic to a sphere. We have:

######  Theorem B.3.

Let $M$ be a closed orientable 3-manifold and
$M_{b}=M~{}\\#_{b}~{}\mathbb{B}^{3}$. Denote the boundary spheres by
$S_{1},\ldots,S_{b}$, and write $T_{S_{i}}$ for the corresponding sphere
twist. Let $\mathbb{T}_{i}<\operatorname{Map}(M_{b})$ be the subgroup
generated by the twists $T_{S_{i}}$ for $1\leq i\leq b$. Then:

  1. (i)

If $\mathbb{T}_{1}=0$, then for any $b>1$ one has
$\mathbb{T}_{b}=(\mathbb{Z}/2)^{b-1}$, where the only relation is given by
$T_{S_{1}}\cdots T_{S_{b}}=1$.

  2. (ii)

If $\mathbb{T}_{1}=\mathbb{Z}/2$, then for any $b>1$ one has
$\mathbb{T}_{b}=(\mathbb{Z}/2)^{b}$.

###### Proof.

First of all, by Corollary B.2, sphere twists have order $2$ when $b\geq 2$.
On the other hand, when $b=1$ the sphere twist could either be trivial or of
order $2$, see [44, Remark 2.4] for more details.

If $\mathbb{T}_{1}=0$, we would have $T_{S_{1}}\cdots T_{S_{b}}=1$; see the
proof of [55, Lemma 2.1] or Lemma B.9 below. The proof that there is no other
relation follows from the same proof of Lemma B.9, using the inclusion
property in dimension $3$ and Corollary B.2.

Finally, the same argument works when $\mathbb{T}_{1}=\mathbb{Z}/2$, with the
only difference that in this case there is no product relation. ∎

We will now use the ideas in [10, Section 4,5] in order to deduce that a given
mapping class is a product of Dehn twists about boundary spheres. We need some
preliminaries first.

Again, let $M$ be a compact connected orientable 3-manifold with spherical
boundary. A diffeomorphism $f\in\operatorname{Diff}(M,\partial M)$ induces a
bundle map $Df:TM\to TM$. Since every 3-manifold is parallelizable, $Df$
further induces a map $M\to\operatorname{SO}_{3}(\mathbb{R})$. Now, if $C$ is
a closed curve on $M$, or an arc connecting two boundary components, then
$\mathcal{D}_{f}(C)\in\pi_{1}(\mathrm{SO}_{3}(\mathbb{R}))\cong\mathbb{Z}/2$.
Moreover, the value of $\mathcal{D}_{f}(C)$ only depends on the isotopy class
(relative to endpoints, if any) of $C$. The following may be deduced from the
description of $\operatorname{Map}(M)$ in terms of automorphisms of
$\pi_{1}(M)$; see [59, Theorem 1.5] or [44, Proposition 2.1]:

######  Theorem B.4.

Let $M$ be a $3$-manifold with spherical boundary and at least two boundary
components. Let $f\in\operatorname{Diff}(M,\partial M)$ such that:

  1. (i)

For every curve $C$, we have that $\mathcal{D}_{f}(C)=0$ and $f$ fixes the
homotopy class of $C$;

  2. (ii)

For every arc $A$ connecting two distinct boundary spheres, $f$ fixes the
homotopy class (relative to endpoints) of $A$.

Then $f$ is the Dehn twist along a separating sphere in $M$.

Now let $N$ be a connected orientable $3$-manifold with spherical boundary,
and assume that $M$ is obtained from $N$ by attaching $P_{1},P_{2}$ along two
of its sphere boundaries, where $P_{i}=P\\#_{d+1}\mathbb{B}^{3}$, with $P$ a
fixed closed prime $3$-manifold or $\mathbb{S}^{3}$ and $d\geq 2$. Let
$M_{i}=N\cup P_{i}$. By the inclusion property (see e.g. [44, Proposition
2.3]), we regard $\operatorname{Map}(N),\operatorname{Map}(M_{1})$ and
$\operatorname{Map}(M_{2})$ as subgroups of $\operatorname{Map}(M)$. We
finally prove:

######  Theorem B.5.

$\operatorname{Map}(M_{1})\cap\operatorname{Map}(M_{2})=\operatorname{Map}(N)$.

###### Proof.

The “$\supseteq$” part is obvious, so it suffices to show the “$\subseteq$”
part. For $i=1,2$, let $S_{i}$ be the boundary sphere of $N$ to which $P_{i}$
is glued, and write $P_{i}^{\prime}=P_{i}\smallsetminus S_{i}$. An element
$f\in\operatorname{Map}(M_{1})\cap\operatorname{Map}(M_{2})$ fixes the isotopy
class of $S_{i}$, and hence the homotopy class of $S_{1}\cup S_{2}$. By
Laudenbach’s theorem [57, Theorem III.1.3], homotopic system of spheres are
isotopic. Thus, up to isotopy, we can assume that
$f\mid_{S_{i}}=\operatorname{id}$ and, as a consequence, $f(P_{i})=P_{i}$.
Hence $f$ decomposes as $f_{1}f_{N}f_{2}$ where
$f_{i}\in\operatorname{Map}(P_{i})$ and $f_{N}\in\operatorname{Map}(N)$. Since
the three maps commute with each other, we have
$ff_{N}^{-1}=f_{1}f_{2}\in\operatorname{Map}(M_{1})\cap\operatorname{Map}(M_{2})$.
Thus it suffices to show that $f_{1}f_{2}\in\operatorname{Map}(N)$.

Note that if two arcs or curves in $P_{i}^{\prime}$ are isotopic in $M$, they
are already isotopic in $P_{i}^{\prime}$. Since
$f\in\operatorname{Map}(M_{i})$, we have that $\mathcal{D}_{f}$ values
trivially at every curve and arc in $P^{\prime}_{i}$. Moreover, as $f$ induces
the identity map on $\pi_{1}(P_{i})$, for $i=1,2$, it follows that $f_{i}$ is
the product of Dehn twists around all boundary spheres of $P_{i}$.
Furthermore, since the Dehn twist around $S_{i}$ can be pushed to $N$, we may
assume $f_{i}$ is the product of Dehn twists along all boundaries of
$P^{\prime}_{i}$, denoted $\mathbb{T}_{i}$.

At this point it suffices to show that following: if
$\mathbb{T}_{1}\mathbb{T}_{2}\in\operatorname{Map}(M_{1})\cap\operatorname{Map}(M_{2})$,
then $\mathbb{T}_{1}\mathbb{T}_{2}\in\operatorname{Map}(N)$. Observe that if
$\mathbb{T}_{1}\mathbb{T}_{2}\in\operatorname{Map}(M_{i})$, we would also have
$\mathbb{T}_{i}\in\operatorname{Map}(M_{3-i})$ since
$\mathbb{T}_{3-i}\in\operatorname{Map}(M_{3-i})$. Thus the theorem finally
boils down to the following claim.

Claim: Fix $i=1,2$. If $\mathbb{T}_{i}\in\operatorname{Map}(M_{3-i})$, then
$\mathbb{T}_{i}=1$ or $\mathbb{T}_{1}\mathbb{T}_{2}\in\operatorname{Map}(N)$.

We prove the claim for $i=2$. Suppose that $\mathbb{T}_{2}\neq 1$. As an
element in $\operatorname{Diff}(M_{1},\partial M_{1})$, $\mathbb{T}_{2}$ fixes
the homotopy class of every curve on $M_{1}$ and
$\mathcal{D}_{\mathbb{T}_{1}\mathbb{T}_{2}}$ values trivially at every curve.
Moreover $\mathbb{T}_{2}$ fixes the homotopy class of every arc in $M_{1}$
connecting boundaries, because $\pi_{1}(N)=\pi_{1}(M_{1})\ast\pi_{1}(M_{2})$.
Thus by Theorem B.4, $\mathbb{T}_{2}$ is a Dehn twist about a separating
sphere in $M_{1}$. We can decompose this sphere further as the connected sum
of two separating spheres $S_{P_{1}}$ and $S_{N}$ with $S_{P_{1}}\subset
P_{1}$ and $S_{N}\subset N$. If $S_{P_{1}}$ is parallel to $S_{1}$, then we
are done. If not, then it is the connected sum of some boundary spheres in
$P_{1}^{\prime}$. Note that $T_{S_{P_{1}}}\cdot T_{N}\cdot\mathbb{T}_{2}=1$.

Assume that $T_{S_{P_{1}}}\neq\mathbb{T}_{1}$. In particular $T_{S_{P_{1}}}$
is not the product of Dehn twists along all spheres of $P_{1}^{\prime}$. Now,
$S_{N}$ cuts off $M$ into two components $K_{1}$ and $K_{2}$, where assume
that $P_{1},P_{2}\subset K_{2}$. By choosing an appropriate separating sphere,
$K_{2}$ may be expressed as the connected sum of a $(2d+1)$-holed sphere
$H_{2d+1}$ with another 3-manifold $K_{3}$, such that the boundary components
of $H_{2d+1}$ correspond to $S_{N}$ and the $2d$ boundaries of
$P_{1}^{\prime},P_{2}^{\prime}$. Now we can decompose the inclusion map
$K_{1}\subset M$ as:

$K_{1}\subset
K_{1}\bigcup_{S_{N}}H_{2d+1}\subset(K_{1}\cup_{S_{N}}H_{2d+1})\\#K_{3}.$

Again, the inclusion property tells us that both inclusion maps induce
injective maps between the corresponding mapping class groups. But restricting
to $K_{1}\bigcup_{S_{N}}H_{2d+1}$, we have
$T_{S_{P_{1}}}\cdot\mathbb{T}_{2}\cdot T_{N}\neq 1$ by Theorem B.3. Thus
$T_{S_{P_{1}}}\cdot T_{N}\cdot\mathbb{T}_{2}\neq 1$ in
$\operatorname{Map}(M)$. Therefore we must have $T_{S_{P_{1}}}=\mathbb{T}_{1}$
which implies $\mathbb{T}_{1}\mathbb{T}_{2}=T_{N}\in\operatorname{Map}(N)$, as
desired. ∎

###  B.3. High dimensions

Finally, we settle the intersection and inclusion properties in higher-
dimensional manifolds. Let
$W_{g}^{b}=\\#_{g}~{}\mathbb{S}^{n}\times\mathbb{S}^{n}~{}\\#_{b}~{}\mathbb{B}^{2n}$,
where $n\geq 3$. When $b=0$, we will simply write $W_{g}$. Before getting
started, we need to quickly review Kreck’s calculation of
$\operatorname{Map}(W_{g})$ [53]; see also [51, Section 1.3]. Let
$H_{n}(W_{g})\cong\mathbb{Z}^{2g}$ be the integral $n$-th homology group of
$W_{g}$, and $G_{g}$ the group of automorphisms of $H_{n}(W_{g})$ that
preserve both the intersection form and Wall’s quadratic form; see [51,
Section 1.2] for an explicit description of $G_{g}$ as a (finite index
subgroup of a) symplectic or orthogonal group. Kreck [53] showed:

######  Theorem B.6.

The natural homomorphism $\mathfrak{h}:\operatorname{Map}(W_{g})\to G_{g}$ is
surjective.

The kernel of the above surjective homomorphism is called the Torelli subgroup
$\mathcal{T}_{g}$ of $\operatorname{Map}(W_{g})$, so we have a short exact
sequence

$1\to\mathcal{T}_{g}\to\operatorname{Map}(W_{g})\xrightarrow[]{\mathfrak{h}}G_{g}\to
1.$

The Torelli subgroup $\mathcal{T}_{g}$ is further described by Kreck [53]
through another short exact sequence, which we now recall. Let
$\Sigma:\pi_{n}(\operatorname{SO}(n))\to\pi_{n}(\operatorname{SO}(n+1))$ be
the map associated to the natural inclusion homomorphism
$\operatorname{SO}(n)\hookrightarrow\operatorname{SO}(n+1)$. Building up on
results of Haefliger [35], Kreck [53] proved that there is a short exact
sequence

$1\to\Theta_{2n+1}\to\mathcal{T}_{g}\to\operatorname{Hom}(\mathbb{Z}^{2g},\Sigma(\pi_{n}(SO(n))))\to
1,$

where $\Theta_{2n+1}$ is the group of homotopy $(2n+1)$-spheres up to
$h$-cobordism, which is a finite abelian group after the work of Kervaire and
Milnor [50].

Recall that the capping homomorphism
$\kappa_{b}:\operatorname{Map}(W_{g}^{b})\to\operatorname{Map}(W_{g})$ is
induced by attaching a disk to every boundary component of $W_{g}^{b}$. In the
special case when $b=1$, a result of Kreck (see [51, Lemma 1.1] for a proof)
shows that $\kappa_{1}$ is an isomorphism. Next, the existence of the capping
homomorphism implies the following:

######  Lemma B.7 (Inclusion property, I).

Let $W_{g}^{b+b^{\prime}}$ be obtained from $W_{g}^{b}$ by attaching
$W_{0}^{b^{\prime}+1}$ along one of its boundary components, where
$b,b^{\prime}\geq 1$. Then, the associated homomorphism
$\operatorname{Map}(W_{g}^{b})\to\operatorname{Map}(W_{g}^{b+b^{\prime}})$ is
injective.

After the above description, we have the following high-dimensional analog of
Theorem B.3:

######  Theorem B.8.

Let $S_{1},\cdots,S_{b}\subset W_{g}^{b}$ be embedded spheres parallel to the
$b\geq 1$ boundary components of $W_{g}^{b}$. Then the product
$T_{S_{1}}\cdots T_{S_{b}}$ is trivial in $\operatorname{Map}(W_{g}^{b})$, and
the subgroup generated by the twists along boundary spheres is isomorphic to
$(\mathbb{Z}/2)^{b-1}$.

###### Proof.

First, assume $b=1$. If $g=0$, the result is trivial. If $g\geq 1$, it holds
because of Kreck’s theorem that $\kappa_{1}$ is an isomorphism.

Hence we assume $b\geq 2$, so that Corollary B.2 gives that $T_{S_{i}}$ is a
non-trivial element of order 2 in $\operatorname{Map}(W_{g}^{b})$ since
$W_{g}^{b}$ is stably parallelizable for any $g$ and $b$. We claim that
$T_{S_{1}}\cdots T_{S_{b}}=1$. If $g=0$, then the claim follows as in [44, p.
214–215]. If $g\geq 1$, let $S$ be a sphere cutting $W_{g}^{b}$ into
$W_{0}^{b+1}$ and $W_{g}^{1}$. Since $T_{S}=T_{S_{1}}\cdots T_{S_{b}}$ in
$\operatorname{Map}(W_{0}^{b+1})$, then the same relation holds in
$\operatorname{Map}(W_{g}^{b})$. But as $T_{S}$ vanishes in
$\operatorname{Map}(W_{g}^{1})$, it vanishes in
$\operatorname{Map}(W_{g}^{b})$ too. This establishes the claim, and the also
result for $b\leq 2$.

Finally, we need to show there is no further relation when $b\geq 3$. Suppose
that some partial product $T_{S_{i_{1}}}\cdots T_{S_{i_{k}}}$ were equal to
the identity in $\operatorname{Map}(W_{g}^{b})$. Let $S^{\prime}$ be a sphere
which cuts off a disk containing exactly $S_{i_{1}},\ldots,S_{i_{k}}$, and
write $W_{g}^{b^{\prime}}$ for the other component of the complement of
$S^{\prime}$. Note that $b^{\prime}\geq 2$. Hence $T_{S^{\prime}}$ is
nontrivial in $W_{g}^{b^{\prime}}$ by Corollary B.2. On the other hand in
$\operatorname{Map}(W_{g}^{b})$, we have $T_{S^{\prime}}=T_{S_{1}}\cdots
T_{S_{k}}$ which equals $1$. But by Lemma B.7, this means $T_{S^{\prime}}=1$
in $\operatorname{Map}(W_{g}^{b^{\prime}})$, which is a contradiction. ∎

As a consequence, we obtain:

######  Corollary B.9.

There is a split short exact sequence:

$1\to(\mathbb{Z}/2)^{b-1}\to\operatorname{Map}(W_{g}^{b})\xrightarrow{\kappa_{b}}\operatorname{Map}(W_{g})\to
1$

where $(\mathbb{Z}/2)^{b-1}$ is generated by the twists along $b-1$ boundary
spheres. In particular, we have
$\operatorname{Map}(W_{g}^{b})\cong\operatorname{Map}(W_{g})\times(\mathbb{Z}/2)^{b-1}$.

###### Proof.

After Theorem B.8, we only need to show that $\kappa_{b}$ splits. To this end,
first recall that
$\operatorname{Map}(W_{g})\cong\operatorname{Map}(W_{g}^{1})$. Now, realize
$W_{g}^{1}$ inside $W_{g}^{b}$ as a connected component of the complement of a
separating sphere that cuts off the $b$ boundary spheres of $W_{g}^{b}$. In
this way, we obtain a map
$\operatorname{Map}(W_{g}^{1})\to\operatorname{Map}(W_{g}^{b})$ which provides
the desired splitting.

∎

The general case of the inclusion property now follows immediately from Lemma
B.7, Proposition B.9 and Kreck’s calculation of $\operatorname{Map}(W_{g})$:

######  Theorem B.10 (Inclusion property, II).

Let $W_{g_{2}}^{b_{2}}$ be obtained from $W_{g_{1}}^{b_{1}}$ by attaching a
copy of $W_{h}^{n}$ ($n\geq 2$) along a boundary component. Then the
associated homomorphism
$\operatorname{Map}(W_{g_{1}}^{b_{1}})\to\operatorname{Map}(W_{g_{2}}^{b_{2}})$
is injective.

Finally, we settle the intersection property for the class of manifolds
$W_{g}^{b}$. For convenience, let $N=W_{g}^{b}$ with $b\geq 2$,
$h\in\\{0,1\\}$ and $M$ the result of gluing two copies of $W_{h}^{d+1}$ along
two distinct boundary spheres of $N$. Write $M_{1}$ and $M_{2}$ for the
submanifolds of $M$ resulting from attaching to $N$ one copy of $W_{1}^{d+1}$.
Again, we use the inclusion property to regard
$\operatorname{Map}(N),\operatorname{Map}(M_{1}),\operatorname{Map}(M_{2})$ as
subgroups of $\operatorname{Map}(M)$. We have:

######  Theorem B.11.

$\operatorname{Map}(N)=\operatorname{Map}(M_{1})\cap\operatorname{Map}(M_{2})$.

###### Proof.

As in the previous subsections, it suffices to prove $\iota$ is surjective. We
have the following composition of maps:

$\operatorname{Map}(N)\xrightarrow[]{\iota}\operatorname{Map}(M_{1})\cap\operatorname{Map}(M_{2})\xrightarrow[]{\kappa}\kappa(\operatorname{Map}(M_{1}))\cap\kappa(\operatorname{Map}(M_{2}))$

We know that $\kappa$ is surjective by Proposition B.9; similarly,
$\kappa\circ\iota$ is surjective by Kreck’s description of the mapping class
group, as described above. At this point, a direct calculation based on Lemma
B.8 and Proposition B.9 shows that the kernel of $\kappa$ lies in the image of
$\iota$. Hence we are done. ∎

##  Appendix C The piece complex in high dimensions, by Oscar Randal-Williams

The purpose of this section is to study the handle complex in high dimensions,
and in particular prove Theorem 7.7 in Section 7. Following the notation
therein, $O$ is a compact, simply-connected orientable manifold of even
dimension $2n\geq 6$. Write $W_{g}$ for the connected sum of $O$ and $g$
copies of $\mathbb{S}^{n}\times\mathbb{S}^{n}$, where $n\geq 3$, and
$W_{g}^{b}$ for the result of removing $b\geq 1$ disjoint $2n$-dimensional
open balls from $W_{g}$.

Let $\mathcal{H}(W_{g}^{b})$ be the handle complex, which can equivalently be
regarded as the simplicial complex whose vertices are isotopy classes of
$(2n-1)$-dimensional separating spheres that bound a handle (i.e. a manifold
diffeomorphic to
$H:={\mathbb{S}}^{n}\times{\mathbb{S}}^{n}\smallsetminus\mathbb{B}^{2n}$). Its
$k$-simplices are then sets of $k+1$ vertices which can be represented
disjointly in $W_{g}^{b}$. We will prove:

######  Theorem C.1.

$\mathcal{H}(W_{g}^{b})$ is
$\left\lfloor\frac{g-4}{2}\right\rfloor$-connected.

Let $\mathcal{H}^{\prime}(W_{g}^{b})$ be simplicial complex whose vertices are
isotopy classes of embeddings $e:H\to W_{g}^{b}$. Its $k$-simplices are sets
of $k+1$ vertices which can be represented disjointly in $W_{g}^{b}$. There is
a map of simplicial complexes

$\psi:\mathcal{H}^{\prime}(W_{g}^{b})\longrightarrow\mathcal{H}(W_{g}^{b})$

given on vertices by sending an isotopy class of embeddings $[e:H\to
W_{g}^{b}]$ to the isotopy class of submanifolds $[e(\partial H)\subset
W_{g}^{b}]$; this clearly sends simplices to simplices.

######  Lemma C.2.

Any embedding $e:H\hookrightarrow H$ is isotopic to a diffeomorphism.

###### Proof.

First note that the map $e_{*}:H_{n}(H;\mathbb{Z})\to H_{n}(H;\mathbb{Z})$
induced by such an embedding is an isomorphism, as it respects the
intersection form and the intersection form of $H$ is non-degenerate.

Given such an embedding, by pushing inwards along a collar we can suppose that
it has image in the interior of $H$. Then the difference
$K:=H\smallsetminus\mathrm{int}(e(H))$ is a cobordism from $e(\partial H)$ to
$H$. It is easy to see using the Seifert–van Kampen theorem that $K$ is
simply-connected, and by excision we have $H_{*}(K,e(\partial
H);\mathbb{Z})\cong H_{*}(H,e(H);\mathbb{Z})$ which vanishes by the discussion
above. Thus $K$ is a simply-connected $h$-cobordism, so by the $h$-cobordism
theorem [60] there is a diffeomorphism $K\cong[0,1]\times e(\partial H)$
relative to $e(\partial H)$. Using this product structure, and a collar of
$H$, we can find an isotopy from $e$ to a diffeomorphism. ∎

######  Lemma C.3.

The map $\psi$ has a section.

###### Proof.

For each vertex $v\in\mathcal{H}(W_{g}^{b})$ we choose a representative
$v=[S\subset W_{g}^{b}]$. As $S$ bounds a manifold diffeomorphic to $H$,
choosing such a diffeomorphism gives an embedding $e:H\to W_{g}^{b}$ with
$[e(\partial H)\subset W_{g}^{b}]=[S\subset W_{g}^{b}]=v$. We make such a
choice for each vertex $v$, and attempt to define a simplicial map by
$\bar{\psi}(v):=[e:H\to W_{g}^{b}]$; if this does define a simplicial map then
it will be a section as required.

To see that $\bar{\psi}$ defines a simplicial map, it suffices to show that if
$\\{e_{i}\\}_{i=0}^{k}$ is a collection of embeddings $H\hookrightarrow
W_{g}^{b}$ such that the submanifolds $\\{e_{i}(\partial H)\\}_{i=0}^{k}$ are
distinct up to isotopy and may be isotoped to be mutually disjoint, then the
$e_{i}$ may be isotoped to be mutually disjoint. By applying isotopy extension
to the isotopies that make the $e_{i}(\partial H)$ mutually disjoint, we may
suppose that the maps $\partial e_{i}$ have mutually disjoint images.

We claim that in this case the $e_{i}$ must have mutually disjoint images. If
not then the image of one is contained in the image of the other: suppose
$e_{i}(H)\subset e_{j}(H)$, so that $e_{j}^{-1}\circ e_{i}:H\hookrightarrow H$
is an embedding. Then by Lemma C.2 this self-embedding is isotopic to a
diffeomorphism, and so $e_{i}$ is isotopic to an embedding with the same image
as $e_{j}$, and hence $e_{i}(\partial H)=e_{j}(\partial H)$. But we supposed
that the submanifolds $\\{e_{i}(\partial H)\\}_{i=0}^{k}$ are distinct up to
isotopy, giving a contradiction. ∎

######  Lemma C.4.

$\mathcal{H}(W_{g}^{b})$ and $\mathcal{H}^{\prime}(W_{g}^{b})$ are flag
complexes.

###### Proof.

We first show that $\mathcal{H}^{\prime}(W_{g}^{b})$ is a flag complex. Let
$\\{[e_{i}]\\}_{i=0}^{k}$ be a set of vertices of
$\mathcal{H}^{\prime}(W_{g}^{b})$ which pairwise span 1-simplices, i.e. each
pair $e_{i}$ and $e_{j}$ may be made disjoint up to isotopy. It suffices to
show that if $e_{0},\ldots,e_{i-1}$ are disjoint embeddings and $e_{i}$ is
disjoint up to isotopy from each of them, then it may be changed by an isotopy
to be simultaneously disjoint from all of them. This follows by an application
of the Whitney trick.

In more detail, let us call the submanifolds ${\mathbb{S}}^{n}\times\\{*\\}$
and $\\{*\\}\times{\mathbb{S}}^{n}$ inside $H$ the _cores_. As $e_{i}$ is
disjoint up to isotopy from each $e_{0},\ldots,e_{i-1}$, the algebraic
intersection number of the cores of $e_{i}(H)$ with those of each $e_{j}(H)$
are zero. As $W_{g}^{b}$ is simply-connected we may therefore use the Whitney
trick to isotope $e_{i}$ so that the cores of $e_{i}(H)$ are disjoint from
those of each $e_{j}(H)$. Now $H$ may be isotoped into an arbitrarily small
neighborhood of its cores, so we may isotope $e_{i}$ so that its image is
disjoint from the cores of each $e_{j}(H)$. Finally, using an isotopy of each
$e_{j}(H)$ to a small neighborhood of its cores, and using isotopy extension,
we may find an isotopy of $e_{i}$ to an embeddings which misses the $e_{j}(H)$
as required.

To see that $\mathcal{H}(W_{g}^{b})$ is also a flag complex, we use Lemma C.3.
Let $v_{0},\ldots,v_{k}\in\mathcal{H}(W_{g}^{b})$ be a set of distinct
vertices such that each pair spans a 1-simplex. Using the section $\bar{\psi}$
from Lemma C.3 it follows that
$\bar{\psi}(v_{0}),\ldots,\bar{\psi}(v_{k})\in\mathcal{H}^{\prime}(W_{g}^{b})$
is a set of distinct vertices in which each pair spans a 1-simplex, and as we
have seen above $\mathcal{H}^{\prime}(W_{g}^{b})$ is a flag complex so this
collection of vertices spans a $k$-simplex. Applying $\psi$ shows that
$v_{0},\ldots,v_{k}$ spans a $k$-simplex in $\mathcal{H}(W_{g}^{b})$ as
required. ∎

###### Proof of Theorem C.1.

As $\psi$ has a section, it suffices to prove that
$\mathcal{H}^{\prime}(W_{g}^{b})$ is
$\left\lfloor\frac{g-4}{2}\right\rfloor$-connected, and to show this we follow
[31, Section 5]. We first construct an algebraic avatar of the simplicial
complex $\mathcal{H}^{\prime}(W_{g}^{b})$ as follows. Let
$\mathcal{I}_{n}^{\text{fr}}(W_{g}^{b})$ denote the set of regular homotopy
classes of immersions $i:{\mathbb{S}}^{n}\times\mathbb{D}^{n}\looparrowright
W_{g}^{b}$, which by the Hirsch–Smale immersion theory is in bijection with
the set of homotopy classes of maps
${\mathbb{S}}^{n}\to\mathrm{Fr}(W_{g}^{b})$ to the frame bundle of
$W_{g}^{b}$. As $W_{g}^{b}$ is simply-connected, the frame bundle
$\mathrm{Fr}(W_{g}^{b})$ is simple, and so based and unbased homotopy classes
of maps to this space agree: thus $\mathcal{I}_{n}^{\text{fr}}(W_{g}^{b})$
agrees with [31, Definition 5.2] and in particular has the structure of an
abelian group. This abelian group is equipped with a $(-1)^{n}$-symmetric
bilinear form
$\lambda:\mathcal{I}_{n}^{\text{fr}}(W_{g}^{b})\otimes\mathcal{I}_{n}^{\text{fr}}(W_{g}^{b})\to\mathbb{Z}$
given by the algebraic intersection number, and with a function

$\mu:\mathcal{I}_{n}^{\text{fr}}(W_{g}^{b})\longrightarrow\begin{cases}\mathbb{Z}&n\text{
even}\\\ \mathbb{Z}/2&n\text{ odd}\end{cases}$

given by counting self-intersections; $\mu$ is a quadratic refinement of
$\lambda$, and this data forms a quadratic module in the sense of [31, Section
3].

Let $\mathsf{H}$ denote the hyperbolic quadratic module, i.e.
$\mathbb{Z}\\{e,f\\}$ with $\lambda$ given in this basis by
$\left(\begin{smallmatrix}0&1\\\ (-1)^{n}&0\end{smallmatrix}\right)$ and $\mu$
determined by $\mu(e)=\mu(f)=0$ and the quadratic property. For a quadratic
module $\mathsf{M}$ let $Q(\mathsf{M})$ denote the simplicial complex having
vertices the morphisms of quadratic modules $\mathsf{H}\to\mathsf{M}$ (which
are automatically injective, as $\mathsf{H}$ is non-degenerate), and where a
collection of such morphisms spans a simplex if their images are mutually
orthogonal. As orthogonality of submodules can be tested pairwise, this is a
flag complex.

There is a map of simplicial complexes

$\phi:\mathcal{H}^{\prime}(W_{g}^{b})\longrightarrow
Q(\mathcal{I}_{n}^{\text{fr}}(W_{g}^{b}),\lambda,\mu)$

given as follows. If $[s:H\hookrightarrow W_{g}^{b}]$ is an isotopy class of
embedding, then restricting this embedding to the two standard cores
$e,f:{\mathbb{S}}^{n}\times\mathbb{D}^{n}\hookrightarrow H$ determines a
hyperbolic pair $[s\circ e],[s\circ
f]\in\mathcal{I}_{n}^{\text{fr}}(W_{g}^{b})$ and hence a map
$s_{\text{alg}}:\mathsf{H}\to\mathcal{I}_{n}^{\text{fr}}(W_{g}^{b})$, and we
declare $\phi([s:H\hookrightarrow W_{g}^{b}])$ to be $s_{\text{alg}}$. If we
change $s$ by an isotopy then the regular homotopy classes $[s\circ e]$ and
$[s\circ f]$ do not change, so $\phi$ is well-defined on vertices. If a pair
of embeddings $s$ and $s^{\prime}$ have disjoint images up to isotopy then the
submodules $s_{\text{alg}}(\mathsf{H})$ and
$s^{\prime}_{\text{alg}}(\mathsf{H})$ of
$\mathcal{I}_{n}^{\text{fr}}(W_{g}^{b})$ are clearly orthogonal: as both are
flag complexes (using Lemma C.4), it follows that $\phi$ is a simplicial map.

Now $\mathcal{I}_{n}^{\text{fr}}(W_{g}^{b})$ contains $\mathsf{H}^{\oplus g}$
as a quadratic submodule, using the $g$ disjoint copies of $H$, so by [31,
Theorem 3.2] the complex
$Q(\mathcal{I}_{n}^{\text{fr}}(W_{g}^{b}),\lambda,\mu)$ is
$\left\lfloor\frac{g-4}{2}\right\rfloor$-connected, and is locally $wCM$ of
dimension $\geq\lfloor\tfrac{g-1}{2}\rfloor$. One can then repeat the argument
of [31, Lemma 5.5] on the map $\phi$. We note that there is a mild difference
in that

  1. (i)

we are considering isotopy classes of embeddings, rather than actual
embeddings, and

  2. (ii)

our embeddings do not come with a tether to the boundary;

however these two differences simplify rather than complicate the argument. ∎

## References

  * [1] J. Aramayona, J. Aroca, M. Cumplido, R. Skipper, X. Wu, Block mapping class groups and their finiteness properties. Preprint, arXiv:2207.06671
  * [2] J. Aramayona, L. Funar. Asymptotic mapping class groups of closed surfaces punctured along Cantor sets. Moscow Mathematical Journal 21 (2021).
  * [3] J. Aramayona, J. Souto, Automorphisms of the graph of free splittings. Michigan Math. J. 60 (2011), no. 3, 483–493.
  * [4] J. Aramayona, N. G. Vlamis. Big mapping class groups: an overview. In Ken’ichi Ohshika and Athanase Papadopoulos, editors,In the Tradition of Thurston: Geometry and Topology, chapter 12, pages 459–496. Springer, 2020.
  * [5] C. A. Athanasiadis. Decompositions and connectivity of matching and chessboard complexes. Discrete Comput. Geom., 31(3):395-403, 2004.
  * [6] G. Baumslag, Automorphism groups of residually finite groups. J. London Math. Soc. 38 (1963), 117-118.
  * [7] M. Bestvina, N. Brady. Morse theory and finiteness properties of groups. Invent. Math. ,129(3):445–470, 1997.
  * [8] A. Björner, Nerves, fibers and homotopy groups. J. Combin. Theory Ser. A 102 (2003), no. 1, 88–93.
  * [9] I. Bokor, D. Crowley, S. Friedl, F. Hebestreit, D. Kasprowski, M. Land, J. Nicholson. Connected sum decompositions of high-dimensional manifolds. 2019-20 MATRIX Annals,5–30, MATRIX Book Ser., 4, Springer, Cham, 2021\.
  * [10] T. Brendle, N. Broaddus, A. Putman, The mapping class group of connect sums of $S^{2}\times S^{1}$. Preprint, arXiv:2012.01529.
  * [11] M. Brin, The algebra of strand splitting. I. A braided version of Thompson’s group V. J. Group Theory 10 (2007), no. 6, 757-788.
  * [12] K. Brown, Finiteness properties of groups. Proceedings of the Northwestern conference on cohomology of groups (Evanston, Ill., 1985). J. Pure Appl. Algebra 44 (1987), no. 1–3, 45–75.
  * [13] K. Brown, The geometry of finitely presented infinite simple groups. Algorithms and classification in combinatorial group theory (Berkeley, CA, 1989), 121-136, Math. Sci. Res. Inst. Publ., 23, Springer, New York, 1992.
  * [14] K. Brown, R. Geoghegan, An infinite-dimensional torsion-free $FP_{\infty}$ group. Invent. Math. 77 (1984), no. 2, 367-381.
  * [15] R. Budney, D. Gabai, Knotted 3-balls in ${\mathbb{S}}^{4}$. Preprint, arxiv:1912.09029.
  * [16] K.- U. Bux, M. Fluch, M. Marschler, S. Witzel, M. Zaremsky, The braided Thompson’s groups are of type $F_{\infty}$. With an appendix by Zaremsky. J. Reine Angew. Math. 718 (2016), 59–101.
  * [17] J. Cerf, La stratification naturelle des espaces de fonctions différentiables réelles et le théoréme de la pseudo-isotopie. (French) Inst. Hautes Études Sci. Publ. Math. No. 39 (1970), 5–173.
  * [18] M. Culler, K. Vogtmann, Moduli of graphs and automorphisms of free groups. Invent. Math. 84 (1986), no. 1, 91–119.
  * [19] P. Dehornoy, The group of parenthesized braids. Adv. Math. 205 (2006), no. 2, 354-409.
  * [20] B. Farb, D. Margalit, A primer on mapping class groups. Princeton Mathematical Series, 49. Princeton University Press, Princeton, NJ, 2012.
  * [21] D. Farley, Finiteness and ${\rm CAT}(0)$ properties of diagram groups. Topology 42 (2003), no. 5, 1065–1082.
  * [22] D. Farley, Homological and finiteness properties of picture groups. Trans. Amer. Math. Soc. 357 (2005), no. 9, 3567–3584.
  * [23] L. Funar, Braided Houghton groups as mapping class groups. An. Ştiinţ. Univ. Al. I. Cuza Iaşi. Mat. (N.S.) 53 (2007), no. 2, 229–240.
  * [24] L. Funar, C. Kapoudjian, On a universal mapping class group of genus zero. Geom. Funct. Anal. 14 (2004), no. 5, 965–1012.
  * [25] L. Funar, C. Kapoudjian, The braided Ptolemy-Thompson group is finitely presented. Geom. Topol. 12 (2008), no. 1, 475-530.
  * [26] L. Funar, C. Kapoudjian, An infinite genus mapping class group and stable cohomology. Comm. Math. Phys. 287 (2009), no. 3, 784-804.
  * [27] L. Funar. C. Kapoudjian, V. Sergiescu. Asymptotically rigid mapping class groups and Thompson’s groups. Handbook of Teichmüller theory. Volume III, 595–664, IRMA Lect. Math. Theor. Phys., 17, Eur. Math. Soc., Zürich, 2012.
  * [28] L. Funar, Y. Neretin, Diffeomorphism groups of tame Cantor sets and Thompson–like groups. Compos. Math. 154 (2018), no. 5, 1066–1110.
  * [29] L. Funar, M. Nguyen, On the automorphism group of the asymptotic pants complex of an infinite surface of genus zero. Math. Nachr. 289 (2016), no. 10, 1189–1207.
  * [30] S. Galatius. Stable homology of automorphism groups of free groups. Ann. of Math. (2), 173(2):705–768, 2011\.
  * [31] S. Galatius, O. Randal-Williams, Homological stability for moduli spaces of high dimensional manifolds. I. J. Amer. Math. Soc. 31 (2018), no. 1, 215–264.
  * [32] A. Genevois, A. Lonjou, and C. Urech. Asymptotically rigid mapping class groups I: Finiteness properties of braided Thompson’s and Houghton’s groups. Geometry and Topology, in press.
  * [33] A. Genevois, A. Lonjou, and C. Urech. Asymptotically rigid mapping class groups II: strand diagrams and nonpositive curvature. Preprint, https://arxiv.org/abs/2110.06721.
  * [34] R. Geoghegan, Topological methods in group theory. Graduate Texts in Mathematics, 243. Springer, New York, 2008. xiv+473 pp.
  * [35] A. Haefliger, Plongements différentiables de variétés dans variétés. Comment. Math. Helv. 36 (1961), 47–82.
  * [36] L. Harer, Stability of the homology of the mapping class groups of orientable surfaces. Ann. of Math. (2) 121 (1985), no. 2, 215-249.
  * [37] W. J. Harvey, On branch loci in Teichmüller space. Trans. Amer. Math. Soc. 153 (1971), 387–399.
  * [38] A. Hatcher, A Proof of the Smale Conjecture, $\operatorname{Diff}({\mathbb{S}}^{3})\simeq O(4)$. Ann. of Math. 117 (1983), no. 3, second series, 553-607.
  * [39] A. Hatcher, Linearization in 3-dimensional topology. Proceedings of the International Congress of Mathematicians (Helsinki 1978), pp. 463-468, Acad. Sci. Fennica, Helsinki, 1980\.
  * [40] A. Hatcher, On triangulations of surfaces. Topology Appl. 40 (1991), no. 2, 189-194.
  * [41] A. Hatcher, D. McCullough, Finite presentation of 3-manifold mapping class groups. Groups of self-equivalences and related topics (Montreal, PQ, 1988), 48–57, Lecture Notes in Math., 1425, Springer, Berlin, 1990.
  * [42] A. Hatcher, K. Vogtmann, Tethers and homology stability for surfaces. Algebr. Geom. Topol. 17 (2017), no. 3, 1871–1916.
  * [43] A. Hatcher, N. Wahl, Stabilization for the automorphisms of free groups with boundaries, Geom. Topol. 9 (2005), 1295–1336.
  * [44] A. Hatcher and N. Wahl. Stabilization for mapping class groups of 3-manifolds. Duke Math.J., 155(2):205–269, 2010.
  * [45] G. Higman, Finitely presented infinite simple groups. Notes on Pure Mathematics, No. 8 (1974). Department of Pure Mathematics, Department of Mathematics, I.A.S. Australian National University, Canberra, 1974. vii+82 pp.
  * [46] G. Higman, Subgroups of finitely presented groups. Proc. Roy. Soc. London Ser. A 262 (1961), 455–475.
  * [47] W. C. Hsiang, R. W. Sharpe, Parametrized surgery and isotopy. Pacific J. Math. 67 (1976), no. 2, 401–459.
  * [48] S. Hong, D. McCullough, Mapping class groups of 3-manifolds, Geometry and topology down under, 53-63, Contemp. Math., 597, Amer. Math. Soc., Providence, RI, 2013.
  * [49] C. Kapoudjian, V. Sergiescu, An extension of the Burau representation to a mapping class group associated to Thompson’s group T. Geometry and dynamics, 141-164, Contemp. Math., 389, Aportaciones Mat., Amer. Math. Soc., Providence, RI, 2005.
  * [50] M.A. Kervaire, J.W. Milnor, Groups of homotopy spheres. I. Ann. of Math. (2) 77 (1963), 504–537.
  * [51] M. Krannich, Mapping class groups of highly connected $(4k+2)$-manifolds. Selecta Math. (N.S.) 26 (2020), no. 5, Paper No. 81, 49 pp.
  * [52] M. Krannich, O. Randal-Williams, Mapping class groups of simply connected high-dimensional manifolds need not be arithmetic. Comptes Rendus Mathématique 2020, 358, p. 469-473
  * [53] M. Kreck, Isotopy classes of diffeomorphisms of $(k-1)$-connected almost-parallelizable $2k$-manifolds. Algebraic topology, Aarhus 1978 (Proc. Sympos., Univ. Aarhus, Aarhus, 1978), pp. 643–663, Lecture Notes in Math., 763, Springer, Berlin, 1979.
  * [54] M. Kreck, Surgery and duality. Ann. of Math. (2) 149 (1999), no. 3, 707–754.
  * [55] J. Landgraf, Cutting and pasting in the Torelli subgroup of $\operatorname{Out}(F_{n})$. Preprint, arXiv:2106.14147.
  * [56] F. Laudenbach, Sur les 2-sphères d’une variété de dimension 3. Ann. of Math. (2) 97 (1973), 57–81.
  * [57] F. Laudenbach, Topologie de la dimension trois: homotopie et isotopie. Astérisque, No. 12. Société Mathématique de France, Paris, 1974. i+152 pp.
  * [58] I. J. Leary, A metric Kan-Thurston theorem. J. Topol. 6 (2013), no. 1, 251–284.
  * [59] D. McCullough, Topological and algebraic automorphisms of 3-manifolds. Groups of self-equivalences and related topics (Montreal, PQ, 1988), 102-113, Lecture Notes in Math., 1425, Springer, Berlin, 1990.
  * [60] J. Milnor, Lectures on the $h$-cobordism theorem, notes by L. Siebenmann and J. Sondow, Princeton University Press, Princeton, NJ, 1965. v+116 pp.
  * [61] D. Quillen, Homotopy properties of the poset of nontrivial p-subgroups of a group. Adv. in Math., 28 (1978), no. 2, 101-128.
  * [62] R. Skipper, X. Wu, Finiteness properties for relatives of braided Higman-Thompson groups. Preprint, arXiv:2103.14589.
  * [63] R. Skipper, X. Wu, Homological stability for the ribbon Higman–Thompson groups. Preprint, arXiv:2103.14589.
  * [64] A. Soulié, Some computations of stable twisted homology for mapping class groups, Comm. Algebra, in press.
  * [65] M. Stein, Groups of piecewise linear homeomorphisms. Trans. Amer. Math. Soc. 332 (1992), no. 2, 477–514.
  * [66] D. Sullivan, Infinitesimal computations in topology, Publ. Math., Inst. Hautes Étud. Sci. (1977), no. 47, p. 269-331.
  * [67] M. Szymik, N. Wahl, The homology of the Higman–Thompson groups. Invent. Math. 216 (2019), no. 2, 445-518.
  * [68] W. Thumann, Operad groups and their finiteness properties. Adv. Math. 307 (2017), 417–487.
  * [69] U. Tillmann, Homology stability for symmetric diffeomorphism and mapping class groups. Math. Proc. Cambridge Philos. Soc. 160 (2016), no. 1, 121–139.
  * [70] N. Wahl, Ribbon braids and related operads. Thesis, Oxford 2001.
  * [71] S. Witzel. M. C. B. Zaremsky, Thompson groups for systems of groups, and their finiteness properties. Groups Geom. Dyn. 12 (2018), no. 1, 289–358.