$\|\alpha_{0}\|\colon\|Y_{\bullet}\|\to\text{Mod-Conf}_{S^{n-1}}(\bar{D}^{n})$

is a weak homotopy equivalence of right $\text{MConf}(S^{n-1})$-modules.

###### Proof.

The image of the map $\|\alpha_{0}\|\colon\|Y_{\bullet}\|\to\text{Mod-
Conf}_{S^{n-1}}(\bar{D}^{n})$ is contained in $\text{Mod-
Conf}_{S^{n-1}}(\bar{D}^{n})^{\leq 1}$. Since the inclusion $\text{Mod-
Conf}_{S^{n-1}}(\bar{D}^{n})^{\leq 1}\hookrightarrow\text{Mod-
Conf}_{S^{n-1}}(\bar{D}^{n})$ is a homotopy equivalence by Lemma 4.23, it
suffices to show that the map
$\|\alpha_{0}\|\colon\|Y_{\bullet}\|\to\text{Mod-
Conf}_{S^{n-1}}(\bar{D}^{n})^{\leq 1}$ is a weak homotopy equivalence.

By Lemma 4.30, $Y_{\bullet}$ is a sub-semi-simplicial space of $\text{Mod-
Conf}_{S^{n-1}}(\bar{D}^{n})^{\leq 1}\times W_{\bullet}$ which is an open
subset of $\text{Mod-Conf}_{S^{n-1}}(\bar{D}^{n})^{\leq 1}\times W_{\bullet}$
in each degree ($W_{\bullet}$ is the semi-simplicial set from Definition
4.29). Therefore, by Proposition 4.28, the map
$\|\alpha_{0}\|\colon\|Y_{\bullet}\|\to\text{Mod-
Conf}_{S^{n-1}}(\bar{D}^{n})^{\leq 1}$ is a Serre microfibration. By
Proposition 4.31, the fibers of the map $\|\alpha_{0}\|$ are weakly
contractible. Since the map $\|\alpha_{0}\|$ is a Serre microfibration with
weakly contractible fibers, by Lemma 4.27, the map
$\|\alpha_{0}\|\colon\|Y_{\bullet}\|\to\text{Mod-
Conf}_{S^{n-1}}(\bar{D}^{n})^{\leq 1}$ is a weak homotopy equivalence. ∎

######  Remark 4.33.

We have given a resolution of $\text{Mod-Conf}_{S^{n-1}}(\bar{D}^{n})$ without
any motivation. We will now show that there is a completely conceptual
explanation for how to come up with a guess for a resolution.

We have that $S\colonequals C_{*}(\text{MConf}(S^{n-1});\mathbb{Z})$ is a
differential graded ring and $C_{*}(\text{Mod-
Conf}_{S^{n-1}}(\bar{D}^{n});\mathbb{Z})$ is a differential graded module over
$S$. We want to find a resolution of $C_{*}(\text{Mod-
Conf}_{S^{n-1}}(\bar{D}^{n});\mathbb{Z})$ by free $S$-modules.

Since the ring $S$ is augmented over $\mathbb{Z}$, the hyper Tor groups

$\text{Tor}^{S}_{*}(C_{*}(\text{Mod-
Conf}_{S^{n-1}}(\bar{D}^{n});\mathbb{Z}),\mathbb{Z})$

are defined. These hyper Tor groups provide lower bounds on the ranks of the
free $S$-modules in a free resolution of $C_{*}(\text{Mod-
Conf}_{S^{n-1}}(\bar{D}^{n});\mathbb{Z})$. We can calculate these hyper Tor
groups by calculating the hyper homology of the following bar construction

$B_{\bullet}(C_{*}(\text{Mod-
Conf}_{S^{n-1}}(\bar{D}^{n});\mathbb{Z}),C_{*}(\text{MConf}(S^{n-1});\mathbb{Z}),\mathbb{Z})$

which is equivalent to $C_{*}(B_{\bullet}(\text{Mod-
Conf}_{S^{n-1}}(\bar{D}^{n}),\text{MConf}(S^{n-1}),\text{pt});\mathbb{Z})$.
The homology of this double complex is isomorphic to the singular homology
$H_{*}(\|B_{\bullet}(\text{Mod-
Conf}_{S^{n-1}}(\bar{D}^{n}),\text{MConf}(S^{n-1}),\text{pt})\|;\mathbb{Z})$.

We can understand the space $\|B_{\bullet}(\text{Mod-
Conf}_{S^{n-1}}(\bar{D}^{n}),\text{MConf}(S^{n-1}),\text{pt})\|$ better by
working with relative configuration spaces. Let $L\subset X$ be a closed
subspace of a space $X$. The relative configuration space $\text{Conf}(X,L)$
is the quotient space of $\text{Conf}(X)$ by the equivalence relation which
identifies two finite subsets $s$ and $s_{1}$ of $X$ if $s\cap(X\setminus
L)=s_{1}\cap(X\setminus L)$. The relative configuration space
$\text{Conf}(D^{n},\text{pt})$ is homotopy equivalent to a point, so
$\text{Mod-Conf}_{S^{n-1}}(\bar{D}^{n},\text{pt})$ is homotopy equivalent to a
point by Lemma 4.7. By a relative version of Proposition 4.19, the space

$\|B_{\bullet}(\text{Mod-
Conf}_{S^{n-1}}(\bar{D}^{n}),\text{MConf}(S^{n-1}),\text{Mod-
Conf}_{S^{n-1}}(\bar{D}^{n},\text{pt}))\|$

is weakly equivalent to the relative configuration space
$\text{Conf}(S^{n},x_{0})$. By pushing all but one point of a configuration in
$\text{Conf}(S^{n},x_{0})$ to $x_{0}$, we have that $\text{Conf}(S^{n},x_{0})$
is homotopy equivalent to $S^{n}$ (this is a classic argument which happens in
the construction of the scanning map–see [McD75]).

Since the homology $H_{*}(S^{n})$ is only nonzero in degrees $0$ and $n$, this
suggests we might be able to give a resolution of $C_{*}(\text{Mod-
Conf}_{S^{n-1}}(\bar{D}^{n});\mathbb{Z})$ from two free $S$-modules based on a
suitable model of $S^{n}$. One such model for $S^{n}$ is given by the homotopy
coequalizer $\text{hocoeq}(S^{n-1}\rightrightarrows S^{0})$. This model for
$S^{n}$ can be viewed as a length two semi-simplicial resolution

$S^{n-1}\to S^{0}\to S^{n}$

of $S^{n}$ (the description mentioned in Section 1.4 involves a different
model for $S^{n}$, the cone on $S^{n-1}\to\text{pt}$, which we will not use).
As a result, it is natural for us to guess that a resolution of
$C_{*}(\text{Mod-Conf}_{S^{n-1}}(\bar{D}^{n});\mathbb{Z})$ of length two can
be built from free $S$-modules using $C_{*}(S^{n-1};\mathbb{Z})$ and
$C_{*}(S^{0};\mathbb{Z})$.

We now construct a map $C_{*}(Y)\to C_{*}(\text{Mod-
Conf}_{S^{n-1}}(\bar{D}^{n}))$ of $C_{*}(\text{MConf}(S^{n-1}))$-modules,
which will allow us to construct a stabilization map on
$C_{*}(\text{Conf}(M))$.

######  Definition 4.34.

Fix an embedding $f^{{}^{\prime}}\colon D^{n}\to S^{n-1}\times(0,1)$. Define
$\text{MConf}(f^{{}^{\prime}})\colon\text{Conf}(D^{n})\to\text{MConf}(S^{n-1})$
and $\text{Mod-Conf}(f^{{}^{\prime}})\colon\text{Mod-
Conf}_{S^{n-1}}(\bar{D}^{n})\times\text{MConf}(D^{n})\to\text{Mod-
Conf}_{S^{n-1}}(\bar{D}^{n})$ to be the maps

$\displaystyle\text{MConf}(f^{{}^{\prime}})\colon\text{Conf}(D^{n})$
$\displaystyle\to\text{MConf}(S^{n-1})$ $\displaystyle\mathcal{C}$
$\displaystyle\mapsto(\text{Conf}(\mathcal{C}),1)$

and

$\displaystyle\text{Mod-Conf}(f^{{}^{\prime}})\colon\text{Mod-
Conf}_{S^{n-1}}(\bar{D}^{n})\times\text{Conf}(D^{n})$
$\displaystyle\to\text{Mod-Conf}_{S^{n-1}}(\bar{D}^{n})$
$\displaystyle\big{(}(\mathcal{C},\lambda),\mathcal{C}_{1}\big{)}$
$\displaystyle\mapsto
m_{r}\big{(}(\mathcal{C},\lambda),\text{MConf}(f^{{}^{\prime}})(\mathcal{C}_{1})\big{)}.$

######  Definition 4.35.

Let $R$ be a ring. Given a class $z\in H_{*}(\text{Conf}(D^{n});R)$, by abuse
of notation, define

$t_{z}\colon C_{*}(\text{Mod-Conf}_{S^{n-1}}(\bar{D}^{n});R)\to
C_{*}(\text{Mod-Conf}_{S^{n-1}}(\bar{D}^{n});R)$

to be the restriction $\text{Mod-Conf}(f^{{}^{\prime}})_{*}(-\otimes z)$.
Define

$g^{{}^{\prime}}_{z}\colon C_{*}(\text{Mod-Conf}_{S^{n-1}}(\bar{D}^{n})_{\leq
1};R)\otimes C_{*}(\text{MConf}(S^{n-1});R)\to C_{*}(\text{Mod-
Conf}_{S^{n-1}}(\bar{D}^{n});R)$

to be the composition
$(m_{r})_{*}\circ(t_{z}\otimes(\text{id}_{\text{MConf}(S^{n-1})})_{*})$.

To obtain a stabilization map from $g^{{}^{\prime}}_{z}$, we will need the
following well-known result.

######  Theorem 4.36 (see e.g. [FHT01, Theorem 4.18]).

Let $X$ be a CW complex and $R$ a ring. Let $C_{*}^{\text{cell}}(X;R)$ denote
the cellular chain complex of $X$ with coefficients in $R$. There is a quasi-
isomorphism $m\colon C_{*}(X;R)\to C_{*}^{\text{cell}}(X;R)$.

######  Proposition 4.37.

Let $R$ be a ring and let $e\in H_{0}(\text{Conf}_{1}(\mathbb{R}^{n});R)$ be
the class of a point. Suppose that $z\in H_{*}(\text{Conf}(\mathbb{R}^{n});R)$
is a homology class such that the Browder bracket $\left[z,e\right]=0$. Then
the maps $g^{{}^{\prime}}_{z}\circ(d_{0})_{*}$ and
$g^{{}^{\prime}}_{z}\circ(d_{1})_{*}$ are chain homotopy equivalent.

###### Proof.

Recall that $Y_{1}=\text{MConf}(S^{n-1})_{1}\times\text{MConf}(S^{n-1})$.
Since $H_{*}(\text{MConf}(S^{n-1})_{1})=H_{*}(S^{n-1})$ is a free $R$-module,
by the Künneth theorem we have that

$C_{i}(Y_{1})=C_{i}(\text{MConf}(S^{n-1})_{1}\times\text{MConf}(S^{n-1}))\cong\bigoplus_{a+b=i}C_{a}(\text{MConf}(S^{n-1})_{1})\otimes
C_{b}(\text{MConf}(S^{n-1})).$

Given $\sigma\otimes\tau\in C_{a}(\text{MConf}(S^{n-1})_{1})\otimes
C_{b}(\text{MConf}(S^{n-1}))$, we can represent
$(g^{{}^{\prime}}_{z}\circ(d_{0})_{*})(\sigma\otimes\tau)$ and
$(g^{{}^{\prime}}_{z}\circ(d_{1})_{*})(\sigma\otimes\tau)$ via the following
figure and the geometric interpretation of the multiplication map:

Figure 12. The maps $g_{z}\circ(d_{0})_{*}$ and $g_{z}\circ(d_{1})_{*}$

Algebraically, we can express the image of $(\sigma,\tau)$ under these two
maps as

$\displaystyle g^{{}^{\prime}}_{z}\circ(d_{0})_{*}\colon$
$\displaystyle\sigma\otimes\tau\mapsto(\emptyset,0)\otimes(\sigma\cdot\tau)\mapsto\big{(}t_{z}(\emptyset,0)\big{)}\otimes(\sigma\cdot\tau)\mapsto(m_{r})_{*}(z,\sigma\cdot\tau),$
$\displaystyle g^{{}^{\prime}}_{z}\circ(d_{1})_{*}\colon$
$\displaystyle\sigma\otimes\tau\mapsto\text{Conf}(\iota)_{*}(\sigma)\otimes\tau\mapsto
t_{z}(\text{Conf}(\iota)_{*}\sigma)\otimes\tau\mapsto(m_{r})_{*}(t_{z}(\text{Conf}(\iota)_{*}\sigma),\tau).$

Since
$(m_{r})_{*}(z,\sigma\cdot\tau)=(m_{r})_{*}((m_{r})_{*}(z,\sigma),\tau)$, it
suffices to prove that the maps

$\displaystyle(m_{r})_{*}(z,-)\colon C_{*}(\text{MConf}(S^{n-1})_{1})$
$\displaystyle\to C_{*}(\text{Mod-Conf}_{S^{n-1}}(\bar{D}^{n}))$
$\displaystyle t_{z}(\text{Conf}(\iota)_{*}(-))\colon
C_{*}(\text{MConf}(S^{n-1})_{1})$ $\displaystyle\to C_{*}(\text{Mod-
Conf}_{S^{n-1}}(\bar{D}^{n}))$

are chain homotopy equivalent.

Since $\text{MConf}(S^{n-1})_{1}$ is homotopy equivalent to $S^{n-1}$, the
chain complex $C_{*}(\text{MConf}(S^{n-1})_{1})$ is chain homotopy equivalent
to $C_{*}(S^{n-1})$, so we may treat $\sigma$ as an element of
$C_{*}(S^{n-1})$. The chain complex $C_{*}(S^{n-1})$ is chain homotopy
equivalent to the cellular chain complex $C^{\text{cell}}_{*}(X)$ by Theorem
4.36. Therefore, we may treat $\sigma$ as an element of
$C_{*}^{\text{cell}}(S^{n-1})$. Since we can give a CW structure on $S^{n-1}$
using one $0$-cell and one $(n-1)$-cell, $C_{*}^{\text{cell}}(S^{n-1})$ is
generated by a cell in degree $0$ and degree $n-1$. As a result, to show that
the maps $(m_{r})_{*}(z,-)$ and $t_{z}(\text{Conf}(\iota)_{*}(-))$ are chain
homotopy equivalent, we only have two cases to consider:

  1. Case 1:

$\sigma$ is the class of a point in
$H_{0}(S^{n-1})=C_{0}^{\text{cell}}(S^{n-1})$, and

  2. Case 2:

$\sigma$ is a generator of $H_{n-1}(S^{n-1})=C_{n-1}^{\text{cell}}(S^{n-1})$.

Case 1: Since $\sigma$ is the class of a point, the maps
$(m_{r})_{*}(z,\sigma)$ and $t_{z}(\text{Conf}(\iota)_{*}\sigma)$ are chain
homotopy equivalent to the chain-level stabilization map $t_{1}(z)$, so we can
construct a chain homotopy in this case.

Case 2: If $\sigma$ is a fundamental class in $H_{n-1}(S^{n-1})$, then
$(g_{z}\circ(d_{1})_{*})(\sigma\otimes\tau)=0$ since $i\colon
S^{n-1}\hookrightarrow\bar{D}^{n}$ being nullhomotopic implies that
$t_{z}(\text{Conf}(\iota)_{*}\sigma)=t_{z}\big{(}i_{*}(\sigma)\big{)}=0$.
Since $(m_{r})_{*}(z,\sigma)=\left[z,e\right]$ (see Figure 6) and by
assumption $\left[z,e\right]=0$, we have that $(m_{r})_{*}(z,\sigma)=0$.
Therefore, fixing a nullhomotopy for $(m_{r})_{*}(z,\sigma)$ gives a chain
homotopy between $(g^{{}^{\prime}}_{z}\circ(d_{1})_{*})(\sigma\otimes\tau)$
and $(g^{{}^{\prime}}_{z}\circ(d_{2})_{*})(\sigma\otimes\tau)$ in this case as
well ∎

The following proposition lists some homology classes $z\in
H_{*}(\text{Conf}(\mathbb{R}^{n});R)$ which satisfy the condition
$\left[z,e\right]=0$.

######  Lemma 4.38.

Let $R$ be a ring and let $e\in H_{0}(\text{Conf}_{1}(\mathbb{R}^{n});R)$ be
the class of a point.

  1. (1)

If $\omega\in H_{*}(\text{Conf}(\mathbb{R}^{n});\mathbb{F}_{p})$ is a homology
class such that $\omega=z^{p}$ for some class $z$, then the Browder bracket
$\left[\omega,e\right]=0$. It follows that $\left[e^{p},e\right]=0$ for all
$n$.

  2. (2)

If $R$ is $\mathbb{F}_{p}$, then for any class $\omega\in
H_{*}(\text{Conf}(\mathbb{R}^{n});\mathbb{F}_{p})$ the Browder bracket
$\left[Q^{s}\omega,e\right]$ is zero.

  3. (3)

If $n$ is odd or $R=\mathbb{F}_{2}$, then $\left[e,e\right]=0$.

  4. (4)

Let $x_{i},y_{i},z_{i}\in H_{*}(\text{Conf}(\mathbb{R}^{2});\mathbb{F}_{p})$
denote the homology classes from Theorem 2.3. We have that
$\left[x_{i},e\right]=\left[y_{i},e\right]=\left[z_{i},e\right]=0$.

###### Proof.

Parts 1 and 3 appear in [KM16, Proposition 2.5] and [Ran13a, p. 16],
respectively. Parts 1 and 2 are immediate consequences of the derivation
property of the Browder bracket (Property 4) and the relationship of a Dyer-
Lashof operation with a Browder bracket (Property 5), respectively. Part 3
follows from the geometric description of the Browder bracket. The class
$\left[e,e\right]$ corresponds to twice a generator of
$H_{n-1}(\mathbb{R}P^{n-1};R)$, so it vanishes when $n$ is odd or
$R=\mathbb{F}_{2}$.

For Part 4, we first show that $\left[x_{i},e\right]$ vanishes. Recall that
$x_{i}$ is the class $\xi\cdots\xi e\in
H_{*}(\text{Conf}(\mathbb{R}^{2});\mathbb{F}_{2})$. Since $\left[x,x\right]=0$
for all $x\in H_{*}(\text{Conf}(\mathbb{R}^{2});\mathbb{F}_{2})$ (Property 3
of the Browder bracket), we have that $\left[e,e\right]=0$. Since

$\left[e,x_{1}\right]=\left[e,\xi e\right]=\left[e,\left[e,e\right]\right]$

(by Property 7 of the Browder bracket) and $\left[e,e\right]=0$, we have that
$\left[x_{1},e\right]=0$. Since $x_{i}=\xi x_{i-1}$, the same argument
recursively shows that $\left[x_{i},e\right]=0$.

We now deal with the case that $p$ is odd. we first show that
$\left[z_{i},e\right]=0$ for all $i$. Recall that
$z_{i}=\xi\cdots\xi\left[e,e\right]$ and
$y_{i}=\beta\xi\cdots\xi\left[e,e\right]$. By the Jacobi identity of the
Browder bracket (Property 3 of the Browder bracket),
$\left[e,\left[e,e\right]\right]=\left[e,z_{0}\right]=0$. By the relation of
the top homology operation $\xi$ and the Browder bracket and the Jacobi
identity (Properties 7 and 3 of the Browder bracket), $\left[e,z_{i}\right]$
can be recursively expressed as

$\displaystyle\left[e,z_{i}\right]$ $\displaystyle=\left[e,\xi z_{i-1}\right]$
$\displaystyle=\text{ad}^{p}(z_{i-1})(e)\text{ by Property }\ref{iterated
Browder bracket}\text{ of the Browder bracket}$
$\displaystyle=\left[z_{i-1},\left[z_{i-1},\cdots,\left[z_{i-1},\left[z_{i-1},e\right]\right]\cdots\right]\right]\text{
by definition of }\text{ad}^{p}.$

Since $\left[z_{0},e\right]=0$, we have that $\left[z_{i-1},e\right]=0$ by
recursion and so $\left[z_{i},e\right]=0$.

We now show that $\left[y_{i},e\right]=0$ for $i>0$. First, we show that
$\left[y_{1},e\right]=0$. Recall the homology operation
$\zeta(-)\colonequals\beta\xi(-)-\text{ad}^{p-1}(-)(\beta(-))$ and that
$\left[x,\zeta y\right]=0$ by Property 7 of the Browder bracket. Since
$y_{i}=\beta\xi z_{i-1}$ and the Browder bracket is linear in both entries, we
have that

$\displaystyle 0$ $\displaystyle=\left[e,\zeta
z_{i-1}\right]=\left[e,\beta\xi(z_{i-1})-\text{ad}^{p-1}(z_{i-1})(\beta(z_{i-1}))\right]$
$\displaystyle=\left[e,y_{i}-\text{ad}^{p-1}(z_{i-1})(\beta(z_{i-1}))\right]$
$\displaystyle=\left[e,y_{i}\right]-\left[e,\text{ad}^{p-1}(z_{i-1})(\beta(z_{i-1}))\right].$

Therefore, it suffices to show that
$\left[e,\text{ad}^{p-1}(z_{i-1})(\beta(z_{i-1}))\right]=0$. For $i=1$, we
have that $\beta(z_{i-1})=\beta(z_{0})=\beta(\left[e,e\right])$. By Property 6
of the Browder bracket,

$\beta\left[e,e\right]=\left[\beta e,e\right]\pm\left[e,\beta e\right].$

We have that $\beta e=0$ since $e$ is a homology class of degree $0$.
Therefore, $\beta z_{0}=0$ and so $\left[e,y_{1}\right]=0$.

Suppose now that $i>1$. We have that $\beta z_{i-1}=y_{i-1}$ since $i>1$. Note
that

$\displaystyle\left[e,\text{ad}^{p-1}(z_{i-1})(y_{i-1})\right]$
$\displaystyle=\left[e,\text{ad}(z_{i-1})(\text{ad}^{p-2}(z_{i-1})(y_{i-1}))\right]$
$\displaystyle=\left[e,\left[z_{i-1},\text{ad}^{p-2}(z_{i-1})(y_{i-1})\right]\right]$

By the Jacobi identity (Property 3 of the Browder bracket), we have that

$\displaystyle 0$
$\displaystyle=\pm\left[e,\left[z_{i-1},\text{ad}^{p-2}(z_{i-1})(y_{i-1})\right]\right]\pm\left[z_{i-1},\left[e,\text{ad}^{p-2}(z_{i-1})(y_{i-1})\right]\right]$
$\displaystyle\pm\left[\text{ad}^{p-2}(z_{i-1})(y_{i-1}),\left[z_{i-1},e\right]\right].$

Since $\left[z_{i-1},e\right]=0$, it suffices to show that
$\left[z_{i-1},\left[e,\text{ad}^{p-2}(z_{i-1})(y_{i-1})\right]\right]=0$. By
iterating the Jacobi identity, it suffices to show that

$\left[e,\text{ad}(z_{i-1})(y_{i-1})\right]=\left[e,\left[z_{i-1},y_{i-1}\right]\right]=0.$

But this follows from an another application of the Jacobi identity

$0=\pm\left[e,\left[z_{i-1},y_{i-1}\right]\right]\pm\left[z_{i-1},\left[e,y_{i-1}\right]\right]\pm\left[y_{i-1},\left[z_{i-1},e\right]\right]$

and the fact that $\left[e,z_{i-1}\right]$ and $\left[e,y_{i-1}\right]$ are
both zero ($\left[e,y_{i-1}\right]$ is zero by recursion). ∎

By Proposition 4.37, the map $g^{{}^{\prime}}_{z}\circ(d_{0})_{*}\colon
C_{*}(Y_{1};R)\to C_{*}(\text{Mod-Conf}_{S^{n-1}}(\bar{D}^{n});R)$ is chain
homotopic to the map $g^{{}^{\prime}}_{z}\circ(d_{1})_{*}$. By fixing a chain
homotopy between these two maps, we can define a map $C_{*}(Y;R)\to
C_{*}(\text{Mod-Conf}_{S^{n-1}}(\bar{D}^{n});R)$.

######  Definition 4.39.

Let $R$ be a ring and let $e\in H_{0}(\text{Conf}_{1}(\mathbb{R}^{n});R)$ be
the class of a point. Suppose that $z\in
H_{i}(\text{Conf}_{k}(\mathbb{R}^{n});R)$ is a homology class such that the
Browder bracket $\left[z,e\right]$ of $z$ and $e$ vanishes. After fixing a
chain homotopy from $g^{{}^{\prime}}_{z}\circ(d_{0})_{*}$ to
$g^{{}^{\prime}}_{z}\circ(d_{1})_{*}$, let $g_{z}\colon C_{*}(Y;R)\to
C_{*}(\text{Mod-Conf}_{S^{n-1}}(\bar{D}^{n});R)$ denote the induced map on
$C_{*}(Y;R)$.

By Proposition 4.37, the map $g_{z}$ is a map of
$C_{*}(\text{MConf}(S^{n-1});R)$-modules. The map $g_{z}$ induces a map of
chain complexes

$C_{*}\big{(}\|B_{\bullet}(Y,\text{MConf}(S^{n-1}),\text{Mod-
Conf}_{S^{n-1}}(M\setminus D^{n}))\|;R\big{)}\to
C_{*}(\|X_{\bullet}({\bar{D}^{n}},{S^{n-1}},{M\setminus D^{n}})\|;R)$

which enables us to define a new chain-level stabilization map for
$C_{*}(\text{Conf}(M);R)$.

######  Definition 4.40.

Let $R$ be a ring and let $e\in H_{0}(\text{Conf}_{1}(\mathbb{R}^{n});R)$ be
the class of a point. Suppose that $z\in
H_{i}(\text{Conf}_{k}(\mathbb{R}^{n});R)$ is a homology class such that the
Browder bracket $\left[z,e\right]$ of $z$ and $e$ vanishes. Define a map of
chain complexes

$t_{z,c}\colon C_{*}\big{(}\|B_{\bullet}(Y,\text{MConf}(S^{n-1}),\text{Mod-
Conf}_{S^{n-1}}(M\setminus D^{n}))\|\big{)}\to
C_{*}(\|X_{\bullet}({\bar{D}^{n}},{S^{n-1}},{M\setminus D^{n}})\|)$

as follows: given chains $\sigma\in C_{*}(Y)$, $\tau_{1},\ldots,\tau_{q}\in
C_{*}(\text{MConf}(S^{n-1}))$, and $\omega\in C_{*}(\text{Mod-
Conf}_{S^{n-1}}(M\setminus D^{n})),$ let

$t_{z,c}(\sigma\otimes\tau_{1}\otimes\cdots\otimes\tau_{q}\otimes\omega)\colonequals
g_{z}(\sigma)\otimes\tau_{1}\otimes\cdots\otimes\tau_{q}\otimes\omega.$

By abuse of notation, we also write $t_{z,c}$ to denote the map on homology.

We now fix a chain homotopy from $g_{z}^{{}^{\prime}}\circ(d_{0})_{*}$ to
$g_{z}^{{}^{\prime}}\circ(d_{1})_{*}$ once and for all time. Since the map
$g_{z}\colon C_{*}(Y)\to C_{*}(\text{Mod-Conf}_{S^{n-1}}(\bar{D}^{n}))$
depends on fixing a chain homotopy, the map $t_{z,c}$ also depends on this.

Let $R$ be a ring and let $e\in H_{0}(\text{Conf}_{1}(\mathbb{R}^{n});R)$ be
the class of a point. Suppose that $z\in
H_{a}(\text{Conf}_{b}(\mathbb{R}^{n});R)$ is a homology class such that
$\left[z,e\right]$ is zero. By Definition 4.40, we have a stabilization map

$t_{z,c}\colon C_{*}(\text{Conf}_{k}(M);R)\to
C_{*+a}(\text{Conf}_{k+b}(M);R).$

If $M$ is an open connected manifold, we also also the stabilization map

$t_{z}\colon C_{*}(\text{Conf}_{k}(M);R)\to C_{*+a}(\text{Conf}_{k+b}(M);R)$

defined in Definition 3.2. The following result relates the new stabilization
map $t_{z,c}$ to the map $t_{z}$.

######  Proposition 4.41.

Suppose that $M$ is an open connected manifold and that $R$ is a ring. Let
$e\in H_{0}(\text{Conf}_{1}(\mathbb{R}^{n});R)$ be the class of a point.
Suppose that $z\in H_{*}(\text{Conf}(\mathbb{R}^{n});R)$ is a class such that
$\left[z,e\right]$ is zero. Let $t_{z}\colon C_{*}(\text{Conf}(M))\to
C_{*}(\text{Conf}(M))$ and $t_{z,c}\colon C_{*}(\text{Conf}(M))\to
C_{*}(\text{Conf}(M))$ denote the stabilization maps defined in Definition 3.2
and Definition 4.40, respectively. The stabilization maps $t_{z}$ and
$t_{z,c}$ are chain homotopic.

###### Proof.

By Lemma 3.8, without loss of generality, we can assume that $M$ is the
interior of a compact manifold $\bar{M}$. The stabilization map $t_{z}\colon
C_{*}(\text{Conf}(M))\to C_{*}(\text{Conf}(M))$ comes from an embedding
$\iota\colon M\sqcup\mathbb{R}^{n}\to M$ as in Lemma 3.1. Fix a closed
$n$-dimensional disk $\bar{E}\subset\bar{M}$ such that the interior $E$
strictly contains $\iota(\mathbb{R}^{n})$ and such that the intersection of
the boundary of $\bar{E}$ and $\bar{M}$ is diffeomorphic to the closed disk
$\bar{D}^{n-1}_{1}\colonequals\bar{D}^{n-1}$. Let
$E^{{}^{\prime}}\colonequals\bar{E}\setminus D^{n-1}_{1}$, let
$M^{{}^{\prime}}\colonequals\bar{M}\setminus D^{n-1}_{1}$, and let $U\subset
E$ be an open $n$-dimensional disk in $E$ disjoint from
$\iota(\mathbb{R}^{n})$. Fix a closed $n$-dimensional disk $\bar{D}\subset U$
and let $G\colonequals U\setminus D\cong S^{n-1}\times\left(-1,0\right]$. We
can summarize our setup with the following picture (see Figure 13).

Figure 13. Regions of the manifold $M^{{}^{\prime}}$

Since the configuration space of a manifold with boundary is homotopy
equivalent to the configuration space of the interior of the manifold, by
Proposition 4.19, we have the following weak homotopy equivalences:

$\text{Conf}(M)\simeq\text{Conf}(M^{{}^{\prime}})\simeq\|X_{\bullet}({M^{{}^{\prime}}\setminus\text{int}(E^{{}^{\prime}})},{\bar{D}^{n-1}_{0}},{E^{{}^{\prime}}})\|.$

We now modify the bar construction
$\|X_{\bullet}({M^{{}^{\prime}}\setminus\text{int}(E^{{}^{\prime}})},{\bar{D}^{n-1}_{0}},{E^{{}^{\prime}}})\|$
by replacing $\text{Mod-Conf}_{\bar{D}^{n-1}_{0}}(E^{{}^{\prime}})$ with a
weakly homotopy equivalent space $Z$ in order to obtain another bar
construction and another weak homotopy equivalence

$\|B_{\bullet}(\text{Mod-
Conf}_{\bar{D}^{n-1}_{0}}(M^{{}^{\prime}}\setminus\text{int}(E^{{}^{\prime}})),\text{MConf}(\bar{D}^{n-1}_{0}),Z)\|\simeq\text{Conf}(M).$

From this new bar construction, we can compare the stabilization maps
$t_{z,c}$ and $t_{z}$ on $C_{*}(\text{Conf}(M))$.

The spaces $\text{Mod-Conf}_{\bar{D}^{n-1}_{0}}(E^{{}^{\prime}})$ and
$\text{Conf}(E^{{}^{\prime}})$ are homotopy equivalent, since they are both
homotopy equivalent to $\text{Conf}(E)$ (the space $\text{Mod-
Conf}_{\bar{D}^{n-1}_{0}}(E^{{}^{\prime}})$ is homotopy equivalent to
$\text{Conf}(E)$ by Lemma 4.7). Since the open disk $E$ is diffeomorphic to
$U=\bar{D}\cup_{S^{n-1}}G=\bar{D}\cup_{S^{n-1}}S^{n-1}\times\left(-1,0\right]$,
the space $\|X_{\bullet}({\bar{D}},{S^{n-1}},{G})\|$ is weakly homotopy
equivalent to $\text{Conf}(E)$ by Proposition 4.19. The inclusion
$\text{MConf}(S^{n-1})\hookrightarrow\text{Mod-Conf}_{S^{n-1}}(G),$ coming
from the inclusion $S^{n-1}\times\left[0,\infty\right)\hookrightarrow
S^{n-1}\times(-1,\infty)$, is a map of $\text{MConf}(S^{n-1})$-modules and it
is a homotopy equivalence because both spaces are homotopy equivalent to
$\text{Conf}(S^{n-1}\times(0,1))$ by Lemma 4.7. Set $Z$ to be the space

$Z\colonequals\|B_{\bullet}(\text{Mod-
Conf}_{S^{n-1}}(\bar{D}),\text{MConf}(S^{n-1}),\text{MConf}(S^{n-1}))\|.$

The map $Z\to\|X_{\bullet}({\bar{D}},{S^{n-1}},{G})\|,$ coming from the
inclusion $\text{MConf}(S^{n-1})\hookrightarrow\text{Mod-Conf}_{S^{n-1}}(G)$,
is a weak homotopy equivalence by Theorem 4.10.

The space $Z$ has the structure of a left module over $\text{MConf}(S^{n-1})$
coming from the left module structure on $\text{Mod-Conf}_{S^{n-1}}(\bar{D})$.
Since $\bar{D}^{n-1}_{0}$ is a subspace of $S^{n-1}$, the space
$\text{MConf}(\bar{D}^{n-1}_{0})$ is a submonoid of the monoid
$\text{MConf}(S^{n-1})$. As a result, the space $Z$ is a left
$\text{MConf}(\bar{D}^{n-1}_{0})$-module. Therefore, we can consider the space

$\|B_{\bullet}(\text{Mod-
Conf}_{\bar{D}^{n-1}_{0}}(M^{{}^{\prime}}\setminus\text{int}(E^{{}^{\prime}})),\text{MConf}(\bar{D}^{n-1}_{0}),Z)\|.$

We have an inclusion of left $\text{MConf}(\bar{D}^{n-1}_{0})$-modules
$\text{Mod-Conf}_{\bar{D}^{n-1}_{0}}(E^{{}^{\prime}})\hookrightarrow\text{Mod-
Conf}_{S^{n-1}}(\bar{D})$ that comes from an inclusion
$E^{{}^{\prime}}\hookrightarrow\bar{D}$. The former inclusion induces a map

$f\colon\|X_{\bullet}({M^{{}^{\prime}}\setminus\text{int}(E^{{}^{\prime}})},{\bar{D}^{n-1}_{0}},{E^{{}^{\prime}}})\|\to\|B_{\bullet}(\text{Mod-
Conf}_{\bar{D}^{n-1}_{0}}(M^{{}^{\prime}}\setminus\text{int}(E^{{}^{\prime}})),\text{MConf}(\bar{D}^{n-1}_{0}),Z)\|.$

By Theorem 4.10, the map $f$ is a weak homotopy equivalence, since the map
$\text{Mod-Conf}_{\bar{D}^{n-1}_{0}}(E^{{}^{\prime}})\to Z$ being a weak
homotopy equivalence implies that $f$ is a levelwise weak homotopy
equivalence. As a result, we have that $\text{Conf}(M)$ is weakly homotopy
equivalent to $\|B_{\bullet}(\text{Mod-
Conf}_{\bar{D}^{n-1}_{0}}(M^{{}^{\prime}}\setminus\text{int}(E^{{}^{\prime}})),\text{MConf}(\bar{D}^{n-1}_{0}),Z)\|$.

The stabilization map $t_{z,c}\colon C_{*}(\text{Conf}(M))\to
C_{*}(\text{Conf}(M))$ comes from the map $g_{z}\colon C_{*}(Y)\to C_{*}(Z)$,
defined in Definition 4.39, which puts the class $z$ inside the disk $D$ (see
Figure 14). The other stabilization map $t_{z}\colon C_{*}(\text{Conf}(M))\to
C_{*}(\text{Conf}(M))$ places the class $z$ in the region $V$ (see Figure 14).
To show that these two stabilization maps are homotopic, it suffices to show
that the map $g_{z}$ is homotopic to a map which puts the class $z$ in the
region $V$.

Since the space $\text{MConf}(S^{n-1})$ is also a right module over itself, we
also have that $Z$ is a right $\text{MConf}(S^{n-1})$-module. Let $S$ denote
the ring $C_{*}(\text{MConf}(S^{n-1}))$. Since $C_{*}(Z)$ is a right
$S$-module, we have that

$C_{*}(Z)\simeq C_{*}(Z)\otimes_{S}S\simeq C_{*}(\text{Mod-
Conf}_{S^{n-1}}(D))\otimes_{S}S.$

Let $\sigma$ and $\tau$ be elements in $C_{*}(\text{Mod-
Conf}_{S^{n-1}}(\bar{D}))$ and $S$, respectively. and consider
$\sigma\otimes\tau\colonequals\sigma\otimes_{S}\tau$. We have that

$\displaystyle g_{z}(\sigma\otimes\tau)$
$\displaystyle=g_{z}(\sigma)\otimes\tau=t_{z}(\sigma)\otimes\tau$
$\displaystyle=z\sigma\otimes\tau=(m_{r})_{*}((z\sigma)\otimes_{F}\tau)\otimes
1$ (since $\left[z,e\right]$=0, the map $g_{z}$ is a map of right $S$-modules
by Proposition 4.37)
$\displaystyle=(z\sigma)\text{Conf}(\iota)_{*}(\tau)\otimes 1.$ Recall that
the configuration space $\text{Conf}(D^{n})$ is a homotopy commutative
H-space. Therefore, up to homotopy we have that
$\displaystyle(z\sigma)\text{Conf}(\iota)_{*}(\tau)\otimes 1$
$\displaystyle=\sigma(\text{Conf}(\iota)_{*}(\tau)z)\otimes 1$
$\displaystyle\sigma\otimes\tau z$ $\displaystyle=\sigma\otimes t_{z}(\tau).$

We can express this argument in terms of the following pictures (see Figure
14).

Figure 14. Comparing the stabilization maps $t_{z}$ and $t_{z,c}$

Therefore, the map $g_{z}$ is homotopic to a map which puts the class $z$ in
the region $V$ and so the stabilization maps $t_{z}$ and $t_{z,c}$ are
homotopic. ∎

######  Corollary 4.42.

Let $R$ be a ring and let $e\in H_{0}(\text{Conf}_{1}(\mathbb{R}^{n});R)$ be
the class of a point. Suppose that $z\in
H_{a}(\text{Conf}_{b}(\mathbb{R}^{n});R)$ is a class such that
$\left[z,e\right]$ is zero. Under the identifications
$Y\simeq\text{Conf}(\mathbb{R}^{n})\simeq\text{Mod-
Conf}_{S^{n-1}}(\bar{D}^{n})$, the stabilization map $g_{z}\colon
C_{*}(Y;R)\to C_{*}(\text{Conf}(\text{Mod-Conf}_{S^{n-1}}(\bar{D}^{n});R)$ is
chain homotopic to the map $t_{z}\colon
C_{*}(\text{Conf}(\mathbb{R}^{n});R)\to C_{*}(\text{Conf}(\mathbb{R}^{n});R)$.

######  Remark 4.43.

Suppose that $M$ is an open connected manifold of dimension $n$ and that $z\in
H_{i}(\text{Conf}_{k}(\mathbb{R}^{n});R)$ is a homology class such that the
Browder bracket $\left[z,e\right]$ of $z$ and $e$ vanishes. Since the
stabilization maps $t_{z}$ and $t_{z,c}$ are chain homotopic by Proposition
4.41, they induce the same map on homology. Since the map $t_{z,c}\colon
H_{*}(\text{Conf}(M);R)\to H_{*}(\text{Conf}(M);R)$ does not depend on an
embedding $\iota\colon M\sqcup\mathbb{R}^{n}\to M$, the stabilization map

$t_{z}\colon H_{*}(\text{Conf}(M);R)\to H_{*}(\text{Conf}(M);R)$

does not depend on a choice of an embedding $M\sqcup\mathbb{R}^{n}\to M$.

###  4.3. Iterated Mapping Cones for Closed Manifolds and the Puncture
Resolution

To formulate secondary (and higher-order) homological stability for the
configuration space of a closed manifold $M$ of dimension $n$, we need to
define an iterated mapping cone for $C_{*}(\text{Conf}(M))$. The construction
of an iterated mapping cone for $C_{*}(\text{Conf}(M))$ when $M$ is an open
manifold (see Section 2 and the beginning of Section 3.2) does not work here
because that construction depends on an embedding
$M\bigsqcup(\bigsqcup_{j=1}^{m}D^{n})\to M$.

After constructing an iterated mapping cone, we prove stability using a
puncture resolution argument inspired by [Ran13a, Section 9.4]. Our puncture
resolution is slightly more similar to the puncture resolution of [KM16,
Section 3] than to that of [Ran13a, Section 9.4]. The puncture resolution is a
semi-simplicial resolution
$(Z_{k,\bullet}(M,\bar{E}_{m}),\text{Conf}_{k}(M),\epsilon)$ of
$\text{Conf}_{k}(M)$ whose simplices consist of configuration spaces of open
manifolds. By a geometric realization spectral sequence, we can compare the
homology of the $p$-simplices $Z_{k,p}(M,\bar{E}_{m})$ to the homology of
$\text{Conf}_{k}(M)$ and leverage stability results for configuration spaces
of open manifolds to prove analogous results for configuration spaces of
closed manifolds.

First, we fix some notation.

######  Definition 4.44.

Let $N$ be a manifold of dimension $n$ and let $m$ be a positive integer. Fix
$m$ disjoint closed disks $\bar{D}^{n}_{1},\ldots,\bar{D}^{n}_{m}$ in $N$ and
let $\bar{E}_{m}\colonequals\bigsqcup_{j=1}^{m}\bar{D}^{n}_{j}\subset N$. Let
$S^{n-1}_{j}$ denote the boundary of $\bar{D}^{n}_{j}$ and let $E_{m}$ denote
the interior of $\bar{E}_{m}$.

We need to the following model for $\text{Conf}(M\setminus E_{m})$.

######  Definition 4.45.

Let $N$ and $\bar{E}_{m}\subset N$ be as in Definition 4.44. Define
$\text{Mod-Conf}^{m}_{S^{n-1}}(N\setminus E_{m})$ to be the subspace of

$\text{Conf}\big{(}(N\setminus
E_{m})\cup_{\partial\bar{E}_{m}}\big{(}\partial\bar{E}_{m}\times\left[0,\infty\right)\big{)}\big{)}\times\left[0,\infty\right)^{m}$

consisting of elements $(\mathcal{C},\lambda_{1},\ldots,\lambda_{m})$, with
$\mathcal{C}\in\text{Conf}\big{(}(N\setminus
E_{m})\cup_{\partial\bar{E}_{m}}\big{(}\partial\bar{E}_{m}\times\left[0,\infty\right)\big{)}\big{)}$
and $\lambda_{1},\ldots,\lambda_{m}\in\left[0,\infty\right)$, such that
$\mathcal{C}\in\text{Conf}\big{(}(N\setminus
E_{m})\cup_{\partial\bar{E}_{m}}\big{(}\bigsqcup_{j=1;\lambda_{j}\neq
0}^{m}S^{n-1}_{j}\times\left[0,\lambda_{j}\right)\big{)}\setminus\big{(}\bigsqcup_{i=1;\lambda_{i}=0}^{m}S^{n-1}_{i})\big{)}$.

Let

$\iota_{j}\colon S^{n-1}\times(0,\infty)\hookrightarrow
S^{n-1}_{j}\times(0,\infty)\subset\big{(}N\setminus
E_{m}\big{)}\cup_{\partial\bar{E}_{m}}\big{(}\partial\bar{E}_{m}\times\left[0,\infty\right)\big{)}$

be the inclusion under the identity map and let

$\text{Conf}(\iota_{j})\colon\text{Conf}(S^{n-1}_{j}\times(0,\infty))\to\text{Conf}\big{(}\big{(}N\setminus
E_{m}\big{)}\cup_{\partial\bar{E}_{m}}\big{(}\partial\bar{E}_{m}\times\left[0,\infty\right)\big{)}\big{)}$

denote the induced map of configuration spaces. The map

$\displaystyle m_{r}\colon\text{Mod-Conf}^{m}_{S^{n-1}}(N\setminus
E_{m})\times\big{(}\prod_{j=1}^{m}\text{MConf}(S^{n-1}_{j})\big{)}$
$\displaystyle\to\text{Mod-Conf}^{m}_{S^{n-1}}(N\setminus E_{m})$
$\displaystyle\big{(}(\mathcal{C}^{{}^{\prime}},\lambda_{1}^{{}^{\prime}},\ldots,\lambda_{m}^{{}^{\prime}}),\big{(}(\mathcal{C}_{1},\lambda_{1}),\ldots,(\mathcal{C}_{m},\lambda_{m})\big{)}\big{)}$
$\displaystyle\mapsto\big{(}\bigcup_{j=1}^{m}\text{Conf}(\iota_{j})\big{(}\text{sh}_{\displaystyle\lambda_{j}}(\mathcal{C}_{j})\big{)}\cup\mathcal{C}^{{}^{\prime}},\lambda_{1}^{{}^{\prime}}+\lambda_{1},\ldots,\lambda_{m}^{{}^{\prime}}+\lambda_{m}\big{)}$

gives the space $\text{Mod-Conf}^{m}_{S^{n-1}}(N\setminus E_{m})$ the
structure of a right module over the monoid
$\prod_{j=1}^{m}\text{MConf}(S^{n-1}_{j})$ (see Figure 15). Similarly, the map

$\displaystyle
m_{l}\colon\big{(}\prod_{j=1}^{m}\text{MConf}(S^{n-1}_{j})\big{)}\times\text{Mod-
Conf}^{m}_{S^{n-1}}(N\setminus E_{m})$ $\displaystyle\to\text{Mod-
Conf}^{m}_{S^{n-1}}(N\setminus E_{m})$
$\big{(}\big{(}(\mathcal{C}_{1},\lambda_{1}),\ldots,(\mathcal{C}_{m},\lambda_{m})\big{)},(\mathcal{C}^{{}^{\prime}},\lambda_{1}^{{}^{\prime}},\ldots,\lambda_{m}^{{}^{\prime}})\big{)}\mapsto\big{(}\bigcup_{j=1}^{m}\text{Conf}(\iota_{j})\big{(}\text{sh}_{\displaystyle\lambda_{j}^{{}^{\prime}}+\lambda_{j}}\text{sc}_{-1}(\mathcal{C}_{j})\big{)}\cup\mathcal{C}^{{}^{\prime}},\lambda_{1}+\lambda_{1}^{{}^{\prime}},\ldots,\lambda_{m}+\lambda_{m}^{{}^{\prime}}\big{)}$

gives the space $\text{Mod-Conf}^{m}_{S^{n-1}}(N\setminus E_{m})$ the
structure of a left module over the monoid
$\prod_{j=1}^{m}\text{MConf}(S^{n-1}_{j})$.

Figure 15. The right $\prod_{j=1}^{m}\text{MConf}(S^{n-1}_{j})$-module
structure of $\text{Mod-Conf}^{m}_{S^{n-1}}(N\setminus E_{m})$

######  Definition 4.46.

Let $M$ and $\bar{E}_{m}\subset M$ be as in Definition 4.44. Define
$\mathcal{X}^{m}_{\bullet}({\bar{D}^{n}},{\partial S^{n-1}},{M\setminus
E_{m}})$ to be the semi-simplicial space

$\mathcal{X}^{m}_{\bullet}({\bar{D}^{n}},{\partial S^{n-1}},{M\setminus
E_{m}})\colonequals B_{\bullet}\big{(}\prod_{j=1}^{m}\text{Mod-
Conf}_{S^{n-1}_{j}}(\bar{D}^{n}_{j}),\prod_{j=1}^{m}\text{MConf}(S^{n-1}_{j}),\text{Mod-
Conf}^{m}_{S^{n-1}}(M\setminus E_{m})\big{)}$

Given an element $(\mathcal{C},\lambda)$ in
$\text{MConf}(\bigsqcup_{j=1}^{m}S^{n-1}_{j})$ (or in $\text{Mod-
Conf}_{\partial\bar{E}_{m}}(N)$) let
$\mathcal{C}(j)\colonequals\mathcal{C}\cap\text{Conf}(S^{n-1}_{j}\times\left[0,\lambda\right))$
$\big{(}$respectively, let
$\mathcal{C}(j)\colonequals\mathcal{C}\cap\text{Conf}\big{(}\bar{D}^{n}_{j}\cup_{S^{n-1}_{j}}(S^{n-1}_{j}\times\left[0,\lambda\right))\big{)}\big{)}$
denote the underlying set of points of the configuration $\mathcal{C}$
contained in $S^{n-1}_{j}\times\left[0,\lambda\right)$ $\big{(}$respectively,
in
$(\bar{D}^{n}_{j}\cup_{S^{n-1}_{j}}(S^{n-1}_{j}\times\left[0,\lambda\right))\big{)}$.
The map

$\displaystyle\text{MConf}(\bigsqcup_{j=1}^{m}S^{n-1}_{j})$
$\displaystyle\to\prod_{j=1}^{m}\text{MConf}(S^{n-1}_{j})$
$\displaystyle(\mathcal{C},\lambda)$
$\displaystyle\mapsto((\mathcal{C}(1),\lambda),\ldots,(\mathcal{C}(m),\lambda))$

is an inclusion of topological monoids, so we can view
$\text{MConf}(\partial\bar{E}_{m})=\text{MConf}(\bigsqcup_{j=1}^{m}S^{n-1}_{j})$
as a submonoid of the monoid $\prod_{j=1}^{m}\text{MConf}(S^{n-1}_{j})$.
Similarly, the map

$\displaystyle\text{Mod-Conf}_{\partial\bar{E}_{m}}(\bar{E}_{m})$
$\displaystyle\to\prod_{j=1}^{m}\text{Mod-
Conf}_{S^{n-1}_{j}}(\bar{D}^{n}_{j})$ $\displaystyle(\mathcal{C},\lambda)$
$\displaystyle\mapsto((\mathcal{C}(1),\lambda),\ldots,(\mathcal{C}(m),\lambda))$

is a map of modules over the monoid
$\text{MConf}(\bigsqcup_{j=1}^{m}S^{n-1}_{j})$. In addition, the map

$\displaystyle\text{Mod-Conf}_{\partial\bar{E}_{m}}(M\setminus E_{m})$
$\displaystyle\to\text{Mod-Conf}^{m}_{S^{n-1}}(M\setminus E_{m})$
$\displaystyle(\mathcal{C},\lambda)$
$\displaystyle\mapsto(\mathcal{C},\lambda,\ldots,\lambda)$

is a map of modules over the monoid
$\text{MConf}(\bigsqcup_{j=1}^{m}S^{n-1}_{j})$. As a consequence, we have a
map of semi-simplicial spaces

$X_{\bullet}(\bar{E}_{m},\partial\bar{E}_{m},M\setminus
E_{m})\to\mathcal{X}^{m}_{\bullet}({\bar{D}^{n}},{\partial
S^{n-1}},{M\setminus E_{m}}).$

######  Lemma 4.47.

Let $M$ and $\bar{E}_{m}\subset M$ be as in Definition 4.44. The map

$\|X_{\bullet}(\bar{E}_{m},\partial\bar{E}_{m},M\setminus
E_{m})\|\to\|\mathcal{X}^{m}_{\bullet}({\bar{D}^{n}},{S^{n-1}},{M\setminus
E_{m}})\|$

is a weak homotopy equivalence.

###### Proof.

The spaces $\text{MConf}(\bigsqcup_{j=1}^{m}S^{n-1}_{j})$ and
$\prod_{j=1}^{m}\text{MConf}(S^{n-1}_{j})$ are homotopy equivalent to the
spaces $\prod_{j=1}^{m}\text{Conf}(S^{n-1}_{j}\times(0,1))$ by Lemma 4.7. The
spaces $\text{Mod-Conf}_{\partial\bar{E}_{m}}(\bar{E}_{m})$ and
$\prod_{j=1}^{m}\text{Mod-Conf}_{S^{n-1}_{j}}(\bar{D}^{n}_{j})$ are homotopy
equivalent to $\prod_{j=1}^{m}\text{Conf}(D^{n}_{j})$ and the space
$\text{Mod-Conf}_{\partial\bar{E}_{m}}(M\setminus E_{m})$ is homotopy
equivalent to $\text{Conf}(M\setminus E_{m})$, again by Lemma 4.7. By a
similar argument to the proof of Lemma 4.7, the space $\text{Mod-
Conf}^{m}_{S^{n-1}}(M\setminus E_{m})$ is homotopy equivalent to
$\text{Conf}(M\setminus E_{m})$. Therefore, the map

$X_{\bullet}(\bar{E}_{m},\partial\bar{E}_{m},M\setminus
E_{m})\to\mathcal{X}^{m}_{\bullet}({\bar{D}^{n}},{\partial
S^{n-1}},{M\setminus E_{m}})$

is a levelwise weak homotopy equivalence. The result immediately follows from
Theorem 4.10. ∎

As a consequence of Lemma 4.47, the space

$\|\mathcal{X}^{m}_{\bullet}({\bar{D}^{n}},{\partial S^{n-1}},{M\setminus
E_{m}})\|$

is weakly homotopy equivalent to $\text{Conf}(M)$. As a result, to define an
iterated mapping cone on $C_{*}(\text{Conf}(M))$, it suffices to define it on
$C_{*}(\|\mathcal{X}^{m}_{\bullet}({\bar{D}^{n}},{\partial
S^{n-1}},{M\setminus E_{m}})\|)$. First, we need to define a stabilization map
$t_{z,c}$ for $C_{*}(\|\mathcal{X}^{m}_{\bullet}({\bar{D}^{n}},{\partial
S^{n-1}},{M\setminus E_{m}})\|)$.

######  Definition 4.48.

Let $M$ and $\bar{E}_{m}\subset M$ be as in Definition 4.44. For
$j=1,\ldots,m$, let $Y_{j}$ denote the space $Y$. Let $R$ be a ring and let
$e\in H_{0}(\mathbb{R}^{n};R)$ be the class of a point. Suppose that
$z_{1},\ldots,z_{m}$ are classes in $H_{*}(\text{Conf}(\mathbb{R}^{n});R)$
such that $\left[z_{j},e\right]=0$ for all $j=1,\ldots,m$.

By abuse of notation, define

$t_{z_{j},c}\colon
C_{*}(\|B_{\bullet}(\prod_{j=1}^{m}Y_{j},\prod_{j=1}^{m}\text{MConf}(S^{n-1}_{j}),\text{Mod-
Conf}^{m}_{S^{n-1}}(M\setminus E_{m}))\|\to
C_{*}(\|\mathcal{X}^{m}_{\bullet}({\bar{D}^{n}},{\partial S^{n-1}},{M\setminus
E_{m}})\|)$

to be the stabilization map induced from the map $g_{z_{j}}\colon
C_{*}(Y_{j};R)\to C_{*}(\text{Mod-Conf}_{S^{n-1}_{j}}(\bar{D}^{n}_{j});R)$,
i.e. given chains

$\sigma_{j}\in C_{*}(Y_{j};R)\text{ for }j=1,\ldots,m,\tau\in
C_{*}(\prod_{i=1}^{p}(\prod_{j=1}^{m}\text{MConf}(S^{n-1}_{j}))),\text{ and
}\omega\in C_{*}(\text{Mod-Conf}^{m}_{S^{n-1}}(M\setminus E_{m})),$

let

$t_{z_{j},c}(\sigma_{1}\otimes\cdots\otimes\sigma_{m}\otimes\tau\otimes\omega)\colonequals\sigma_{1}\otimes\cdots\otimes\sigma_{j-1}\otimes
g_{z_{j}}(\sigma_{j})\otimes\sigma_{j+1}\cdots\otimes\sigma_{m}\otimes\tau\otimes\omega.$

Qualitatively, the stabilization map $t_{z_{j},c}$ can be described as
stabilizing in the $j$-th disk $D^{n}_{j}$. Given classes $z_{1},\ldots,z_{m}$
in $H_{*}(\text{Conf}(\mathbb{R}^{n});R)$ such that $\left[z_{j},e\right]=0$
for all $j=1,\ldots,m$, the stabilization maps
$t_{z_{1},c},\ldots,t_{z_{m},c}$ from Definition 4.48 commute with each other.
Therefore, we can define the following iterated mapping cone on
$C_{*}(\|\mathcal{X}^{m}_{\bullet}({\bar{D}^{n}},{\partial
S^{n-1}},{M\setminus E_{m}})\|;R)$ as follows.

######  Definition 4.49.

Let $M$ and $\bar{E}_{m}\subset M$ be as in Definition 4.44. Let $e\in
H_{0}(\text{Conf}_{1}(\mathbb{R}^{n}))$be the class of a point. Suppose that
$z_{1},\ldots,z_{m}$ are classes in $H_{*}(\text{Conf}(\mathbb{R}^{n}))$ such
that $\left[z_{j},e\right]=0$ for all $j=1,\ldots,m$. By abuse of notation,
define $\text{Cone}^{m}(M,(t_{z_{j},c}))$ to be the iterated mapping cone
$\text{Cone}^{m}(\|\mathcal{X}^{m}_{\bullet}({\bar{D}^{n}},{\partial
S^{n-1}},{M\setminus E_{m}})\|,(t_{z_{j},c}))$ associated to the collection
$(t_{z_{1},c},\ldots,t_{z_{m},c})$, where $t_{z_{j},c}$ is the stabilization
map from Definition 4.48.

We now describe the puncture resolution argument for proving stability for the
configuraiton space of a closed manifold.

######  Definition 4.50.

Given a space $X$, let

$\text{OConf}_{k}(X)\colonequals\\{(x_{1},\ldots,x_{k})\subset X^{k}\colon
x_{i}\neq x_{j}\text{ if }i\neq j\\}$

denote the _set_ of ordered configurations of $X$ of cardinality $k$. We do
not use use the topology on $\text{OConf}_{k}(X)$ as a subspace of $X^{k}$.

######  Definition 4.51.

Let $M$ and $\bar{E}_{m}\subset M$ be as in Definition 4.44. Define
$Z_{k,\bullet}(M,\bar{E}_{m})$ to be the semi-simplicial space whose
$i$-simplices are

$\displaystyle Z_{k,i}(M,\bar{E}_{m})\colonequals\bigsqcup_{(m_{0},\ldots
m_{i})\in\text{OConf}_{i+1}(M\setminus\bar{E}_{m})}\text{Conf}_{k}(M\setminus\\{m_{0},\ldots,m_{i}\\}).$

The inclusions $M\setminus\\{m_{0},\ldots,m_{i}\\}\hookrightarrow
M\setminus\\{m_{0},\ldots,\widehat{m_{j}},\ldots,m_{i}\\},$ for
$j=0,\ldots,i$, induce the face maps for $Z_{k,\bullet}(M,\bar{E}_{m})$.

The map $\epsilon\colon Z_{k,0}(M,\bar{E}_{m})\to\text{Conf}_{k}(M)$ obtained
by forgetting the ordered configuration $(m_{0})$ satisfies $\epsilon
d_{0}=\epsilon d_{1}$ so
$(Z_{k,\bullet}(M,\bar{E}_{m}),\text{Conf}_{k}(M),\epsilon)$ is an augmented
semi-simplicial space.

######  Lemma 4.52.

Let $M$ and $\bar{E}_{m}\subset M$ be as in Definition 4.44. Let
$(Z_{k,\bullet}(M,\bar{E}_{m}),\text{Conf}_{k}(M),\epsilon)$ be the augmented
semi-simplicial space from Definition 4.51. The augmentation map induces a
weak homotopy equivalence
$\|\epsilon\|\colon\|Z_{k,\bullet}(M,\bar{E}_{m})\|\to\text{Conf}_{k}(M)$.

###### Proof.

Our argument is the same as [KMT16, Proposition 5.8], so we just summarize it.
The map
$\|\epsilon\|\colon\|Z_{k,\bullet}(M,\bar{E}_{m})\|\to\text{Conf}_{k}(M)$ is a
Serre microfibration. The fiber of this map is contractible because it is
isomorphic to the complex of injective words for an infinite set
$M\setminus\bar{E}_{m}\setminus\\{m_{1},\ldots m_{k}\\}$, with
$m_{1},\ldots,m_{k}$ a finite set of points in $M\setminus\bar{E}_{m}$.
Therefore, by Lemma 4.27, the map $\|\epsilon\|$ is a weak homotopy
equivalence. ∎

######  Lemma 4.53.

Let $M$ and $\bar{E}_{m}\subset M$ be as in Definition 4.44. Let
$(Z_{k,\bullet}(M,\bar{E}_{m}),\text{Conf}_{k}(M),\epsilon)$ be the augmented
semi-simplicial space from Definition 4.51. Given a stabilization map
$s_{z}\colon C_{*}(\text{Conf}_{k}(M))\to C_{*+a}(\text{Conf}_{k+b}(M))$
induced from a class $z\in H_{a}(\text{Conf}_{b}(D^{n}))$, with
$D^{n}\subseteq\bar{D}_{j}^{n}$ for some $j$, there is a map

$s_{z,\bullet}\colon C_{*}(Z_{k,\bullet}(M,\bar{E}_{m}))\to
C_{*+a}(Z_{k+b,\bullet}(M,\bar{E}_{m}))$

of chain complexes of semi-simplicial spaces extending $s_{z}$ to
$C_{*}(Z_{k,\bullet}(M,\bar{E}_{m}))$.

###### Proof.

Let $(\epsilon_{i})_{*}\colon C_{*}(Z_{k,i}(M,\bar{E}_{m}))\to
C_{*}(\text{Conf}_{k}(M))$ denote the chain map obtained from the chain map
$\epsilon_{i}\colon Z_{k,i}(M,\bar{E}_{m})\to\text{Conf}_{k}(M)$ which forgets
the ordered configuration $(m_{0},\ldots,m_{i})$. We can represent
$C_{*}(Z_{k,i}(M,\bar{E}_{m}))$ by a pair $(\sigma,(m_{0},\ldots,m_{i}))$ with
$(m_{0},\ldots,m_{i})\in\text{OConf}_{i+1}(M\setminus\bar{E}_{m})$ and
$\sigma\in
C_{*}\big{(}\text{Conf}(M\setminus\\{m_{0},\ldots,m_{i}\\})\big{)}$. Since the
points $m_{0},\ldots,m_{i}$ are not in $\bar{E}_{m}$, the chain
$s_{z}\big{(}(\epsilon_{i})_{*}(\sigma)\big{)}\in
C_{*+a}(\text{Conf}_{k+b}(M))$ can be treated as a chain in
$\text{Conf}(M\setminus\\{m_{0},\ldots,m_{i}\\})$. Therefore,
$s_{z,i}(\sigma,(m_{0},\ldots,m_{i}))\colonequals(s_{z}\circ(\epsilon_{i})_{*}(\sigma),(m_{0},\ldots,m_{i}))$
is a chain in $C_{*+a}(Z_{k+b,i}(M,\bar{E}_{m})).$

To show that the maps $s_{z,i}$ assemble to give a map $s_{z,\bullet}\colon
C_{*}(Z_{k,\bullet}(M,\bar{E}_{m}))\to
C_{*+a}(Z_{k+b,\bullet}(M,\bar{E}_{m}))$ of chain complexes of semi-simplicial
spaces, we need to show that the following diagram of chain complexes

${C_{*}(Z_{k,i}(M,\bar{E}_{m}))}$${C_{*}(Z_{k,i-1}(M,\bar{E}_{m}))}$${C_{*+a}(Z_{k+b,i}(M,\bar{E}_{m}))}$${C_{*+a}(Z_{k+b,i-1}(M,\bar{E}_{m}))}$$\scriptstyle{s_{i,z}}$$\scriptstyle{d_{j}}$$\scriptstyle{s_{i-1,z}}$$\scriptstyle{d_{j}}$

commutes. Since for all $m_{0},\ldots,m_{i}\in M\setminus E_{m}$, the
following diagram commutes

${C_{*}\big{(}\text{Conf}(M\setminus\\{m_{0},\ldots,m_{i}\\})\big{)}}$${C_{*}\big{(}\text{Conf}(M\setminus\\{m_{0},\ldots,\widehat{m}_{j},\ldots,m_{i}\\})\big{)}}$${C_{*}\big{(}\text{Conf}(M\setminus\\{m_{0},\ldots,m_{i}\\})\big{)}}$${C_{*}\big{(}\text{Conf}(M\setminus\\{m_{0},\ldots,\widehat{m}_{j},\ldots,m_{i}\\})\big{)}}$$\scriptstyle{d_{j}}$$\scriptstyle{s_{i,z}}$$\scriptstyle{s_{i-1,z}}$$\scriptstyle{d_{j}}$

it follows that the original diagram commutes as well. ∎

Using the puncture resolution $Z_{k,\bullet}(M,\bar{E}_{m})$ for
$\text{Conf}_{k}(M)$, we can show that the stabilization map $t_{z,c}$ is an
isomorphism of (relative) homology groups in the same range where $t_{z}$ is
an isomorphism.

######  Proposition 4.54.

Suppose that $M$ is a closed connected surface. Let $p$ be an odd prime and
let $D(p,m,k)$ denote the constant from Proposition 2.6. The stabilization map

$t_{e^{p},c}\colon H_{i}(\text{Conf}_{k}(M);\mathbb{F}_{p})\to
H_{i}(\text{Conf}_{k+p}(M);\mathbb{F}_{p})$

is an isomorphism for $i<D(p,1,k)$ and a surjection for $i=D(p,1,k)$.

###### Proof.

Fix a closed disk $\bar{D}^{n}$ in $M$. Showing that the map
$t_{e^{p},c}\colon H_{i}(\text{Conf}_{k}(M);\mathbb{F}_{p})\to
H_{i}(\text{Conf}_{k+p}(M);\mathbb{F}_{p})$ is an isomorphism for $i<D(p,1,k)$
and a surjection for $i=D(p,1,k)$ is equivalent to showing that
$H_{i}(\text{Conf}_{k+p}(M),\text{Conf}_{k}(M);\mathbb{F}_{p})$ is zero for
$i\leq D(p,1,k)$. By definition of the stabilization map $t_{e^{p},c}$ and
Lemma 4.53, the stabilization map $t_{e^{p},c}$ extends to a map of chain
complexes

$(t_{e^{p},c})_{\bullet}\colon
C_{i}(Z_{k,\bullet}(M,\bar{D}^{n});\mathbb{F}_{p})\to
C_{i}(Z_{k+p,\bullet}(M,\bar{D}^{n});\mathbb{F}_{p}).$

We can apply a relative version of the geometric realization spectral sequence
to $Z_{k,\bullet}(M,\bar{D}^{n})$:

$E^{1}_{s,t}=H_{t}(Z_{k+p,s}(M,\bar{D}^{n}),Z_{k,s}(M,\bar{D}^{n});\mathbb{F}_{p})\Longrightarrow
H_{s+t}(\|Z_{k+p,s}(M,\bar{D}^{n})\|,\|Z_{k,s}(M,\bar{D}^{n})\|;\mathbb{F}_{p}).$

By Lemma 4.52, the space $\|Z_{k,\bullet}(M,\bar{D}^{n})\|$ is weakly homotopy
equivalent to $\text{Conf}_{k}(M)$. Therefore, to show that
$H_{i}(\text{Conf}_{k+p}(M),\text{Conf}_{k}(M);\mathbb{F}_{p})$ vanishes for
$i\leq D(p,1,k)$, it suffices to show that the relative homology
$H_{t}(Z_{k+p,s}(M,\bar{D}^{n}),Z_{k,s}(M,\bar{D}^{n});\mathbb{F}_{p})$
vanishes for $s+t=i\leq D(p,1,k)$.

Since the manifolds $\text{Conf}_{k}(M\setminus\\{m_{0},\ldots,m_{s}\\})$
appearing in $Z_{k,s}(M,\bar{D}^{n})$ are open for all nonnegative $s$, the
chain map

$(t_{e^{p},c})_{s}\colon C_{i}(Z_{k,s}(M,\bar{D}^{n});\mathbb{F}_{p})\to
C_{i}(Z_{k+p,s}(M,\bar{D}^{n});\mathbb{F}_{p})$

is chain homotopy equivalent to the map

$(t_{e^{p}})_{s}\colon C_{i}(Z_{k,s}(M,\bar{D}^{n});\mathbb{F}_{p})\to
C_{i}(Z_{k+p,s}(M,\bar{D}^{n});\mathbb{F}_{p})$

by Proposition 4.41. Therefore, the relative homology

$H_{t}(Z_{k+p,s}(M,\bar{D}^{n}),Z_{k,s}(M,\bar{D}^{n});\mathbb{F}_{p})=H_{t}(\text{Cone}((t_{e^{p},c})_{s}\colon
Z_{k,s}(M,\bar{D}^{n})\to Z_{k+p,s}(M,\bar{D}^{n}));\mathbb{F}_{p})$

is isomorphic to $H_{t}(\text{Cone}((t_{e^{p}})_{s}\colon
Z_{k,s}(M,\bar{D}^{n})\to Z_{k+p,s}(M,\bar{D}^{n}));\mathbb{F}_{p})$, which
vanishes for $s+t\leq D(p,1,k)$ by Proposition 3.6. ∎

By applying the argument of Proposition 4.54, we can obtain additional
stability results for the configuration spaces of a closed manifold and prove
Theorem 1.2 in the case that homology is with $\mathbb{F}_{p}$ coefficients.

######  Proposition 4.55.

Suppose that $M$ is a closed connected manifold of dimension $n\geq 2$. Let
$D(p,m,k)$ denote the constant from Proposition 2.6. Let $e\in
H_{0}(\text{Conf}_{1}(\mathbb{R}^{n}))$ be the class of a point.

  1. (1)

The map

$t_{e,c}\colon H_{i}(\text{Conf}_{k}(M);\mathbb{F}_{2})\to
H_{i}(\text{Conf}_{k+1}(M);\mathbb{F}_{2})$

is an isomorphism for $i<k/2$ and a surjection for $i=k/2$.

  2. (2)

Suppose that $n$ is $2$. Let $x_{j}\in
H_{*}(\text{Conf}(\mathbb{R}^{2});\mathbb{F}_{2})$ denote the classes from
Theorem 2.3. Let $\text{Cone}_{k}^{m}(M,(t_{\displaystyle x_{j},c}))$ denote
the m-th iterated mapping cone associated to the collection
$(t_{\displaystyle e^{p},c},t_{\displaystyle x_{1},c},\ldots,t_{\displaystyle
x_{m-1},c})$ as defined in Definition 4.49.

The map

$t_{\displaystyle x_{m},c}\colon
H_{i}\big{(}\text{Cone}_{k}^{m}(M,(t_{\displaystyle
x_{j},c}));\mathbb{F}_{2}\big{)}\to
H_{i+2^{m}-1}\big{(}\text{Cone}_{k+2^{m}}^{m}(M,(t_{\displaystyle
x_{j},c}));\mathbb{F}_{2}\big{)}$

is an isomorphism for $i<D(2,m+1,k)$ and a surjection for $i=D(2,m+1,k)$.

  3. (3)

Suppose that $n$ is $2$ and that $p$ is an odd prime. Let $y_{j}\in
H_{*}(\text{Conf}(\mathbb{R}^{2});\mathbb{F}_{p})$ denote the classes from
Theorem 2.3. Let $\text{Cone}_{k}^{m}(M,(t_{\displaystyle y_{j},c}))$ denote
the m-th iterated mapping cone associated to the collection $(t_{\displaystyle
e^{p},c},t_{\displaystyle y_{1},c},\ldots,t_{\displaystyle y_{m-1},c})$.

The map

$t_{\displaystyle y_{m},c}\colon
H_{i}\big{(}\text{Cone}_{k}^{m}(M,(t_{\displaystyle
y_{j},c}));\mathbb{F}_{p}\big{)}\to
H_{i+2p^{m}-2}\big{(}\text{Cone}_{k+2p^{m}}^{m}(M,(t_{\displaystyle
y_{j},c}));\mathbb{F}_{p}\big{)}$

is an isomorphism for $i<D(p,m+1,k)$ and a surjection for $i=D(p,m+1,k)$.

  4. (4)

Suppose that $n$ is odd. The map

$t_{e,c}\colon H_{i}(\text{Conf}_{k}(M);\mathbb{Z})\to
H_{i}(\text{Conf}_{k+1}(M);\mathbb{Z})$

is an isomorphism for $i<k/2$ and a surjection for $i=k/2$.

  5. (5)

Suppose that that $n$ is odd. Suppose that $R$ is $\mathbb{Q}$ or
$\mathbb{F}_{p}$ with $p>2$. The map

$t_{e,c}\colon H_{i}(\text{Conf}_{k}(M);R)\to H_{i}(\text{Conf}_{k+1}(M);R)$

is an isomorphism for $i<k$ and a surjection for $i=k$.

  6. (6)

Suppose that $n$ is greater than $2$. Let $\omega_{j}\in
H_{*}(\text{Conf}(\mathbb{R}^{n});\mathbb{F}_{2})$ denote the classes from
Proposition 3.16. Let $\text{Cone}_{k}^{m}(M,(t_{\displaystyle\omega_{j},c}))$
denote the m-th iterated mapping cone associated to the collection
$(t_{\displaystyle
e,c},t_{\displaystyle\omega_{1},c},\ldots,t_{\displaystyle\omega_{m-1},c})$.

The map

$t_{\displaystyle\omega_{m},c}\colon
H_{i}\big{(}\text{Cone}_{k}^{m}(M,(t_{\displaystyle\omega_{j},c}));\mathbb{F}_{2}\big{)}\to
H_{i+2^{m}-1}\big{(}\text{Cone}_{k+2^{m}}^{m}(M,(t_{\displaystyle\omega_{j},c}));\mathbb{F}_{2}\big{)}$

is an isomorphism for $i<D(2,m+1,k)$ and a surjection for $i=D(2,m+1,k)$. is
an isomorphism for $i<D(2,m+1,k)$ and a surjection for $i=D(2,m+1,k)$.

  7. (7)

Suppose that $n$ is even and greater than $2$ and that $p$ is an odd prime.
The map

$t_{e^{p},c}\colon H_{i}(\text{Conf}_{k}(M);\mathbb{F}_{p})\to
H_{i}(\text{Conf}_{k+p}(M);\mathbb{F}_{p})$

is an isomorphism for $i<k$ and a surjection for $i=k$.

###### Proof.

The argument is exactly the same as the proof of Proposition 4.54. The only
difference is that to prove Parts 2, 3, and 6, we replace
$Z_{k,\bullet}(M,\bar{D}^{n})$ with $Z_{k,\bullet}(M,\bar{E}_{m})$. In
addition, we also implicitly use the fact that for an open manifold $M$, the
stabilization map

$t_{\displaystyle y_{m},c}\colon
C_{i}\big{(}\text{Cone}_{k}^{m}(M,(t_{\displaystyle
y_{j},c}));\mathbb{F}_{p}\big{)}\to
C_{i+2p^{m}-2}\big{(}\text{Cone}_{k+2p^{m}}^{m}(M,(t_{\displaystyle
y_{j},c}));\mathbb{F}_{p}\big{)}$

is chain homotopy equivalent to the map

$t_{\displaystyle y_{m}}\colon
H_{i}\big{(}\text{Cone}_{k}^{m}(M,(t_{\displaystyle
y_{j},c}));\mathbb{F}_{p}\big{)}\to
H_{i+2p^{m}-2}\big{(}\text{Cone}_{k+2p^{m}}^{m}(M,(t_{\displaystyle
y_{j}}));\mathbb{F}_{p}\big{)}$

by a stronger version of Proposition 4.41 for iterated mapping cones (the
proof of this stronger version is similar to the proof of Proposition 4.41, so
we omit it). The range for Part 1 follows from Proposition 3.6 and Theorem
3.18. The ranges for Parts 2 and 3 follow from Corollary 3.13. The ranges for
Parts 4, 5, and 7 follow from Theorem 3.18. The ranges for Part 6 follow from
Proposition 3.17. ∎

When the dimension of the manifold is odd, one can apply Parts 5 and 6 to
obtain the following secondary homological stability result for the integral
relative homology $H_{*}(\text{Conf}_{k}(M),\text{Conf}_{k-1}(M);\mathbb{Z})$,
which proves Theorem 1.2 in the case that homology is integral homology,
exactly like Proposition 3.19

######  Corollary 4.56.

Suppose that $M$ is a closed connected manifold of odd dimension greater than
$2$. The map

$t_{\displaystyle\omega_{1},c}\colon
C_{i}(\text{Conf}_{k}(M),\text{Conf}_{k-1}(M);\mathbb{Z})\to
C_{i+1}(\text{Conf}_{k+2}(M),\text{Conf}_{k+1}(M);\mathbb{Z})$

induces an isomorphism for $i<D(2,2,k)$ and a surjection for $i=D(2,2,k)$.

We now prove Theorem 1.2.

###### Proof of Theorem 1.2.

Theorem 1.1 immediately follows from Proposition 4.55 and Corollary 4.56 in
the case $m=1$. ∎

######  Remark 4.57.

If $M$ is a closed connected manifold of even dimension, we do not know if
there is a similar statement to Corollary 4.56. In this case, the issue is
that we do not know how to define relative homology
$H_{*}(\text{Conf}_{k}(M),\text{Conf}_{k-1}(M);\mathbb{Z})$. When $M$ is a
closed connected manifold of even dimension, we can only consider
$H_{*}(\text{Conf}_{k}(M),\text{Conf}_{k-p}(M);\mathbb{F}_{p})$, not
$H_{*}(\text{Conf}_{k}(M),\text{Conf}_{k-1}(M);\mathbb{F}_{p})$, and there is
no analogue of $H_{*}(\text{Conf}_{k}(M),\text{Conf}_{k-p}(M);\mathbb{F}_{p})$
for integral homology.

It might still be possible to obtain a secondary stability statement result by
working in the setting of divided power algebras. Let $D$ denote the divided
power algebra over $\mathbb{Z}$ in a single variable. Suppose that $M$ were an
open manifold. The cochain complex $C^{*}(\text{Conf}(M);\mathbb{Z})$ can be
viewed as a $\mathbb{Z}[x]$-module, where the variable $x$ acts on
$C^{*}(\text{Conf}(M);\mathbb{Z})$ by stabilization, and we have that

$\text{Tor}_{i}^{\mathbb{Z}[x]}(C^{*}(\text{Conf}(M);\mathbb{Z}),\mathbb{Z})\cong\bigoplus_{k}H_{i}(\text{Conf}_{k}(M),\text{Conf}_{k-1}(M);\mathbb{Z}).$

When $M$ is a closed manifold, $\mathbb{Z}[x]$ no longer acts on
$C^{*}(\text{Conf}(M);\mathbb{Z})$, so instead consider the hyper Tor groups
$\text{Tor}_{i}^{D}(C^{*}(\text{Conf}(M);\mathbb{Z}),\mathbb{Z})$. These hyper
Tor groups are an analogue of the relative homology
$H_{i}(\text{Conf}_{k}(M),\text{Conf}_{k-1}(M);\mathbb{Z})$. We conjecture
that the hyper Tor groups
$\text{Tor}_{*}^{D}(C^{*}(\text{Conf}(M);\mathbb{Z}),\mathbb{Z})$ stabilize
for a closed manifold $M$. We also ask if there is an analogue of the
secondary stabilization map

$t_{2}\colon C_{*}(\text{Conf}_{k}(M),\text{Conf}_{k-1}(M);\mathbb{Z})\to
C_{*+1}(\text{Conf}_{k+2}(M),\text{Conf}_{k+1}(M);\mathbb{Z})$

for the hyper Tor groups
$\text{Tor}_{*}^{D}(C^{*}(\text{Conf}(M);\mathbb{Z}),\mathbb{Z})$.

######  Remark 4.58.

In the case that $M$ is a manifold of finite type and homology is with
coefficients over the field $\mathbb{F}_{2}$, the secondary stability results
for this case can also be deduced from [BCT89]. Bödigheimer–Cohen–Taylor show
that the homology $H_{*}(\text{Conf}(M);\mathbb{F}_{2})$ can be explicitly
calculated from a graded vector space $\mathscr{C}(H_{*}(M;\mathbb{F}_{2}))$
depending only on the homology $H_{*}(M;\mathbb{F}_{2})$ and the dimension of
$M$ (see [BCT89, Theorem A]). They give an explicit filtration of
$\mathscr{C}(H_{*}(M;\mathbb{F}_{2}))$ which calculates
$H_{*}(\text{Conf}_{k}(M);\mathbb{F}_{2})$ (see [BCT89, Theorem B and Section
4]).

As a result, we can use the vector space
$\mathscr{C}(H_{*}(M;\mathbb{F}_{2}))$ to deduce secondary stability in this
case. Since the stabilization map $t_{1}\colon
H_{*}(\text{Conf}_{k}(M);\mathbb{F}_{2})\to
H_{*}(\text{Conf}_{k+1}(M);\mathbb{F}_{2})$ is injective when $M$ is an open
manifold, the relative homology
$H_{*}(\text{Conf}_{k}(M),\text{Conf}_{k-1}(M);\mathbb{F}_{2})$ is the
cokernel of this stabilization map. Therefore, secondary homological stability
can almost certainly be deduced from the vector space
$\mathscr{C}(H_{*}(M;\mathbb{F}_{2}))$ in this case. The only hard part is
checking that the obvious guess for the secondary stabilization map on
$\mathscr{C}(H_{*}(M;\mathbb{F}_{2}))$ agrees with the secondary stabilization
map $t_{2}\colon
H_{*}(\text{Conf}_{k}(M),\text{Conf}_{k-1}(M);\mathbb{F}_{2})\to
H_{*+1}(\text{Conf}_{k+2}(M),\text{Conf}_{k+1}(M);\mathbb{F}_{2})$.

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# Overlap Cycles for Permutations:
Necessary and Sufficient Conditions

Victoria Horan
Air Force Research Laboratory
Rome, NY 13441 USA<EMAIL_ADDRESS>
###### Abstract

Universal cycles are generalizations of de Bruijn cycles and Gray codes that
were introduced originally by Chung, Diaconis, and Graham in 1992. They have
been developed by many authors since, for various combinatorial objects such
as strings, subsets, permutations, partitions, vector spaces, and designs. One
generalization of universal cycles, which require almost complete overlap of
consecutive words, is $s$-overlap cycles, which relax such a constraint. In
this paper we study permutations and some closely related class of strings,
namely juggling sequences and functions. We prove the existence of $s$-overlap
cycles for these objects, as they do not always lend themselves to the
universal cycle structure.

##  1 Introduction

††Approved for public release; distribution unlimited: 88ABW-2013-4003, 9 Sep
13

Listing structures for combinatorial objects are quickly becoming useful in
more and more interesting applications. Gray codes, first defined in 1947 by
Frank Gray [6], are used in many different surprising places from position
encoders [11] to genetic algorithms [4]. More recently universal cycles are
being used in areas such as rank modulation for multilevel flash memories
[12], however overlap cycles are still being explored and have the potential
to be useful in many applications.

An $s$-overlap cycle, or $s$-ocycle, is an ordering of a set of objects
$\mathcal{C}$, each represented as a string of length $n$. The ordering
requires that object $b=b_{0}b_{1}\ldots b_{n-1}$ follow object
$a=a_{0}a_{1}\ldots a_{n-1}$ only if $a_{n-s}a_{n-s+1}\ldots
a_{n-1}=b_{0}b_{1}\ldots b_{s-1}$. Ocycles were introduced by Godbole,
Knisley, and Norwood in 2010 [5]. Universal cycles, or ucycles are
$(n-1)$-ocycles and were originally introduced in 1992 by Chung, Diaconis, and
Graham [2].

To find $s$-ocycles and ucycles on a set of strings, most proofs employ the
same method. For a given string $X=x_{1}x_{2}\ldots x_{n}$, let
$X^{s-}=x_{1}x_{2}\ldots x_{s}$ denote the $s$-prefix of $X$ and
$X^{s+}=x_{n-s}x_{n-s+1}\ldots x_{n}$ denote the $s$-suffix of $X$. The first
step in the proof is to construct the transition digraph for the set of
strings as follows. Vertices represent $s$-prefixes and $s$-suffixes of
strings (the overlaps), while each edge represents a string, traveling from
its $s$-prefix to its $s$-suffix. Note that the transition digraph is a
directed multigraph in which an Euler tour (a closed walk in which every edge
is traversed exactly once) corresponds bijectively to an $s$-ocycle. To prove
the existence of an Euler tour, we use the following well-known theorem.

######  Theorem 1.1.

_([13], p. 60)_ A directed graph $G$ is eulerian if and only if it is both
balanced and weakly connected.

In this paper, we consider using the ocycle listing structure for
permutations, as well as functions and juggling sequences. We represent
permutations of an $n$-set $\\{0,1,\ldots,n-1\\}$ as a string
$\Pi=\pi_{0}\pi_{1}\ldots\pi_{n-1}$, where the functional representation is
used, i.e. $\pi(0)=\pi_{0}$. Closely related to permutations, juggling
sequences have been an active research area since the 1980’s [1]. These
sequences are used to determine what patterns a fixed set of balls can be
juggled in, where only one ball may be thrown at a time.

######  Definition 1.2.

_[3]_ A juggling sequence is a string $T=t_{0}t_{1}\ldots t_{n-1}$ where each
$t_{i}\geq 0$ such that

$\left|\\{i+t_{i}\pmod{n}\mid 0\leq i\leq n-1\\}\right|=n.$

This sequence illustrates that at time $i$, we should throw a ball high enough
that it is in the air for $t_{i}$ beats, or to height $t_{i}$. The number of
balls used in a given juggling sequence $T=t_{0}t_{1}\ldots t_{n-1}$ is given
by

$b=\frac{1}{n}\sum_{i=0}^{n-1}t_{i}.$

The period of a juggling sequence $t_{0}t_{1}\ldots t_{n-1}$ is $n$, the
length of the string.

An alternative definition considers the corresponding permutation sequence for
a juggling sequence. Given a string $T=t_{0}t_{1}\ldots t_{n-1}$, the
permutation sequence is the string $\Pi_{T}=\pi_{0}\pi_{1}\ldots\pi_{n-1}$
where $\pi_{i}=t_{i}+i\pmod{n}$. Then $\Pi_{T}$ is a permutation of the
$n$-set $\\{0,1,\ldots,n-1\\}$ if and only if $T$ is a valid juggling
sequence. From this definition it is clear that there is a very close
relationship between juggling sequences and permutations, which is further
illustrated by their similar results explored in this paper. The permutation
sequence clarifies the point that a juggler cannot catch or throw two objects
simultaneously.

For example, when $n=3$ and $b=2$ the pattern $015$ is valid. We can see this
by drawing the juggling diagram, shown in Figure 1. However, the sequence
$105$ is not valid, as shown in Figure 2. In this second example, note that on
the second beat we are required two catch two balls simultaneously - an
operation that is not allowed.

015015015 Figure 1: Juggling Diagram for Sequence $015$ 105105105 Figure 2:
Juggling Diagram for Sequence $105$

In [3], Chung and Graham show that it is not always possible to find a single
universal cycle that contains all juggling sequences of period $n$ and at most
$b$ balls. However they do prove that we can use several disjoint universal
cycles to cover each sequence exactly once. This result is closely related to
the long studied problem of universal cycles for permutations. Since ucycles
for permutations are not possible using the standard representation, a
modification of the representation can often provide an effective solution [9,
10].

An alternative that has only recently been explored is to modify the listing
structure rather than the object representation. By utilizing $s$-ocycles
rather than ucycles, partial results for permutations have been obtained in
[7], however we obtain a more complete solution in this paper. In Section 2 we
establish a complete result on the existence of $s$-ocycles for permutations,
and in Section 3 we explore similar results for functions and juggling
sequences.

##  2 Permutations

We begin with a general lemma that will be used for both permutations and
juggling sequences. For a string $X=x_{0}x_{1}\ldots x_{n-1}$, define the
rotation function as follows. A rotation by $s$ is given by:

$\rho^{s}(X)=x_{s}x_{s+1}\ldots x_{n-1}x_{0}x_{1}\ldots x_{s-1}.$

The following lemma shows that for a string of length $n$, rotations by $s$
partition the string into blocks of length gcd$(n,s)$, and we can always
perform repeated $s$-rotations to start $X$ with any block desired.

######  Lemma 2.1.

Let $n,s\in\mathbb{Z}^{+}$ with $1\leq s\leq n-1$ and gcd$(n,s)=d$. Consider a
string $X=x_{0}x_{1}\ldots x_{n-1}$, also written as $X=Y_{0}Y_{1}\ldots
Y_{m-1}$ where $n=md$ and $Y_{i}=x_{id}x_{id+1}\ldots x_{id+d-1}$. Then for
any $i\in\\{0,1,\ldots,m-1\\}$ there is some $j\in\\{0,1,\ldots,m-1\\}$ such
that:

$Y_{i}Y_{i+1}\ldots Y_{m-1}Y_{0}Y_{1}\ldots Y_{i-1}=\rho^{js}(X).$

###### Proof.

Suppose that $s=kd$. Then each rotation $\rho^{s}(X)$ advances through $k$
blocks of $X$. Thus if we can show that gcd$(m,k)=1$, we are done. However we
note that gcd$(n,s)=d$ implies that there are integers $p,q$ such that
$pn+qs=d$. This implies the following.

$\displaystyle pn+qs$ $\displaystyle=$ $\displaystyle d$ $\displaystyle
pmd+qkd$ $\displaystyle=$ $\displaystyle d$ $\displaystyle pm+qk$
$\displaystyle=$ $\displaystyle 1$

Thus the same integers $p,q$ provide confirmation that gcd$(m,k)=1$.

∎

######  Theorem 2.2.

Let $n,s\in\mathbb{Z}^{+}$ with $1\leq s\leq n-1$ and let $M$ be a multiset of
size $n$. There exists an $s$-ocycle on permutations of $M$ if and only if
$n-s>\hbox{gcd}(n,s)$.

###### Proof.

Define gcd$(n,s)=d$, and suppose that $n-s=kd>d$. Construct the transition
digraph $D$ with vertices representing $s$-permutations of $M$ and edges
representing permutations of $M$. We will show that this graph is balanced and
connected, and hence eulerian. Recall that an Euler tour in this graph
corresponds to an $s$-ocycle for permutations of $M$.

Balanced:

Let $X^{s-}=x_{0}x_{1}\ldots x_{s-1}$ be an arbitrary vertex in the transition
graph. For each possible out-edge corresponding to an $n$-permutation
$X=x_{0}x_{1}\ldots x_{n-1}$, we also have the in-edge
$X^{\prime}=x_{s}x_{s+1}\ldots x_{n-1}x_{0}x_{1}\ldots x_{s-1}$. Thus there is
a one-to-one correspondence between in- and out-edges at each vertex, and so D
is balanced.

Connected:

Consider an arbitrary vertex $X^{s-}=x_{0}x_{1}\ldots x_{s-1}$, which is an
$s$-prefix of some permutation $X=x_{0}x_{1}\ldots x_{n-1}$. We will consider
a partition of $X$ into blocks of size $d$, i.e. $X=X_{0}X_{1}\ldots X_{m-1}$
where $X_{i}=x_{id}x_{id+1}\ldots x_{(i+1)d-1}$. Our goal is to show that we
can permute elements in any $k$ consecutive blocks $X_{i}X_{i+1}\ldots
X_{i+k-1}$, which illustrates that any adjacent transpositions are possible,
and hence we can reach all permutations.

In D, rotations of $X$ correspond to the following cycle, where subscripts are
computed modulo $n$.

$\displaystyle x_{0}x_{1}\ldots x_{s-1}$ $\displaystyle\rightarrow$
$\displaystyle x_{n-s}x_{n-s+1}\ldots x_{n-1}$ $\displaystyle\rightarrow$
$\displaystyle x_{n-2s}x_{n-2s+1}\ldots x_{n-s-1}$ $\displaystyle\rightarrow$
$\displaystyle x_{n-3s}x_{n-3s+1}\ldots x_{n-2s-1}$ $\displaystyle\vdots$
$\displaystyle\rightarrow$ $\displaystyle x_{n-is}x_{n-is+1}\ldots
x_{n-(i-1)s-1}$ $\displaystyle\vdots$ $\displaystyle\rightarrow$
$\displaystyle x_{0}x_{1}\ldots x_{s-1}$

Clearly from the vertex $X^{s-}$ we can permute all elements in
$\\{x_{s},x_{s+1},\ldots,x_{n-1}\\}$ to determine an out-edge, and at any
point we can permute the $(n-s)$-suffix of a permutation. We would like to
show that, through rotations, the $(n-s)$-suffix may contain any $k$
consecutive blocks from $\\{X_{0},X_{1},\ldots,X_{m-1}\\}$. Consider the
vertex $x_{n-is}x_{n-is+1}\ldots x_{n-(i-1)s-1}$. How does the starting index
$n-is$ correspond to $d$?

$\displaystyle n-is$ $\displaystyle\equiv$ $\displaystyle in-is\pmod{n}$
$\displaystyle\equiv$ $\displaystyle i(n-s)\pmod{n}$ $\displaystyle\equiv$
$\displaystyle ikd\pmod{n}$

Thus the vertices in this rotation cycle always begin with a multiple of $kd$.
Our final step is to show that any multiple of $d$, say $id$, may be written
as a multiple of $kd$ modulo $n$, which is done by Lemma 3.8. Then we can
rotate to start with any block desired, which is equivalent to pushing any $k$
consecutive blocks to the $(n-s)$-suffix.

To summarize, we can perform adjacent transpositions $x_{i}\leftrightarrow
x_{i+1}$ within $X^{s-}$ by rotating $X$ until $x_{i}$ and $x_{i+1}$ fall into
the $k$ blocks in the $(n-s)$-suffix and then transposing. Finally, by
continuing along through rotations we will arrive at the vertex
$x_{0}x_{1}\ldots x_{i-1}x_{i+1}x_{i}x_{i+2}\ldots x_{s-1}$. Thus adjacent
transpositions are always connected, hence all permutations can be reached.

For the converse, suppose that $n-s=d=\hbox{gcd}(n,s)$. In this case,
rotations of the permutation $X=x_{0}x_{1}\ldots x_{n-1}$ provide an
$(n-s)$-suffix of length $d$ – just one block from $X=X_{0}X_{1}\ldots
X_{m-1}$. Thus we can always permute elements within each block, however the
cyclic order of the blocks is fixed and we can perform the swap
$x_{i}\leftrightarrow x_{j}$ if and only if $x_{i}$ and $x_{j}$ are in the
same block. Thus we are able to permute elements within blocks, but not
permute blocks (only rotate the block order). Hence the transition digraph
connects only permutations with the same block order (rotations allowed).
Permutations with block order that are not simple rotations are not connected,
so the graph is disconnected and no Euler tour exists. ∎

Note that Theorem 2.2 agrees with the following well-known fact about
universal cycles, or $(n-1)$-ocycles.

######  Corollary 2.3.

There is no universal cycle for permutations of $M$.

###### Proof.

In this case, we apply Theorem 2.2 with $s=n-1$. Then gcd$(n,s)=1=n-s$, so no
$(n-1)$-ocycle exists. ∎

##  3 Related Objects

###  3.1 Functions

Many times injective, surjective, and bijective functions are represented by
permutations. We have the following facts about functions and theorems
corresponding to their alternate representations.

######  Fact 3.1.

An injective function $f:[k]\rightarrow[n]$ may be represented by
$x_{1}x_{2}\ldots x_{k}$, the $k$-permutation of $[n]$ defined so that
$x_{i}=f(i)$.

######  Theorem 3.2.

_[7]_ For all $n,s,k\in\mathbb{Z}^{+}$ with $1\leq s<k<n$, there is an
$s$-ocycle for $k$-permutations of $[n]$.

######  Fact 3.3.

A surjective function $f:[n]\rightarrow[h]$ may be represented by
$x_{1}x_{2}\ldots x_{n}$, the string with ground set $[h]$ defined so that
$x_{i}=f(i)$.

In [8], it is shown that surjective functions are also represented by weak
orders of $[n]$ with height $h-1$. This observation is followed by the
following theorem.

######  Theorem 3.4.

_[8]_ For all $n,s,h\in\mathbb{Z}^{+}$ with $1\leq s\leq n-2$, gcd$(s,n)=1$,
and $0\leq h\leq n-1$, there is an $s$-ocycle for $\mathcal{W}(n,h)$.

We are able to improve this theorem as follows.

######  Theorem 3.5.

For all $n,s,h\in\mathbb{Z}^{+}$ with $1\leq s\leq n-2$ and $h\leq n-1$ there
is an $s$-ocycle for strings with ground set $[h]$.

###### Proof.

We will show that the corresponding transition graph is eulerian.

Balanced:

Consider a vertex $X^{s-}=x_{1}x_{2}\ldots x_{s}$. $X^{s-}$ is an $s$-prefix
of the string $X=x_{1}x_{2}\ldots x_{n}$ where $X$ has ground set $[h]$. Since
$x_{s+1}x_{s+2}\ldots x_{n}x_{1}x_{2}\ldots x_{s}$ is also a string with
ground set $[h]$, there is a bijection between in- and out-edges at $X^{s-}$.
Hence the graph is balanced.

Connected:

Define the minimum vertex $V^{s-}$ to be the $s$-prefix of the permutation
$V=12\cdots mm\cdots m$. Let $X=x_{1}x_{2}\ldots x_{n}$ be an arbitrary
multiset permutation with ground set $[h]$, and let $X^{s-}$ be the
$s$-prefix. We will show a path from $X^{s-}$ to $V^{s-}$ exists in the
transition graph.

Compare $X^{s-}$ and $V^{s-}$, and let $i$ be the least index such that
$X^{s-}(i)\neq V^{s-}(i)$. Note that since $h\leq n-1$, some element from
$[h]$ must appear at least twice in $X$. We will refer to any element
appearing more than once as a duplicate. We have two cases.

  1. 1.

If the letter $x_{i}\in[h]$ appears twice in $X$:

Let $d=\hbox{gcd}(n,s)$, and rotate $X$ until the $d$-block containing $x_{i}$
is first. If we rotate $X$ again by following an out-edge in the graph, we
have arrived at a vertex $A$ representing an $s$-substring of $X$ without
$x_{i}$. Since $x_{i}$ also appears elsewhere in $X$, we can follow
(backwards) the in-edge that is identical to $A$ except with $x_{i}$ replaced
by $v_{i}$. Now we are at an edge corresponding to an $s$-substring of
$x_{1}x_{2}\ldots x_{i-1}v_{i}x_{i+1}\ldots x_{n}$, so we are one step closer
to the minimum vertex.

  2. 2.

If the letter $x_{i}\in[h]$ appears exactly once in $X$:

Since $x_{i}$ is not a duplicate in $X$, some other letter $x_{j}\in[h]$ is a
duplicate. In this case, we proceed as in case 1 to replace $x_{j}$ with the
letter $x_{i}$. Then $x_{i}$ is a duplicate so we may follow case 1 again to
replace $x_{i}$ with $v_{i}$. At this point we have moved closer to the
minimum vertex.

Continuing until we have transformed $X^{s-}$ to $V^{s-}$ produces a path from
$X^{s-}$ to the minimum vertex. Hence the graph is connected.

As the transition graph is balanced and connected, it is eulerian by Theorem
1.1. ∎

######  Fact 3.6.

A bijective function $f:[n]\rightarrow[n]$ may be represented by
$x_{1}x_{2}\ldots x_{n}$, the permutation of $[n]$ defined so that
$x_{i}=f(i)$.

######  Theorem 3.7.

Let $n,s\in\mathbb{Z}^{+}$ with $1\leq s\leq n-1$. There exists an $s$-ocycle
on permutations of $[n]$ if and only if $n-s>\hbox{gcd}(n,s)$.

###### Proof.

Use Theorem 2.2 with $M=[n]$. ∎

###  3.2 Juggling Sequences

We begin with some lemmas that will help us to prove our main result.

######  Lemma 3.8.

Let $R=r_{0}r_{1}\ldots r_{n-1}$ be a string, and let $1\leq s\leq n$. Then
$R^{\prime}=\rho^{s}(R)=r_{0-s}r_{1-s}\ldots r_{n-1-s}$ (where addition is
modulo $n$) is a valid juggling sequence if and only if $R$ is a valid
juggling sequence.

###### Proof.

We will check the corresponding permutation sequence for $R^{\prime}$ and show
that it is valid. Suppose for a contradiction that there are
$i,j\in\\{0,1,\ldots,n-1\\}$ with

$r^{\prime}_{i}+i\equiv r^{\prime}_{j}+j\pmod{n}.$

Then we have:

$\displaystyle r^{\prime}_{i}+i$ $\displaystyle\equiv$ $\displaystyle
r^{\prime}_{j}+j\pmod{n}$ $\displaystyle r_{i-s}+i$ $\displaystyle\equiv$
$\displaystyle r_{j-s}+j\pmod{n}$ $\displaystyle r_{i-s}+i-s$
$\displaystyle\equiv$ $\displaystyle r_{j-s}+j-s\pmod{n}$ $\displaystyle
r_{k}+k$ $\displaystyle\equiv$ $\displaystyle r_{\ell}+\ell\pmod{n}$

Thus $\Pi_{R^{\prime}}$ is valid if and only if $\Pi_{R}$ is valid. ∎

######  Lemma 3.9.

Let $T=t_{0}t_{1}\ldots t_{n-1}$ be a juggling sequence, and let $0\leq
i,s\leq n-1$. Then

$\Pi_{\rho^{s}(T)}(i)=\rho^{s}(\Pi_{T})(i)-s\pmod{n}.$

###### Proof.

$\displaystyle\Pi_{\rho^{s}(T)}(i)$ $\displaystyle=$
$\displaystyle\rho^{s}(T)(i)+i\pmod{n}$ $\displaystyle=$ $\displaystyle
T(i+s)+i\pmod{n}$ $\displaystyle=$ $\displaystyle T(i+s)+i+s-s\pmod{n}$
$\displaystyle=$ $\displaystyle\Pi_{T}(i+s)-s\pmod{n}$ $\displaystyle=$
$\displaystyle\rho^{s}(\Pi_{T})(i)-s\pmod{n}$

∎

######  Lemma 3.10.

Fix $n,s,b\in\mathbb{Z}^{+}$ with $1\leq s\leq n-1$. Define $D$ to be the
$s$-ocycle transition digraph for juggling sequences of length $n$ using at
most $b$ balls. From any vertex $v_{0}v_{1}\ldots v_{s-1}$, there exists a
path to any vertex $v_{0}^{\prime}v_{1}^{\prime}\ldots v_{s-1}^{\prime}$ where
$v_{i}^{\prime}\equiv v_{i}\pmod{n}$.

###### Proof.

Since adding/subtracting $n$ to any digit in a juggling sequence of length $n$
does not invalidate the sequence, if $v_{i}\geq n$ we can simply rotate some
juggling sequence $X$ with $s$-prefix $v_{0}v_{1}\ldots v_{s-1}$ until $v_{i}$
is in the $(n-s$)-suffix, replace with $v_{i}$ with $v_{i}-n$, and then rotate
back to our original $s$-prefix with $v_{i}$ replaced by $v_{i}-n$. Repeating
this eventually will find a path to $v_{1}^{\prime}v_{2}^{\prime}\ldots
v_{s}^{\prime}$. ∎

######  Theorem 3.11.

Fix $n,s,b\in\mathbb{Z}^{+}$ such that $1\leq s\leq n-1$. There exists an
$s$-ocycle for the set of juggling sequences with period $n$ and at most $b$
balls if and only if $n-s>\hbox{gcd}(n,s)$.

###### Proof.

We prove the forward direction by showing that the $s$-ocycle transition
digraph $D$ has an Euler tour. By Theorem 1.1 this is done by showing that the
graph is balanced and connected.

Balanced:

Consider a vertex $R=r_{0}r_{1}\ldots r_{s-1}$ in $D$. We will show that any
$(n-s)$-string $Q=q_{0}q_{1}\ldots q_{n-s-1}$ that is a valid $s$-suffix for
$R$ is also a valid $s$-prefix for $R$. Note that strings $RQ$ and $QR$ are
simply rotations of each other, so by Lemma 3.8 either both strings are valid
juggling sequences or neither string is. In this manner, there is a bijection
between in- and out-edges, hence all vertices are balanced.

Connected:

Consider an arbitrary vertex $T^{s-}=t_{0}t_{1}\ldots t_{s-1}$ that is an
$s$-prefix to some juggling sequence $T=t_{0}t_{1}\ldots t_{n-1}$. First, by
Lemma 3.10 we may assume that $t_{i}\in\\{0,1,\ldots,n-1\\}$ for all
$i\in\\{0,1,\ldots,n-1\\}$. We will show that this arbitrary vertex is
connected to the minimum vertex, which we define to be $V^{s-}=0^{s}$ (a
prefix of $V=0^{n}$). In doing so, we will have shown that every vertex is
connected to $V^{s-}$ and hence the graph is connected.

Compare permutation sequences $\Pi_{T}$ and $\Pi_{V}$ corresponding to
juggling sequences $T$ and $V$, respectively. Note that
$\Pi_{V}=012\ldots(n-1)$, so suppose that for all $x\in\\{0,1,\ldots,i-1\\}$
we have $\Pi_{T}(x)=\Pi_{V}(x)=x$, but that $\Pi_{T}(i)=j$ for some
$j\in\\{i+1,i+2,\ldots,s-1\\}$. We will find a path from the $s$-prefix of $T$
to the $s$-prefix of some juggling sequence $T^{\prime}$ with permutation
sequence $\Pi_{T^{\prime}}$ that agrees with $\Pi_{V}$ in the first $i+1$
positions. Repeating this procedure until $i=n$ will construct a path through
$D$ from $T^{s-}$ to $V^{s-}$.

Assume $n-s=kd$ and $n=md$ for integers $m,k$ where $d=\hbox{gcd}(n,s)$, and
let $T=Y_{0}Y_{1}\ldots Y_{m-1}$ be a partition of $T$ into $d$-blocks, i.e.
$Y_{a}=t_{ad}t_{ad+1}\ldots t_{ad+d-1}$. Suppose that $t_{i}\in Y_{a}$ and
$t_{j}\in Y_{b}$. We have two cases.

  1. 1.

If $b-a<k$:

In this case, from Lemma 3.8 we may perform $s$-rotations on $T$ so that we
arrive at a vertex with both $t_{i}$ and $t_{j}$ in the $(n-s)$-suffix of $T$.
Suppose that we performed a total rotation of size $R$, i.e. we are now at the
vertex that represents the $s$-prefix of $\rho^{R}(T)$.

At this point, the values $t_{i}$ and $t_{j}$ are located in positions $i-R$
and $j-R$ of $\rho^{R}(T)$, respectively. By Lemma 3.9 the corresponding
permutation sequence entries are:

$\displaystyle\Pi_{\rho^{R}(T)}(i-R)$ $\displaystyle=$
$\displaystyle\rho^{R}(\Pi_{T})(i-R)-R\pmod{n}$ $\displaystyle=$
$\displaystyle\Pi_{T}(i)-R\pmod{n}$

and

$\displaystyle\Pi_{\rho^{R}(T)}(j-R)$ $\displaystyle=$
$\displaystyle\Pi_{T}(j)-R\pmod{n}.$

We now note that the vertex $(\rho^{R}(T))^{s-}$, defined as the $s$-prefix of
$\rho^{R}(T)$, is also the $s$-prefix of the juggling sequence
$\rho^{R}(T^{\prime})$ that is obtained from $\rho^{R}(T)$ by performing the
swap $\Pi_{\rho^{R}(T)}(i-R)\leftrightarrow\Pi_{\rho^{R}(T)}(j-R)$ and
adjusting the values $\rho^{R}(T)(i-R)$ and $\rho^{R}(T)(j-R)$ appropriately.
Then rotating backwards by $R$ we reach a juggling sequence $T^{\prime}$ with
$\Pi_{T^{\prime}}(x)=x$ for all $x\in\\{0,1,\ldots,i-1,i\\}$. This means that
we have found a path from $T^{s-}$ to $T^{\prime s-}$, which is one step
closer to the minimum vertex.

  2. 2.

If $b-a\geq k$:

In this case the blocks containing $t_{i}$ and $t_{j}$ are more than $k$
apart, so we cannot rotate to have both $t_{i}$ and $t_{j}$ in the
$(n-s)$-suffix of $T$ simultaneously. Instead, we pick a point $t_{z}$ in the
block $Y_{c}$ that is $k$ blocks preceding $Y_{b}$ and follow case (1) with
$t_{z}$ in place of $t_{i}$. This produces a juggling sequence $T^{\prime}$
with $\Pi_{T^{\prime}}(z)=i$, where $i\leq z<j$. Repeating, we will eventually
have moved the permutation sequence value $i$ to a block that is close enough
to block $Y_{a}$ to apply Case (1).

By repeating the above procedures, we will eventually have transitioned to the
vertex corresponding to a juggling sequence with $s$-prefix $01\ldots(s-1)$.
At this point we have reached the minimum vertex and we are done.

For the reverse direction, suppose that $n-s=\hbox{gcd}(n,s)=d$. Recall that
the number of balls $b\in\mathbb{Z}^{+}$ is given by:

$b=\frac{1}{n}\sum_{i=0}^{n-1}t_{i}.$

Thus juggling sequences of period $n$ must always satisfy
$\sum_{i=0}^{n-1}t_{i}\equiv 0\pmod{n}$. Equivalently, when we partition a
juggling sequence $T=t_{0}t_{1}\ldots t_{n-1}$ into blocks of length $d$, i.e.
$T=Y_{0}Y_{1}\ldots Y_{m-1}$, where $n=md$, we must have:

$\sum_{i=0}^{m-1}w(Y_{i})=\sum_{i=0}^{m-1}\sum_{j=0}^{d}t_{id+j}=\sum_{i=0}^{n-1}t_{i}\equiv
0\pmod{n},$

where, for a given block $Y_{i}$, we call $w(Y_{i})=\sum_{j=0}^{d}t_{id+j}$
the weight of block $Y_{i}$.

Now since $n-s=\hbox{gcd}(n,s)$, from vertex $T^{s-}=t_{0}t_{1}\ldots t_{s-1}$
in $D$ we may only move to vertices in which the $(n-s)$-suffix has weight
equivalent to $w(Y_{m-1})$ mod $n$. Thus if we can show that, for any $n,s,$
and $b$, there exists a juggling sequence with a block with weight
$w\not\equiv 0\pmod{n}$, then we are done. This is witnessed by the juggling
sequence

$T=\begin{array}[]{ccccccccc}d&0&0&\ldots&0&(n-d)&0&\ldots&0,\end{array}$

with permutation sequence

$\Pi_{T}=\begin{array}[]{cccccccccc}d&1&2&\ldots&(d-1)&0&(d+1)&(d+2)&\ldots&(n-1).\end{array}$

This juggling sequence utilizes one ball (recall we required
$b\in\mathbb{Z}^{+}$) and the weight of the first block of length $d$ is
$d\not\equiv 0\pmod{n}$. Thus this first block must always have weight equal
to $d$ modulo $n$ if $n-s=d$, and so the vertex $T^{s-}$ that represents the
$s$-prefix of $T$ is not connected to the vertex $0^{s}$. Hence no Euler tour
can exist and so no $s$-ocycle exists. ∎

While Theorem 3.11 completes the question of when $s$-ocycles for juggling
sequences of period $n$ and at most $b$ balls exist, several variations remain
open. For example, one might consider juggling sequences with:

  * •

exactly $b$ balls,

  * •

at least $b$ balls,

  * •

fixed minimum period, or

  * •

period $n$ with no restriction on number of balls.

## References

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