Lemma \ref{lem:improved-range} tells us in particular that for odd primes $p$ the stabilisation map $C_k(\bR^n) \to C_{k+1}(\bR^n)$ induces an isomorphism on homology with $\bF_p$ coefficients in the range $*\leq k$. We now show that this implies that the same is true for the stabilisation map $C_k(M) \to C_{k+1}(M)$ for any smooth, connected, open manifold $M$ of dimension at least $3$. Our method for this is a slight variation of an argument due to Oscar Randal-Williams in \cite[\S 8]{RW-hs-for-ucs}.

\begin{proposition}\label{pImprovedRange}
Let $M$ be a smooth, connected, open $n$-manifold with $n\geq 3$ and let $A$ be an abelian group. If the stabilisation map on $A$-homology
\[
H_*(C_k(\bR^n);A) \longrightarrow H_*(C_{k+1}(\bR^n);A)
\]
is an isomorphism in the range $*\leq k$ then so is the stabilisation map on $A$-homology
\[
H_*(C_k(M);A) \longrightarrow H_*(C_{k+1}(M);A).
\]
So by Lemma \ref{lem:improved-range} the stabilisation map $C_k(M)\to C_{k+1}(M)$ induces isomorphisms on homology with $\bF_p$ coefficients in the range $*\leq k$ for any odd prime $p$.
\end{proposition}

This result has also been recently proved by \cite{KupersMiller2014} using a different method along the lines of \cite{SegalRational}.

\begin{proof}
We will just write $H_*(-)$ for $H_*(-;A)$. Define $R_k(M)$ to be the homotopy cofibre of the stabilisation map $C_k(M)\to C_{k+1}(M)$. Now the stabilisation map $C_k(M)\to C_{k+1}(M)$ is split-injective on homology (see \cite[page 103]{McDuff}) so it induces an isomorphism on homology in degree $*$ if and only if $\widetilde{H}_*(R_k(M))=0$. So the hypothesis of the proposition says that $\widetilde{H}_*(R_k(\bR^n))=0$ for $*\leq k$ and we would like to show that $\widetilde{H}_*(R_k(M))=0$ for $*\leq k$. We refer to \cite{RW-hs-for-ucs} for background and any details which we omit in this proof -- the line of argument is very similar. The proof is by induction on $k$. The base case $k=0$ is obvious so we now fix $k\geq 1$ for the inductive step.

For $i\geq 0$ let $C_l(M)^i$ be the space of $l$-point subsets $c$ of $M$ together with an injection $\{0,\dotsc,i\}\to c$. These fit together to form an semi-simplicial space $C_l(M)^\bullet$ augmented by $C_l(M)$. The stabilisation map lifts to a map $C_l(M)^\bullet \to C_{l+1}(M)^\bullet$ of augmented semi-simplicial spaces. There is a fibre bundle $\pi\colon C_l(M)^i \to \widetilde{C}_{i+1}(M)$, where $\widetilde{C}$ denotes the \emph{ordered} configuration space, given by sending an injection $\{0,\dotsc,i\}\to c$ to its image and remembering the induced ordering. Its fibre over a point is homeomorphic to $C_{l-i-1}(M_{i+1})$, where $M_{i+1}$ denotes the manifold $M$ with $i+1$ points removed. Moreover the projection $\pi$ commutes with the stabilisation map $C_l(M)^i \to C_{l+1}(M)^i$ and the map of fibres over a point is the stabilisation map $C_{l-i-1}(M_{i+1}) \to C_{l-i}(M_{i+1})$. Any map of Serre fibrations over a fixed base space has an associated relative Serre spectral sequence; in this case it has second page
\[
{}^i \widetilde{E}^2_{s,t} \cong H_s( \widetilde{C}_{i+1}(M) ; \widetilde{H}_t( R_{l-i-1}(M_{i+1}) ) )
\]
and converges to $\widetilde{H}_*(R_l(M)^i)$, where $R_l(M)^i$ denotes the homotopy cofibre of the lift $C_l(M)^i \to C_{l+1}(M)^i$ of the stabilisation map.

For $1\leq j\leq k$ there are maps of augmented semi-simplicial spaces $C_{k-j}(M) \times C_j(\bR^n)^\bullet \to C_k(M)^\bullet$ defined similarly to the stabilisation map, except one stabilises by adding the given configuration in $\bR^n$ instead of just a single point. In \cite[\S 8]{RW-hs-for-ucs} it is explained how these induce maps of semi-simplicial spaces $R_{k-j-1}(M)\wedge R_j(\bR^n)^\bullet \to R_k(M)^\bullet$ for $1\leq j\leq k$. Note that when $j=k$ we have $R_{-1}(M)=S^0$ and this is just the map $R_k(\bR^n)^\bullet \to R_k(M)^\bullet$ induced by an embedding $\bR^n \hookrightarrow M$. Each semi-simplicial space has an associated spectral sequence so we obtain a map ${}^j\bar{E}\to E$ of spectral sequences whose first pages are
\begin{align*}
{}^j\bar{E}^1_{s,t} &\cong \widetilde{H}_t(R_{k-j-1}(M)\wedge R_j(\bR^n)^s) \\
E^1_{s,t} &\cong \widetilde{H}_t(R_k(M)^s).
\end{align*}
Note that these are first quadrant plus an extra column $\{s=-1,t\geq 0\}$.

The spectral sequence $E$ converges to $\widetilde{H}_{*+1}$ of the homotopy cofibre of the map $\lVert R_k(M)^\bullet \rVert \to R_k(M)$ induced by the augmentation map. Since taking homotopy cofibres commutes with taking geometric realisation of semi-simplicial spaces this space can also be obtained as follows: first take the homotopy cofibres of the maps $\lVert C_k(M)^\bullet \rVert \to C_k(M)$ and $\lVert C_{k+1}(M)^\bullet \rVert \to C_{k+1}(M)$; these are related by a map induced by stabilisation; then take the homotopy cofibre of this map. Now the augmented semi-simplicial space $C_k(M)^\bullet$ is a $(k-1)$-resolution \cite[Proposition 6.1]{RW-hs-for-ucs}, i.e.\ the map $\lVert C_k(M)^\bullet \rVert \to C_k(M)$ is $(k-1)$-connected. Hence the spectral sequence $E$ converges to zero in total degree $*\leq k-1$.

The inductive hypothesis says that
\begin{equation}\tag{IH}\label{IH}
\widetilde{H}_*(R_l(M))=0 \text{ for } *\leq l<k
\end{equation}
and the hypothesis of the proposition says that
\begin{equation}\tag{Hyp}\label{Hyp}
\widetilde{H}_*(R_l(\bR^n))=0 \text{ for } *\leq l.
\end{equation}
From (IH) we deduce that ${}^i\widetilde{E}^2_{s,t}=0$ for $t\leq l-i-1$ so the spectral sequence ${}^i\widetilde{E}$ converges to zero in total degree $*\leq l-i-1$, so
\begin{equation}\label{Eq1}
\widetilde{H}_*(R_l(M)^i)=0 \text{ for } *\leq l-i-1 \text{ and } i\geq 0.
\end{equation}
In other words:
\begin{equation}\label{Eq1a}
E^1_{s,t}=0 \text{ for } t\leq k-s-1 \text{ and } s\geq 0.
\end{equation}
Also, using the K{\"u}nneth theorem, \eqref{Eq1} and \eqref{IH} we deduce that
\begin{equation}\label{Eq2}
{}^j\bar{E}^1_{s,t}=0 \text{ for } t\leq k-s-1,
\end{equation}
where for the case $\{s=-1 \text{ and } j=k\}$ we also need to use \eqref{Hyp}. We now make the following:

\begin{claim}
For $1\leq j\leq k$ the map ${}^j\bar{E}^1_{j,k-j} \to E^1_{j,k-j}$ is surjective.
\end{claim}

The verification of this claim is delayed until the end of the proof. Now a diagram chase in the following:
\begin{center}
\begin{tikzpicture}
[x=1mm,y=1mm]
\node (t1) at (0,15) {${}^j\bar{E}^1_{j,k-j}$};
\node (t2) at (20,15) {${}^j\bar{E}^{j+1}_{j,k-j}$};
\node (t3) at (50,15) {${}^j\bar{E}^{j+1}_{-1,k}$};
\node (t4) at (70,15) {${}^j\bar{E}^1_{-1,k}$};
\node (b1) at (0,0) {$E^1_{j,k-j}$};
\node (b2) at (20,0) {$E^{j+1}_{j,k-j}$};
\node (b3) at (50,0) {$E^{j+1}_{-1,k}$};
\node (b4) at (70,0) {$E^1_{-1,k}$};
\node at (79,15) {$=0$};
\draw[->>] (t1) to (t2);
\draw[->] (t2) to node[above,font=\small]{$\bar{d}_{j+1}$} (t3);
\draw[->>] (t4) to (t3);
\draw[->>] (b1) to (b2);
\draw[->] (b2) to node[below,font=\small]{$d_{j+1}$} (b3);
\draw[->>] (b4) to (b3);
\draw[->>] (t1) to (b1);
\draw[->] (t2) to (b2);
\draw[->] (t3) to (b3);
\draw[->] (t4) to (b4);
\end{tikzpicture}
\end{center}
shows that the differential $d_{j+1}\colon E^{j+1}_{j,k-j} \to E^{j+1}_{-1,k}$ is zero for $1\leq j\leq k$.

Now we can deduce that the first differential $d_1\colon E^1_{0,t} \to E^1_{-1,t}$ is surjective in a range. First, for $t\leq k-1$ note that the differentials hitting $E^{\square}_{-1,t}$ have source $E^j_{j-1,t-j+1}$ for $1\leq j\leq t+1$. By \eqref{Eq1a} these groups are all zero, so $E^1_{-1,t} = E^{\infty}_{-1,t}$. The spectral sequence $E$ converges to zero in total degree $t-1$ so $E^1_{-1,t}=E^{\infty}_{-1,t}=0$ and so the first differential $d_1\colon E^1_{0,t} \to E^1_{-1,t}$ is vacuously surjective. For $t=k$ we use the result of the diagram chase above, which tells us that the only possible non-zero differential hitting $E^{\square}_{-1,k}$ is the first differential. We know that $E^{\infty}_{-1,k}=0$ since $E$ converges to zero in total degree $k-1$ so the first differential $d_1\colon E^1_{0,k} \to E^1_{-1,k}$ must be surjective. This can be identified as the map on homology induced by the augmentation map $R_k(M)^0 \to R_k(M)$. Hence we have established:

\begin{fact}\label{Fact1}
The augmentation map $a\colon R_k(M)^0 \to R_k(M)$ induces surjections on $A$-homology up to degree $k$.
\end{fact}

Now consider the maps $p\colon C_k(M) \to C_k(M_1)$ and $u\colon C_k(M_1) \to C_k(M)$, defined as follows. The map $p$ is defined similarly to the stabilisation map. Write $M = \mathrm{int}(\mbar)$ for a manifold $\mbar$ with non-empty boundary and choose a self-embedding $e^\prime\colon \mbar\hookrightarrow\mbar$ which is isotopic to the identity and whose image does not contain the missing point of $M_1$. Then $p$ is defined by applying $e^\prime$ to each point of the configuration. The map $u$ is simply the map induced by the inclusion $M_1 \hookrightarrow M$. Since $e^\prime$ is isotopic to the identity the composition $u\circ p$ is homotopic to the identity, and so the induced maps $u_*$ and $p_*$ on homology are semi-inverses: $u_*\circ p_*={\id}$. If we are careful to define $p$ using a self-embedding $e^\prime\colon\mbar\hookrightarrow\mbar$ whose support is disjoint from the self-embedding $e\colon\mbar\hookrightarrow\mbar$ used to define the stabilisation map $s$, then $p$ commutes on the nose with $s$ and there are induced maps $p\colon R_k(M)\to R_k(M_1)$ and $u\colon R_k(M_1) \to R_k(M)$ on mapping cones. Again we have $u\circ p \simeq {\id}$ so $u_*\circ p_*={\id}$.

The methods of the proof of Proposition 6.3 in \cite{RW-hs-for-ucs} show that
\[
\hconn_A(u\colon R_{k-1}(M_1) \to R_{k-1}(M)) \geq \hconn(s\colon C_{k-2}(M) \to C_{k-1}(M)) + \dim(M)
\]
where $\hconn_A(f)$ is the $A$-homology-connectivity of $f$, i.e.\ the largest $*$ such that $\widetilde{H}_*(\mathit{mc}(f);A)=0$, where $\mathit{mc}(f)$ is the mapping cone of $f$. By inductive hypothesis the right-hand side is at least $k-2+\dim(M)\geq k+1$ since we have assumed that $M$ is at least $3$-dimensional. Therefore the $A$-homology-connectivity of $p\colon R_{k-1}(M) \to R_{k-1}(M_1)$ is at least $k$. In particular we have:

\begin{fact}\label{Fact2}
The map $p\colon R_{k-1}(M) \to R_{k-1}(M_1)$ induces surjections on $A$-homology up to degree $k$.
\end{fact}

For our third and final fact, consider the spectral sequence ${}^0\widetilde{E}$ with $l=k$ and recall from just before \eqref{Eq1} that ${}^0\widetilde{E}^2_{s,t}=0$ for $t\leq k-1$. This is the relative Serre spectral sequence for the map of fibre bundles $C_k(M)^0 \to C_{k+1}(M)^0$ over $\widetilde{C}_1(M)=M$. The inclusion of the fibre over a point $*\in M$ is the map $C_{k-1}(M_1)=C_{k-1}(M\smallsetminus\{*\}) \to C_k(M)^0$ which adds the point $*$ to a configuration and labels it by $0$. This induces a map $f\colon R_{k-1}(M_1) \to R_k(M)^0$ on mapping cones. The map on $\widetilde{H}_*$ induced by $f$ can be identified with the composition of the edge homomorphism
\[
\widetilde{H}_*(R_{k-1}(M_1)) = {}^0\widetilde{E}^2_{0,*} \twoheadrightarrow {}^0\widetilde{E}^{\infty}_{0,*}
\]
and the inclusion
\[
{}^0\widetilde{E}^{\infty}_{0,*} \hookrightarrow \widetilde{H}_*(R_k(M)^0)
\]
given by all the extension problems in total degree $*$. But since the second page is trivial for $t\leq k-1$ there are no extension problems in total degree $*\leq k$, and so this inclusion is an isomorphism. Hence we have:

\begin{fact}\label{Fact3}
The map $f\colon R_{k-1}(M_1) \to R_k(M)^0$ induces surjections on $A$-homology up to degree $k$.
\end{fact}

The composition $s^\prime \coloneqq a\circ f\circ p\colon R_{k-1}(M) \to R_k(M)$ is defined exactly like the stabilisation map $s\colon R_{k-1}(M) \to R_k(M)$ except that it uses the self-embedding $e^\prime$ of $\mbar$ instead of $e$. Since we chose $e$ and $e^\prime$ to have disjoint support, the maps $s$ and $s^\prime$ commute. If we now ensure that we picked $e$ and $e^\prime$ to be isotopic, we have that $s$ and $s^\prime$ are homotopic. The square $s\circ s^\prime = s^\prime\circ s$ induces a map of long exact sequences:
\begin{center}
\begin{tikzpicture}
[x=1mm,y=1mm]
\node (t1) at (0,15) {$\widetilde{H}_t(C_k(M))$};
\node (t2) at (30,15) {$\widetilde{H}_t(R_{k-1}(M))$};
\node (t3) at (60,15) {$\widetilde{H}_{t-1}(C_{k-1}(M))$};
\node (t4) at (95,15) {$\widetilde{H}_{t-1}(C_k(M))$};
\node (b1) at (0,0) {$\widetilde{H}_t(C_{k+1}(M))$};
\node (b2) at (30,0) {$\widetilde{H}_t(R_k(M))$};
\draw[->] (t1) to node[above,font=\small]{$c$} (t2);
\draw[->] (t2) to (t3);
\draw[->] (t3) to node[above,font=\small]{$b=s_*$} (t4);
\draw[->] (b1) to node[below,font=\small]{$d$} (b2);
\draw[->] (t1) to node[left,font=\small]{$s^{\prime}_*$} (b1);
\draw[->] (t2) to node[right,font=\small]{$a=s^{\prime}_*$} (b2);
\draw[->] (-15,15) to node[above,font=\small]{$s_*$} (t1);
\draw[->] (-15,0) to node[below,font=\small]{$s_*$} (b1);
\draw[->] (b2) to (45,0);
\draw[->] (t4) to (110,15);
\end{tikzpicture}
\end{center}
Let $t\leq k$ -- our aim is to show that $\widetilde{H}_t(R_k(M))=0$. By Facts \ref{Fact1}, \ref{Fact2} and \ref{Fact3} above, the map $a$ in this diagram is surjective. As mentioned at the beginning of the proof, the stabilisation map is split-injective on homology in all degrees \cite[page 103]{McDuff}, so the map $b$ is injective, and so by exactness the map $c$ is surjective. Hence the composite $a\circ c$ is surjective. But
\[
a\circ c = d\circ s^{\prime}_* = d\circ s_* = 0,
\]
so its codomain $\widetilde{H}_t(R_k(M))$ must be trivial.

It now remains to prove the claim we made earlier in the proof, namely that the map
\[
{}^j\bar{E}^1_{j,k-j} = \widetilde{H}_{k-j}(R_{k-j-1}(M)\wedge R_j(\bR^n)^j) \longrightarrow \widetilde{H}_{k-j}(R_k(M)^j) = E^1_{j,k-j}
\]
is surjective. In fact we will show that the map $R_{k-j-1}(M)\wedge R_j(\bR^n)^j \to R_k(M)^j$ induces surjections on homology in degrees $t\leq k-j$. First note that
\begin{align*}
R_j(\bR^n)^j &= \mathit{mc}(C_j(\bR^n)^j \to C_{j+1}(\bR^n)^j) \\
&= \mathit{mc}(\varnothing \to \widetilde{C}_{j+1}(\bR^n)) \\
&= \widetilde{C}_{j+1}(\bR^n)_+
\end{align*}
and the map $R_{k-j-1}(M)\wedge \widetilde{C}_{j+1}(\bR^n)_+ \to R_k(M)^j$ is given by taking mapping cones of the horizontal arrows in the commutative square:
\begin{center}
\begin{tikzpicture}
[x=1mm,y=1mm]
\node (tl) at (0,15) {$C_{k-j-1}(M)\times \widetilde{C}_{j+1}(\bR^n)$};
\node (tr) at (60,15) {$C_{k-j}(M)\times \widetilde{C}_{j+1}(\bR^n)$};
\node (bl) at (0,0) {$C_k(M)^j$};
\node (br) at (60,0) {$C_{k+1}(M)^j$};
\draw[->] (tl) to node[above,font=\small]{$s\times{\id}$} (tr);
\draw[->] (bl) to node[below,font=\small]{$s$} (br);
\draw[->] (tl) to (bl);
\draw[->] (tr) to (br);
\end{tikzpicture}
\end{center}

To do this we begin by defining some more explicit models for various maps. As before, write $M=\mathrm{int}(\mbar)$ for a manifold $\mbar$ with non-empty boundary and choose two isotopic self-embeddings $e,e^\prime \colon \mbar\hookrightarrow\mbar$ which are both non-surjective and have disjoint support. Choose an embedding $\phi\colon \bR^n \hookrightarrow M\smallsetminus e^\prime(\mbar)$ and pairwise disjoint points $p_0,\dotsc,p_j \in \bR^n$. Write $M_{j+1} = M\smallsetminus\{\phi(p_0),\dotsc,\phi(p_j)\}$. We have a square of maps
\begin{equation}\label{abcd}
\centering
\begin{split}
\begin{tikzpicture}
[x=1mm,y=1mm]
\node (tl) at (0,15) {$C_{k-j-1}(M)$};
\node (tr) at (40,15) {$C_{k-j-1}(M)\times \widetilde{C}_{j+1}(\bR^n)$};
\node (bl) at (0,0) {$C_{k-j-1}(M_{j+1})$};
\node (br) at (40,0) {$C_k(M)^j$};
\draw[->] (tl) to node[above,font=\small]{$\alpha$} (tr);
\draw[->] (bl) to node[below,font=\small]{$\beta$} (br);
\draw[->] (tl) to node[left,font=\small]{$\gamma$} (bl);
\draw[->] (tr) to node[right,font=\small]{$\delta$} (br);
\end{tikzpicture}
\end{split}
\end{equation}
defined by
\begin{align*}
\alpha(c) &= (c,(p_0,\dotsc,p_j)) \\
\gamma(c) &= e^\prime(c) \\
\beta(c) &= c\cup \{\phi(p_0),\dotsc,\phi(p_j)\};i\mapsto \phi(p_i) \\
\delta(c,(q_0,\dotsc,q_j)) &= e^\prime(c)\cup \{\phi(q_0),\dotsc,\phi(q_j)\};i\mapsto \phi(q_i).
\end{align*}

Choose a point $*\in M\smallsetminus e(\mbar)$ and take an explicit model for the stabilisation map to be defined by $c\mapsto e(c)\cup\{*\}$. Since $e$ and $e^\prime$ have disjoint support this induces a map of squares from \eqref{abcd} to $\eqref{abcd}[k\mapsto k+1]$. Taking mapping cones along this map of squares gives us the following:
\begin{center}
\begin{tikzpicture}
[x=1mm,y=1mm]
\node (tl) at (0,15) {$R_{k-j-1}(M)$};
\node (tr) at (40,15) {$R_{k-j-1}(M)\wedge \widetilde{C}_{j+1}(\bR^n)_+$};
\node (bl) at (0,0) {$R_{k-j-1}(M_{j+1})$};
\node (br) at (40,0) {$R_k(M)^j$};
\draw[->] (tl) to node[above,font=\small]{$\bar{\alpha}$} (tr);
\draw[->] (bl) to node[below,font=\small]{$\bar{\beta}$} (br);
\draw[->] (tl) to node[left,font=\small]{$\bar{\gamma}$} (bl);
\draw[->] (tr) to node[right,font=\small]{$\bar{\delta}$} (br);
\end{tikzpicture}
\end{center}

We need to show that $\bar{\delta}$ induces surjections on homology in degrees $t\leq k-j$. This will follow if we can prove this for $\bar{\gamma}$ and $\bar{\beta}$. But $\bar{\gamma}$ is the composition of $j+1$ instances of the map $p$ from Fact \ref{Fact2}, and so this does induce surjections on homology up to degree $k-j$ by Fact \ref{Fact2}.\footnote{The proofs of Facts \ref{Fact2} and \ref{Fact3} earlier did not depend on the claim which we are currently proving, so this is not circular.} When $j=0$ the map $\bar{\beta}$ is surjective on homology up to degree $k$ by Fact \ref{Fact3}. Moreover, the argument proving Fact \ref{Fact3} generalises (using the spectral sequence ${}^j\widetilde{E}$ instead of ${}^0\widetilde{E}$) to prove precisely that the map $\bar{\beta}$ is surjective on homology up to degree $k-j$ in general.
\end{proof}

\begin{remark}\label{rSESofcoeffs}
When $\dim(M)\geq 3$ we have homological stability in the range $*\leq k$ for $\bQ$ coefficients (by \cite[Theorem B]{RW-hs-for-ucs}) and for $\bZ/p$ coefficients with $p$ odd (by Proposition \ref{pImprovedRange} above). Using the short exact sequences of coefficients $0\to \bZ/p \to \bZ/p^{l+1} \to \bZ/p^l \to 0$ and
\[
0\to \bZ[\tfrac12]\to \bQ\to \textstyle{\bigoplus}_{p\neq 2}\colim_{l\to\infty}\bZ/p^l \to 0
\]
this implies homological stability in the range $*\leq k-1$ for $\bZ[\frac12]$ coefficients. This recovers Theorem 1.4 of \cite{KupersMiller2014}, except without surjectivity in degree $k$.
\end{remark}

\section{Homological stability for configuration spaces with labels in a fibre bundle}\label{appendixB}


\begin{df}[Configuration spaces and stabilisation maps with labels in a fibre bundle]\label{dConfigLabels}
Let $\theta\colon E\to M$ be a fibre bundle with path-connected fibres $F$ and define
\[
C_k(M;\theta) \coloneqq \{ \{p_1,\dotsc,p_k\} \subset E \mid \theta(p_i) \neq \theta(p_j) \text{ for } i\neq j \}.
\]
Choose a self-embedding $e\colon\mbar\hookrightarrow\mbar$ which is non-surjective and isotopic to the identity. Choose an open neighbourhood $U\subseteq \mbar$ containing the support of $e$, write $V=U\smallsetminus \partial\mbar$ and choose a trivialisation $\phi\colon \theta^{-1}(V) \to V\times F$ of $E$ over $V$. Define a self-embedding $\widetilde{e}\colon E\hookrightarrow E$ by
\[
p\mapsto \begin{cases}
p & p\notin \theta^{-1}(V) \\
\phi^{-1}\circ (e\times\mathrm{id})\circ \phi(p) & p\in \theta^{-1}(V)
\end{cases}
\]
and note that $\theta\circ \widetilde{e} = e\circ\theta$. Also choose points $*\in M\smallsetminus e(\mbar) \subseteq V$ and $x\in F$. We can then define the stabilisation map $C_k(M;\theta) \to C_{k+1}(M;\theta)$ by
\[
\{p_1,\dotsc,p_k\} \mapsto \{\widetilde{e}(p_1),\dotsc,\widetilde{e}(p_k),\phi^{-1}(*,x)\}.
\]
\end{df}

We may generalise Proposition \ref{pImprovedRange} to configuration spaces with labels in $\theta$ using the following fact.

\begin{remark}\label{rTwistedCoefficients}
Configuration spaces also satisfy homological stability with respect to finite-degree twisted coefficient systems: for the case of symmetric groups this was proved by \cite{Betley2002}, and the general case was proved in \cite{Palmer2013a}. A twisted coefficient system for $M$ is a functor from the partial braid category $\cB(M)$ to $\bZ\text{-mod}$. The partial braid category $\cB(M)$ has objects $\{0,1,2,\dotsc\}$ and a morphism from $m$ to $n$ is a path in $C_k(M)$ from a subset of $\{p_1,\dotsc,p_m\}$ to a subset of $\{p_1,\dotsc,p_n\}$, up to endpoint-preserving homotopy, where $\{p_1,p_2,p_3,\dotsc\}$ is a fixed injective sequence in $M$.

If the twisted coefficient system has degree $d$ the stable range obtained is $*\leq\frac{k-d}{2}$, which arises since homological stability with untwisted $\bZ$ coefficients in the range $*\leq \frac{k}{2}$ is an input for the proof. However, if the twisted coefficient system takes values in the subcategory $\bZ[\frac12]$-mod of $\bZ$-mod and $\dim(M)\geq 3$, then we may instead input \cite{KupersMiller2014} or Proposition \ref{pImprovedRange} to obtain a stable range of $*\leq k-d$ for $C_k(M)$ with coefficients in a functor $\cB(M)\to \bZ[\frac12]\text{-mod}$ of degree $d$.
\end{remark}


\begin{proposition}\label{pImprovedRangeTwisted}
Let $M$ be a smooth, connected, open $n$-manifold with $n\geq 2$ and $\theta\colon E\to M$ a fibre bundle with path-connected fibres. Then the stabilisation map $C_k(M;\theta) \to C_{k+1}(M;\theta)$ induces isomorphisms on $H_*(-;\bZ)$ in the range $*\leq \frac{k}{2}-1$. It induces isomorphisms in the range $*\leq k$ on $H_*(-;\bQ)$, unless $M$ is an orientable surface in which case the range is only $*\leq k-1$. If $n\geq 3$ it induces isomorphisms on $H_*(-;\bZ[\frac12])$ in the range $*\leq k-1$.
\end{proposition}

The worse range $*\leq k-1$ for rational homology of configurations on orientable surfaces is necessary: for example $H_1(C_1(\bR^2);\bQ) = 0 \not\cong \bQ \cong H_1(C_2(\bR^2);\bQ)$.


\begin{remark}\label{rImprovedRangeTwisted}
A version of Proposition \ref{pImprovedRangeTwisted} is also proved in the appendix of \cite{KupersMiller2014Encellattachments}. One of the proofs given there is essentially the same as the proof we give below, and a sketch proof using semi-simplicial resolutions by collections of disjoint arcs in $M$ is also given. This latter method has the advantage that it gives a range of $*\leq \frac{k}{2}$ for $\bZ$ coefficients (at least when $M$ is orientable), rather than the smaller range $*\leq \frac{k}{2}-1$ for $\bZ$ coefficients obtained in Proposition \ref{pImprovedRangeTwisted}.
\end{remark}


\begin{proof}
This will follow by the same considerations as in Remark \ref{rSESofcoeffs} if we show that it induces isomorphisms on $H_*(-;A)$ in the range $*\leq \frac{k}{2}$ when $A=\bF_p$, in the range $*\leq k$ if either (a) $A=\bQ$ and $M$ is not an orientable surface or (b) $A=\bF_p$ for $p$ odd and $n\geq 3$, and in the range $*\leq k-1$ when $A=\bQ$ and $M$ is an orientable surface. The loss of one degree from the range occurs when going from $\bQ$ and $\bQ/\bZ$ coefficients to $\bZ$ coefficients (resp.\ $\bQ$ and $\bQ/\bZ[\frac12]$ coefficients to $\bZ[\frac12]$ coefficients).

Let $A = \bQ \text{ or } \bF_p$ for a prime $p$. There are fibre bundles $C_k(M;\theta) \to C_k(M)$, given by forgetting labels, with fibre $F^k$. The stabilisation maps $C_k(M) \to C_{k+1}(M)$ and $C_k(M;\theta) \to C_{k+1}(M;\theta)$ commute with these fibre bundles and the map of fibres is the inclusion $F^k \hookrightarrow F^{k+1}$. There is then a map of Serre spectral sequences
\begin{center}
\begin{tikzpicture}
[x=1mm,y=1mm]
\node (tl) at (0,15) [anchor=west] {$E^2_{s,t}\cong H_s(C_k(M);H_t(F^k;A))$};
\node (tr) at (80,15) {$H_*(C_k(M;\theta);A)$};
\node (bl) at (0,0) [anchor=west] {$E^2_{s,t}\cong H_s(C_{k+1}(M);H_t(F^{k+1};A))$};
\node (br) at (80,0) {$H_*(C_{k+1}(M;\theta);A)$};
\coordinate (ya) at (bl.north);
\coordinate (yb) at ($ (bl.north)+(1,0) $);
\coordinate (xa) at (tl.south);
\coordinate (xb) at ($ (tl.south)+(0,1) $);
\draw[->] (tl.south) to (intersection of ya--yb and xa--xb);
\draw[->] (tr) to (br);
\node at (60,0) {$\Rightarrow$};
\node at (60,15) {$\Rightarrow$};
\end{tikzpicture}
\end{center}
and our aim is to prove that the map in the limit is an isomorphism in a certain range depending on $A$ and $M$.

Now by Lemma 4.2 of \cite{Palmer2013a} and since $A$ is a field the assignment $k\mapsto H_t(F^k;A)$ extends to form a twisted coefficient system of degree at most $\frac{t}{h+1}\leq t$ where $h=\hconn_A(F)$. Hence the map on the second page is an isomorphism in the range $s\leq \frac{k-t}{2}$ by Theorem 1.3 of \cite{Palmer2013a}. In particular it is an isomorphism in total degree at most $\frac{k}{2}$ and therefore the same holds for the map in the limit.

To obtain the improved range in certain cases note that, by Remark 6.5 of \cite{Palmer2013a}, if homological stability with (untwisted) $A$ coefficients holds for (unlabelled) configuration spaces on $M$ in the range $*\leq f(k)$, then twisted homological stability will hold in the range $*\leq f(k-d)$ for any twisted coefficient system of degree $d$ which factors through the forgetful functor $A\text{-mod} \to \bZ\text{-mod}$ (c.f.\ Remark \ref{rTwistedCoefficients}).

If $A=\bQ$ then the above twisted coefficient system factors through the inclusion $\bQ\text{-mod} \to \bZ\text{-mod}$. By Theorem C of \cite{RW-hs-for-ucs} we may take $f(k)=k$ if $n=\dim(M)\geq 3$. For orientable surfaces we may take $f(k)=k-1$ by Corollary 3 of \cite{Church} or Theorem 1.3 of \cite{Knudsen2014}, and for non-orientable surfaces we may take $f(k)=k$ by Theorem 1.3 of \cite{Knudsen2014}. By the above paragraph the map of spectral sequences is an isomorphism on the second page in the range $s\leq f(k-t)$, and therefore in total degree at most $k$ (resp.\ total degree at most $k-1$ for orientable surfaces). Hence so is the map in the limit.

If $A=\bF_p$ for $p$ odd and $n\geq 3$ then the twisted coefficient system factors through the inclusion $\bZ[\frac12]\text{-mod} \to \bZ\text{-mod}$. By Proposition \ref{pImprovedRange} (or Theorem 1.4 of \cite{KupersMiller2014}) we may take $f(k)=k$. So as above the map of spectral sequences is an isomorphism in total degree at most $k$, and therefore so is the map in the limit.
\end{proof}
{"config": "arxiv", "file": "1406.4916/appendixApart2andB.tex"}