text:
\subsection{Ray partitions}
\renewcommand{\ind}{{T}}
\noindent We fix a partition
$\mathbf{k}=(k_1,\dots,k_r)$ for the entire section. Before we state our main result about the cohomology of $\tilde{V}_{\mathbf{k}}(\R^{p,q})$, we need to introduce a few combinatorial notions.

\begin{defi}
\label{defi:indset}
  The \emph{table} associated with the partition $\mathbf{k}$ is the set
  \[
    \ind_{\mathbf{k}}\coloneqq \{(i,j);\, 1\le i\le r\text{ and }1\le j\le k_i\}.
  \]
 We order $\ind_\mathbf{k}$ lexicographically, which means we write $(i,j)<(i',j')$ if either $i<i'$, or $i=i'$ and $j<j'$ holds.
\end{defi}
\begin{nota}
\label{nota:lengthagility}
For each partition $\Q$ of $\ind_\mathbf{k}$ into non-empty subsets $\Q_1,\dotsc,\Q_l$ we consider two positive integers:
  \begin{itemize}
  \item The number $l(\Q)\coloneqq l$ is called the \emph{length} of the partition, and in general we have $1\le l(\Q)\le \abs{\mathbf{k}}$.
  \item Consider on $\{1,\dotsc,l\}$ the equivalence relation spanned by $\beta\sim\beta'$ if there are $1\le i\le r$ and $1\le j,j'\le k_i$ with $(i,j)\in \Q_\beta$ and $(i,j')\in \Q_{\beta'}$ (i.e.\ the $i$\textsuperscript{th} cluster intersects both $\Q_\beta$ and $\Q_{\beta'}$). The number of equivalence classes $1\le a(\Q)\le \min(l(\Q),r)$  will be called the \emph{agility} of the partition.
  \end{itemize}
\end{nota}

\begin{defi}\label{defi:raypartition}
A \emph{ray partition $\Q$ of type $\mathbf{k}$} is a partition $\Q_1,\dotsc,\Q_l$ of $\ind_\mathbf{k}$, with a total order $\prec_{\beta}$ on each piece $\Q_{\beta}$ (called \emph{ray}), such that the following hold:
  \begin{enumerate}
  \item [{\small\bfseries R1.}] the components are labelled from $1$ to $l$ according to their minimum with respect to
  the global order $<$, i.e.
  \[\min(\Q_1,<)<\dotsb<\min(\Q_l,<);\]
  \item [{\small\bfseries R2.}] for each $1\le\beta\le l$, the minima with respect to $<$ and $\prec_\beta$ coincide
    \[\min(\Q_\beta,\prec_\beta) = \min(\Q_\beta,<).\]
  \end{enumerate}
\end{defi}

\begin{defi}
  \label{defi:witnessed}
  Let $Z=(z_{1}^1,\dotsc,z_r^{\smash{k_r}})\in \tilde{V}_\mathbf{k}(\R^{p,q})$. %
  We say that a ray partition $\Q$ is \emph{witnessed by $Z$} if the following conditions hold:%
  %
  \begin{enumerate}
  \item[{\small \bfseries W1.}] all $z_{i}^{\smash j}$ with $(i,j)\in \Q_\beta$ project along $\pr_{\zeta}$
    to the same point in $\R^{d-1}$.\vspace*{-5px}

  \item[{\small \bfseries W2.}] if $(i,j)\prec_{\beta}(i',j')$ in $\Q_{\beta}$,
    then $\pr_t(z_{i}^{\smash j})<\pr_t(z_{i'}^{\smash{j'}})$ in $\R$.
  \end{enumerate}
\end{defi}
Condition {\small\textbf{W1}} says that the points $z_{i,j}$ with $(i,j)\in \Q_\beta$ lie on a
line in $\R^d$ parallel to the $t$-axis;
 %
condition {\small\textbf{W2}} ensures that the same points are assembled on this line according to the order $\prec_{\beta}$
of their indices. In particular the points $z_{i,j}$ with $(i,j)\in \Q_\beta$ lie on a \emph{ray},
namely the half-line starting at $z_{\min(\Q_\beta,\prec_\beta)}$ and running in the positive $t$-direction.
See \textsc{Figure~\ref{fig:ray}} for an example.

\begin{figure}%
  \centering
  \begin{tikzpicture}[xscale=4.5,yscale=2.7]
    \node [thin,grey] at (1.45,1.2) {\tiny $\R^3$};
    \draw [very thin,bgrey] (0,0) -- (.6,.4) -- (1.4,.4);
    \draw [very thin,bgrey]  (.6,.4) -- (.6,1.2);
    \draw [grey,thin] (.3,1.033) -- (.3,0.072);
    \draw [dgrey,very thick] (.3,.25) -- (.3,.5);
    \draw [grey,thin] (.4,1.07) -- (.4,0.139);
    \draw [grey,thin] (.533,1.12) -- (.533,0.233);
    \draw [grey,thin] (.8,1.043) -- (.8,0.082);
    \draw [grey,thin] (1.023,1.14) -- (1.023,0.283);
    \draw [dgrey,very thick] (.8,.5) -- (.8,.7);
    \node [dgrey] at (.399,.496) {$\bullet$};
    \node [dgrey] at (.5315,.8475) {$\bullet$};
    \node [dgrey] at (.5315,.496) {$\bullet$};
    \node [dgrey] at (.8,.8) {$\bullet$};
    \node [dgrey] at (1.0235,.5975) {$\bullet$};
    \draw[very thick,white] (.2,.8) -- (.2,.1);
    \draw [blila,thin] (.2,0) -- (.76,.4) -- (.76,1.2);
    \draw[very thick,white] (.7,.55) -- (.7,0);
    \draw [lila] (.2,0) -- (.2,1) -- (.76,1.2);
    \draw [blila,thin] (.7,0) -- (1.16,.4) -- (1.16,1.2);
    \draw[very thick,white] (.25,1) -- (.95,1);
    \draw[very thick,white] (1,1) -- (1.4,1.2);
    \draw[very thick,white] (1,.55) -- (1,0);
    \draw [lila] (.7,0) -- (.7,1) -- (1.16,1.2);
    \draw [thin,grey] (0,0) rectangle (1,1);
    \draw [thin,grey] (1,0) -- (1.4,.4) -- (1.4,1.2) -- (1,1);
    \draw [thin,grey] (0,1) -- (.6,1.2) -- (1.4,1.2);
    \node [dred] at (.3,.25) {\tiny $\bullet$};
    \node [blue] at (.3,.4) {\tiny $\bullet$};
    \node [dred] at (.3,.5) {\tiny $\bullet$};
    \node [blue] at (.4,.5) {\tiny $\bullet$};
    \node [blue] at (.533,.5) {\tiny $\bullet$};
    \node [dred] at (.533,.85) {\tiny $\bullet$};
    \node [dgreen] at (1.023,.6) {\tiny $\bullet$};
    \node [dgreen] at (.8,.5) {\tiny $\bullet$};
    \node [dgreen] at (.8,.8) {\tiny $\bullet$};
    \node [dgreen] at (.8,.7) {\tiny $\bullet$};
    \node [dred] at (.24,.25) {\tiny $1,2$};
    \node [blue] at (.24,.4) {\tiny $3,3$};
    \node [dred] at (.24,.5) {\tiny $1,3$};
    \node [blue] at (.46,.5) {\tiny $3,2$};
    \node [blue] at (.593,.5) {\tiny $3,1$};
    \node [dred] at (.593,.85) {\tiny $1,1$};
    \node [dgreen] at (1.083,.6) {\tiny $2,1$};
    \node [dgreen] at (.86,.5) {\tiny $2,2$};
    \node [dgreen] at (.86,.8) {\tiny $2,3$};
    \node [dgreen] at (.86,.7) {\tiny $2,4$};
    %
    \node[dgrey] at (.34,0.35) {\tiny $\Q_2$};
    \node[dgrey] at (.48,0.85) {\tiny $\Q_1$};
    \node[dgrey] at (.37,0.58) {\tiny $\Q_7$};
    \node[dgrey] at (.513,0.413) {\tiny $\Q_6$};
    \node[dgrey] at (.74,0.6) {\tiny $\Q_4$};
    \node[dgrey] at (.74,0.8) {\tiny $\Q_5$};
    \node[dgrey] at (.98,0.55) {\tiny $\Q_3$};
  \end{tikzpicture}
  \caption{A configuration in the space $\tilde{V}_{(\textcolor{red}{3},\textcolor{dgreen}{4},\textcolor{blue}{3})}(\R^{1,2})$ which witnesses the ray par\-ti\-tion $(\Q_1,\dotsc,\Q_7)$, where e.g.\ $\Q_2=\{(1,2)\prec (3,3)\prec (1,3)\}$. The components $\Q_\beta$ are numbered according to their smallest label ({\small\textbf{R1}}), and the point carrying the minimal label lies at the bottom of each ray ({\small\textbf{R2}}). Recall that the verticality condition demands that all points belonging to the same cluster have to lie in the same purple plane. Note that this ray partition seems not to be the most 'efficient' one: we may merge $\Q_4$ and $\Q_5$. We will introduce a measure for 'efficiency' soon.}\label{fig:ray}
\end{figure}

\begin{rem}\label{rem:q1}
  If $q=1$, %
  the space $\tilde{V}_\mathbf{k}(\R^{p,1})$ is disconnected, with path components indexed by tuples $\Sigma\in\prod_{i=1}^r\mathfrak{S}_{k_i}$ (see \textsc{Remark \ref{rem:pi0Vk}}), and
  we would like to calculate the homology of a single path component. In order to do so, we assign to each ray partition $\Q$ of type $\mathbf{k}$ such a tuple $\Sigma$ as follows.
\end{rem}

\begin{defi}
\label{defi:stackedprec}
 Given a ray partition $\Q$, the 'stacked' total order $\prec$ on
  \[
  \ind_\mathbf{k}=(\Q_l,\prec_l)\sqcup \dotsb\sqcup (\Q_1,\prec_1),
  \]
 is determined by the property that it restricts on $\Q_\beta$ to $\prec_\beta$ and that all elements from $\Q_{\beta+1}$ are $\prec$-smaller than all elements from $\Q_\beta$.

 For each $1\le i\le r$, there is a unique $\sigma_i\in\mathfrak{S}_{k_i}$ such that $(i,\sigma(j))\prec (i,\sigma(j+1))$ for all $1\le j< k_i$: we define $\Sigma(\Q)\coloneqq(\sigma_1,\dotsc,\sigma_r)\in\prod_{i=1}^r\mathfrak{S}_{k_i}$.

 The rationale for the previous definition is the following: a configuration $z\in\tilde{V}_{\mathbf{k}}(\R^{p,1})_{\Sigma}$
can only witness ray partitions $\Q$ with $\Sigma(\Q)=\Sigma$.
\end{defi}

The following is the main theorem of the section.
\begin{theo}\label{theo:hom}
  Let $p\ge 0$, $q\ge 1$ and $\mathbf{k}=(k_1,\dotsc,k_r)$ with $k_i\ge1$.%
  \begin{enumerate}
  \item [\textup{\bfseries 1.}] The integral cohomology $H^*\pa{\tilde{V}_\mathbf{k}(\R^{p,q})}$ is freely generated by classes $u_\Q$ for each ray partition, and the cohomological degree of $u_\Q$ is
    \[|u_\Q| = p\cdot \big(r-a(\Q)\big)+(q-1)\cdot \big(|\mathbf{k}| - l(\Q)\big).\]
  \item [\textup{\bfseries 2.}] For $q=1$, the cohomology class $u_\Q$ is supported on the component $\tilde{V}_\mathbf{k}(\R^{p,1})_{\Sigma(\Q)}$.
  \end{enumerate}
\end{theo}
\noindent The rest of the section is devoted to the proof of \textsc{Theorem \ref{theo:hom}}.

\subsection{The weight filtration and the proof of Theorem \ref{theo:hom}}

Throughout this section we fix $\mathbf{k}$, $p$ and $q$ as before. We treat simultaneously the cases $q\ge2$ and $q=1$,
putting in parentheses the differences needed in the case $q=1$.
For $q\ge2$ we abbreviate $\tV\coloneqq \tilde{V}_\mathbf{k}(\R^{p,q})$; for $q=1$ we fix  $\Sigma\in\prod_i\mathfrak{S}_{k_i}$
throughout the section and abbreviate $\tV\coloneqq \tilde{V}_\mathbf{k}(\R^{p,q})_\Sigma$.

\begin{nota}
  \label{nota:bbP}
  For a positive integer $\Lambda\geq 0$ we denote by $\mathbb{P}(\Lambda)$ the set of all sequences $\lambda=\pa{\lambda_1,\dots,\lambda_l}$
  of integers $\lambda_i\geq 1$, for some $1\leq l\leq \Lambda$, satisfying $\lambda_1+\dotsb+\lambda_l=\Lambda$.
  The number $l$ is called the \emph{length} of the sequence.

  We have a natural injection $\mathbb{P}(\Lambda)\hookrightarrow \{0,\dotsc,\Lambda\}^\Lambda$, by adding a suitable number of zeroes at the end of each
  sequence; we consider on $\mathbb{P}(K)$ the inherited lexicographic order.

  We denote by $\mathbb{P}(\mathbf{k})$ the set $\mathbb{P}(\abs{\mathbf{k}})$, and by $N$ its cardinality.
\end{nota}

\begin{defi}
  The \emph{weight} of a ray partition $\Q$ is defined as
  \[
    \omega(\Q)\coloneqq\big(\abs{\Q_1},\dots,\abs{\Q_l}\big)\in\mathbb{P}(\mathbf{k}).
  \]
\end{defi}

In the following we state three lemmata and postpone their
proofs to the next subsection.
\begin{lem}
  \label{lem:rayz}
  Let $Z\in \tV$. There is a unique ray partition, called $\Q^Z$,
  which is witnessed by $Z$ and has maximal weight among all ray partitions witnessed by $Z$. (If $q=1$, we have moreover
  that $\Sigma(\Q^Z)=\Sigma$.)
\end{lem}
\begin{defi}%
  \label{defi:rayfiltration}
  %
  %
  %
  %
  Given a ray partition $\Q$, we denote by $W_\Q\subset \tV$
  the subspace containing all points $z$ with $\Q^Z=\Q$ (see \textsc{Lemma \ref{lem:rayz}}).

  %
  We define a filtration $F_{\bullet}$ on $\tV\cpt$ (see \textsc{Remark \ref{rem:PLD}})
  indexed by the linearly ordered set $\mathbb{P}(\mathbf{k})$: for all $\lambda\in\mathbb{P}(\mathbf{k})$
  %
  %
  define the $\lambda$\textsuperscript{th}
  filtration level
  $F_{\lambda}= F_{\lambda}\tV\cpt$ as the subspace containing $\infty$ and all $Z\in \tV$ with $\omega(\Q^Z)\geq\lambda$.
  Note that for $\lambda<\lambda'$ in $\mathbb{P}(\mathbf{k})$
  we have an inclusion $F_{\lambda'}\subset F_{\lambda}$.
\end{defi}

\begin{lem}
  \label{lem:filtrationclosed}
  %
  Let $\lambda\in\mathbb{P}(\mathbf{k})$. %
  Then the inclusion $F_{\lambda}\subseteq \tV\cpt$ is closed.
\end{lem}

\begin{nota}
  \label{nota:filtrationstrata}
  We can switch our indexing set of the filtration $F_{\bullet}$ from \textsc{Definition~\ref{defi:rayfiltration}}
  from $\mathbb{P}(\mathbf{k})$ to the natural numbers $1\leq \nu\leq N$ in the following way: Let $\chi\colon\set{1,\dots,N}\to\mathbb{P}(\mathbf{k})$
  be the unique order-\emph{reversing} bijection; then for $1\leq \nu\leq N$ we define
  $F_{\nu}=F_{\chi(\nu)}$.
  Moreover we set $F_0\coloneqq\set{\infty}\subset \tV\cpt$. We obtain an \emph{ascending} filtration of $\tV\cpt$ with closed levels
  (see \textsc{Lemma~\ref{lem:filtrationclosed}}):
  \[
    \set{\infty}=F_0 \subseteq F_1\,\subseteq\dotsb\subseteq F_N=\tV\cpt.
  \]
  We also denote $F_{-1}\coloneqq\emptyset$, and for $0\leq \nu\leq N$ we denote by $\mathfrak{F}_{\nu}$ the $\nu$\textsuperscript{th} filtration stratum
  of the filtration $F_{\bullet}$, i.e.\ the difference $\mathfrak{F}_{\nu}=F_{\nu}\setminus F_{\nu-1}$.
\end{nota}

\pagebreak
\begin{lem}
  \label{lem:stratacontractible}
  The strata satisfy the following properties:
  \begin{enumerate}
  \item [\textup{\bfseries\small 1.}] For each ray partition $\Q$ (with $\Sigma(\Q)=\Sigma$), the subspace
    $W_\Q$ is a contractible open manifold of dimension $\abs{\mathbf{k}}+p\cdot a(\Q) + (q-1)\cdot l(\Q)$ and a path
    component of the stratum $\mathfrak{F}_{\nu}$, where $1\leq\nu\leq N$ satisfies $\chi(\nu)=\omega(\Q)$.%
  \item [\textup{\bfseries\small 2.}] All connected components of a stratum $\mathfrak{F}_{\nu}$ with $\nu\geq 1$ arise in this way.
  \item [\textup{\bfseries\small 3.}] The closure $\ol{W}_\Q$ of $W_\Q$
    inside $\tV$ is also a smooth, orientable submanifold of dimension $\abs{\mathbf{k}}+p\cdot a(\Q) + (q-1)\cdot l(\Q)$,
  \end{enumerate}
\end{lem}
We are now ready to prove \textsc{Theorem \ref{theo:hom}}.
\begin{proof}[Proof of Theorem \ref{theo:hom}]
  We consider the Leray spectral sequence associated with the filtered space $\tV\cpt$
  and compute its reduced homology. The $E^1$-page %
  reads%
  \[
    E^1_{\nu,\mu}=H_{\nu+\mu}\pa{F_{\nu},F_{\nu-1}}=\tH_{\nu+\mu}\pa{F_{\nu}/ F_{\nu-1}}.
  \]
  By \textsc{Lemma~\ref{lem:stratacontractible}}, for all $1\leq \nu\leq N$, $\mathfrak{F}_{\nu}$ is the disjoint union of the open manifolds
  $W_{\Q}$ for $\Q$ varying among all ray partitions with $\omega(\Q)=\chi(\nu)$. In particular we have homeomorphisms
  \[
    F_{\nu}/ F_{\nu-1}\cong \mathfrak{F}_{\nu}\cpt\cong \bigvee_{\omega(\Q)=\chi(\nu)}W_{\Q}\cpt.
  \]
  Even for $\nu=0$ we have that $F_0=F_0/F_{-1}=\set{\infty}$ is formally homeomorphic to the \emph{empty wedge}.
  By \textsc{Lemma~\ref{lem:stratacontractible}}, $W_{\Q}$ is an open manifold of dimension
  $d(\Q)\coloneqq \abs{\mathbf{k}}+p\cdot a(\Q) + (q-1)\cdot l(\Q)$ for all ray partitions; hence we can apply Poincaré–Lefschetz duality and obtain for all $\nu,\mu\geq 0$ an isomorphism
  \[
    E^1_{\nu,\mu}\cong\bigoplus_{\omega(\Q)=\chi(\nu)} H_{\nu+\mu}\pa{W_{\Q}\cpt,\infty}\cong\bigoplus_{\omega(\Q)=\chi(\nu)} H^{d(\Q)-\nu-\mu}\pa{W_{\Q}}.
  \]
  Again by \textsc{Lemma~\ref{lem:stratacontractible}}, $W_{\Q}$ is contractible for all
  ray partitions with $\Q$; hence $H^{d(\Q)-\nu-\mu}\pa{W_{\Q}}$
  contributes to the first page of the spectral sequence only in the case $\mu+\nu=d(\Q)$.
  We can rewrite, for all $\nu\geq 0$ and considering all degrees $\mu$ at the same time
  \[
    E^1_{\nu,*}\cong\bigoplus_{\omega(\Q)=\chi(\nu)} H_{d(\Q)}\pa{W_{\Q}\cpt,\infty}.
  \]
  For all ray partitions $\Q$, we can now replace the relative homology of the pair $(W_{\Q}\cpt,\infty)$ with
  the relative homology of the pair $\pa{F_{\nu},F_{\nu}\setminus W_{\Q}}$ or, by excision,
  the relative homology of the pair $\pa{\ol{W}{}_{\Q}\cpt,\ol{W}{}_{\Q}\cpt\setminus W{}_{\Q}\cpt}$.
  Here, as in \textsc{Lemma~\ref{lem:stratacontractible}}, we denote by $\ol{W}_{\Q}$ the closure
  in $\tV$ of $W_{\Q}$, and by $\ol{W}{}_{\Q}\cpt$ the one-point compactification of $\ol{W}_{\Q}$ (it coincides,
  for $\nu\ge1$, with the closure of $W_{\Q}$ in $\tV\cpt$).
  We obtain
  \[
    E^1_{\nu,*}\cong\bigoplus_{\omega(\Q)=\chi(\nu)}
    H_{d(\Q)}   \pa{\ol{W}{}_{\Q}\cpt,\ol{W}{}_{\Q}\cpt\setminus W{}_{\Q}\cpt}.
  \]
  Each direct summand in the previous decomposition is isomorphic to $\Z$, generated by the
  fundamental class of the relative manifold $\pa{\ol{W}{}_{\Q}\cpt,\ol{W}{}_{\Q}\cpt\setminus \tV{}_{\Q}\cpt}$.

  By \textsc{Lemma~\ref{lem:stratacontractible}}, also $\pa{\ol{W}{}_{\Q}\cpt,\infty}$ is a relative manifold, and its fundamental
  class projects to that of $\pa{\ol{W}{}_{\Q}\cpt,\ol{W}{}_{\Q}\cpt\setminus W{}_{\Q}\cpt}$
  under the natural map
  \[
    H_{d(\Q)}\pa{\ol{W}{}_{\Q}\cpt,\infty} \to
    H_{d(\Q)}\pa{\ol{W}{}_{\Q}\cpt,\ol{W}{}_{\Q}\cpt\setminus W{}_{\Q}\cpt}.
  \]
  The previous analysis shows in particular that for all $\nu\geq 0$ the natural map $H_*\pa{F_{\nu},\infty}\to H_*\pa{F_{\nu},F_{\nu-1}}$ is surjective. This suffices to prove that the spectral sequence
  collapses on its first page. This shows that $H_*(\tilde{V}^\infty,\infty)$ is freely generated by the
  fundamental classes of the relative submanifolds $(\ol{W}{}^\infty_\Q,\infty)$, so by Poincar\'e–Lefschetz duality, $H^*(\tilde{V})$ is generated by their duals, which we call $u_\Q$; for each ray partition $\Q$ we finally see
  \begin{align*}
    |u_\Q| &= p\cdot r + (q-1)\cdot |\mathbf{k}| - d(\Q)
    \\ &= p\cdot (r-a(\Q)) + (q-1)\cdot (|\mathbf{k}| - l(\Q)).
  \end{align*}
  For $q=1$ and a fixed component $\Sigma\in\prod_i\mathfrak{S}_{k_i}$, the entire argument takes place inside $\tilde{V}=\tV_\mathbf{k}(\R^{p,q})_\Sigma$;
  more precisely, for each ray partition $\Q$ with $\Sigma(\Q)=\Sigma$, we have $\ol{W}{}_{\Q}\subset\tV_\mathbf{k}(\R^{p,q})_\Sigma$.
  Thus, the second claim of the theorem follows.
\end{proof}

\subsection{Proofs of the three lemmata}

\begin{proof}[Proof of Lemma~\ref{lem:rayz}]
  We construct $\Q^Z$ by recursively constructing a sequence $(\Q^Z_1,\prec^Z_1),\dotsc,(\Q^Z_\gamma,\prec^Z_\gamma)$ with non-empty and disjoint subsets $\Q^Z_1,\dotsc,\Q^Z_\gamma$ of $\ind_\mathbf{k}$ satisfying the axioms {\small\bfseries R1} and {\small\bfseries R2}.%
  \begin{itemize}
  \item {
      For '$\gamma=1$', we write $z_1^1=\pa{\zeta_1^1,t_1^1}\in \R^d$, and let $\Q^Z_1$ contain all $(i,j)\in\ind_{\mathbf{k}}$ such that $\smash{z_i^{\smash{j}}}\in \R^d$ has the form $\pa{\zeta_1^1,t}$ for some $t\ge t_1^1$; in other words,
      $\Q^Z_1$ contains all $(i,j)\in\ind_{\mathbf{k}}$ such that $\smash{z_i^{\smash{j}}}$ lies on the \emph{ray} starting
      at $z_1^1$ and running in the \emph{positive} $t$-direction. The order $\prec^Z_1$ on $\Q^Z_1$ is defined
      in such a way that condition {\small\bfseries W2} holds.
    }
  \item {
      For '$\gamma-1\to \gamma$', if $\smash{\Q^Z_1\sqcup\dotsb\sqcup \Q^Z_{\gamma-1}\ne \ind_\mathbf{k}}$ have been constructed, let $(i_\gamma,j_\gamma)$ be the minimal pair in $\smash{\ind_\mathbf{k}\smallsetminus (\Q^Z_1\sqcup\dotsb\sqcup \Q^Z_{\gamma-1})}$.

      Write $\smash{z_{i_\gamma}^{\smash{j_\gamma}}=(\zeta_{i_\gamma}^{\smash{j_\gamma}},t_{i_\gamma}^{\smash{j_\gamma}})}$ and let $\Q_\gamma^Z$ contain all $(i,j)\in\ind_\mathbf{k}\smallsetminus (\Q_1^Z\sqcup\dotsb\sqcup \Q^Z_{\gamma-1})$ such that $z_{i,j}\in\R^d$ has the form $\smash{(\zeta_{i_\gamma}^{\smash{j_\gamma}},t)}$ for some $t\ge t_{i_\gamma,j_\gamma}$, and define the order $\smash{\prec^Z_\gamma}$ on $\Q^Z_\gamma$ in such a way that condition {\small\bfseries W2} holds.
    }
  \end{itemize}
  Since $\ind_\mathbf{k}$ is finite, this algorithm terminates, and the resulting sequence $\Q^Z\coloneqq(\Q^Z_1,\prec^Z_1),\dotsc,(\Q^Z_{l^Z},\prec^Z_{l^Z})$ is a ray partition, which is witnessed by $Z$.

  \begin{figure}[h]
    \centering
    \begin{tikzpicture}[xscale=2.5,yscale=3]
      \draw[thick,dgrey] (.7,.6) -- (.7,.9);
      \node[dgrey] at (.6,.83) {\tiny $\Q^Z_1$};
      \draw[dgrey] (.82,.675) node[rotate=90] {\tiny $\prec$};
      \draw[dgrey] (.82,.825) node[rotate=90] {\tiny $\prec$};
      \draw[dgrey,thin] (.1,0.05) rectangle (.93,1);
      \node[dyellow] at (.3,.7) {\tiny $\bullet$};
      \node[blue] at (.3,.45) {\tiny $\bullet$};
      \node[blue] at (.3,.3) {\tiny $\bullet$};
      \node[dyellow] at (.3,.15) {\tiny $\bullet$};
      \node[red] at (.7,.9) {\tiny $\bullet$};
      \node[dgreen] at (.7,.75) {\tiny $\bullet$};
      \node[red] at (.7,.6) {\tiny $\bullet$};
      \node[red] at (.7,.45) {\tiny $\bullet$};
      \node[dgreen] at (.7,.3) {\tiny $\bullet$};
      %
      \node[dyellow] at (.42,.7) {\tiny $4,1$};
      \node[blue] at (.42,.45) {\tiny $2,2$};
      \node[blue] at (.42,.3) {\tiny $2,1$};
      \node[dyellow] at (.42,.15) {\tiny $4,2$};
      \node[red] at (.82,.9) {\tiny $1,3$};
      \node[dgreen] at (.82,.75) {\tiny $3,2$};
      \node[red] at (.82,.6) {\tiny $1,1$};
      \node[red] at (.82,.45) {\tiny $1,2$};
      \node[dgreen] at (.82,.3) {\tiny $3,1$};
    \end{tikzpicture}
    \begin{tikzpicture}[xscale=2.5,yscale=3]
      \draw[thick,dgrey] (.7,.6) -- (.7,.9);
      \node[dgrey] at (.6,.83) {\tiny $\Q^Z_1$};
      \draw[dgrey] (.82,.675) node[rotate=90] {\tiny $\prec$};
      \draw[dgrey] (.82,.825) node[rotate=90] {\tiny $\prec$};
      \node[dgrey] at (.7,.448) {$\bullet$};
      \node[dgrey] at (.6,.45) {\tiny $\Q^Z_2$};
      \draw[dgrey,thin] (.1,0.05) rectangle (.93,1);
      \node[dyellow] at (.3,.7) {\tiny $\bullet$};
      \node[blue] at (.3,.45) {\tiny $\bullet$};
      \node[blue] at (.3,.3) {\tiny $\bullet$};
      \node[dyellow] at (.3,.15) {\tiny $\bullet$};
      \node[red] at (.7,.9) {\tiny $\bullet$};
      \node[dgreen] at (.7,.75) {\tiny $\bullet$};
      \node[red] at (.7,.6) {\tiny $\bullet$};
      \node[red] at (.7,.45) {\tiny $\bullet$};
      \node[dgreen] at (.7,.3) {\tiny $\bullet$};
      %
      \node[dyellow] at (.42,.7) {\tiny $4,1$};
      \node[blue] at (.42,.45) {\tiny $2,2$};
      \node[blue] at (.42,.3) {\tiny $2,1$};
      \node[dyellow] at (.42,.15) {\tiny $4,2$};
      \node[red] at (.82,.9) {\tiny $1,3$};
      \node[dgreen] at (.82,.75) {\tiny $3,2$};
      \node[red] at (.82,.6) {\tiny $1,1$};
      \node[red] at (.82,.45) {\tiny $1,2$};
      \node[dgreen] at (.82,.3) {\tiny $3,1$};
    \end{tikzpicture}
    \begin{tikzpicture}[xscale=2.5,yscale=3]
      \draw[thick,dgrey] (.7,.6) -- (.7,.9);
      \node[dgrey] at (.6,.83) {\tiny $\Q^Z_1$};
      \draw[dgrey] (.82,.675) node[rotate=90] {\tiny $\prec$};
      \draw[dgrey] (.82,.825) node[rotate=90] {\tiny $\prec$};
      \node[dgrey] at (.7,.448) {$\bullet$};
      \node[dgrey] at (.6,.45) {\tiny $\Q^Z_2$};
      \draw[dgrey,thin] (.1,0.05) rectangle (.92,1);
      \draw[thick,dgrey] (.3,.3) -- (.3,.7);
      \draw[dgrey] (.42,.575) node[rotate=90] {\tiny $\prec$};
      \draw[dgrey] (.42,.375) node[rotate=90] {\tiny $\prec$};
      \node[dgrey] at (.2,.6) {\tiny $\Q^Z_3$};
      \node[dyellow] at (.3,.7) {\tiny $\bullet$};
      \node[blue] at (.3,.45) {\tiny $\bullet$};
      \node[blue] at (.3,.3) {\tiny $\bullet$};
      \node[dyellow] at (.3,.15) {\tiny $\bullet$};
      \node[red] at (.7,.9) {\tiny $\bullet$};
      \node[dgreen] at (.7,.75) {\tiny $\bullet$};
      \node[red] at (.7,.6) {\tiny $\bullet$};
      \node[red] at (.7,.45) {\tiny $\bullet$};
      \node[dgreen] at (.7,.3) {\tiny $\bullet$};
      %
      \node[dyellow] at (.42,.7) {\tiny $4,1$};
      \node[blue] at (.42,.45) {\tiny $2,2$};
      \node[blue] at (.42,.3) {\tiny $2,1$};
      \node[dyellow] at (.42,.15) {\tiny $4,2$};
      \node[red] at (.82,.9) {\tiny $1,3$};
      \node[dgreen] at (.82,.75) {\tiny $3,2$};
      \node[red] at (.82,.6) {\tiny $1,1$};
      \node[red] at (.82,.45) {\tiny $1,2$};
      \node[dgreen] at (.82,.3) {\tiny $3,1$};
    \end{tikzpicture}
    \begin{tikzpicture}[xscale=2.5,yscale=3]
      \draw[thick,dgrey] (.7,.6) -- (.7,.9);
      \node[dgrey] at (.6,.83) {\tiny $\Q^Z_1$};
      \draw[dgrey] (.82,.675) node[rotate=90] {\tiny $\prec$};
      \draw[dgrey] (.82,.825) node[rotate=90] {\tiny $\prec$};
      \node[dgrey] at (.7,.448) {$\bullet$};
      \node[dgrey] at (.7,.3) {$\bullet$};
      \node[dgrey] at (.6,.45) {\tiny $\Q^Z_2$};
      \node[dgrey] at (.6,.3) {\tiny $\Q^Z_4$};
      \draw[dgrey,thin] (.1,0.05) rectangle (.92,1);
      \draw[thick,dgrey] (.3,.3) -- (.3,.7);
      \draw[dgrey] (.42,.575) node[rotate=90] {\tiny $\prec$};
      \draw[dgrey] (.42,.375) node[rotate=90] {\tiny $\prec$};
      \node[dgrey] at (.2,.6) {\tiny $\Q^Z_3$};
      \node[dyellow] at (.3,.7) {\tiny $\bullet$};
      \node[blue] at (.3,.45) {\tiny $\bullet$};
      \node[blue] at (.3,.3) {\tiny $\bullet$};
      \node[dyellow] at (.3,.15) {\tiny $\bullet$};
      \node[red] at (.7,.9) {\tiny $\bullet$};
      \node[dgreen] at (.7,.75) {\tiny $\bullet$};
      \node[red] at (.7,.6) {\tiny $\bullet$};
      \node[red] at (.7,.45) {\tiny $\bullet$};
      \node[dgreen] at (.7,.3) {\tiny $\bullet$};
      %
      \node[dyellow] at (.42,.7) {\tiny $4,1$};
      \node[blue] at (.42,.45) {\tiny $2,2$};
      \node[blue] at (.42,.3) {\tiny $2,1$};
      \node[dyellow] at (.42,.15) {\tiny $4,2$};
      \node[red] at (.82,.9) {\tiny $1,3$};
      \node[dgreen] at (.82,.75) {\tiny $3,2$};
      \node[red] at (.82,.6) {\tiny $1,1$};
      \node[red] at (.82,.45) {\tiny $1,2$};
      \node[dgreen] at (.82,.3) {\tiny $3,1$};
    \end{tikzpicture}
    \begin{tikzpicture}[xscale=2.5,yscale=3]
      \draw[thick,dgrey] (.7,.6) -- (.7,.9);
      \node[dgrey] at (.6,.83) {\tiny $\Q^Z_1$};
      \draw[dgrey] (.82,.675) node[rotate=90] {\tiny $\prec$};
      \draw[dgrey] (.82,.825) node[rotate=90] {\tiny $\prec$};
      \node[dgrey] at (.7,.448) {$\bullet$};
      \node[dgrey] at (.7,.3) {$\bullet$};
      \node[dgrey] at (.3,.15) {$\bullet$};
      \node[dgrey] at (.6,.45) {\tiny $\Q^Z_2$};
      \node[dgrey] at (.6,.3) {\tiny $\Q^Z_4$};
      \draw[dgrey,thin] (.1,.05) rectangle (.92,1);
      \draw[thick,dgrey] (.3,.3) -- (.3,.7);
      \draw[dgrey] (.42,.575) node[rotate=90] {\tiny $\prec$};
      \draw[dgrey] (.42,.375) node[rotate=90] {\tiny $\prec$};
      \node[dgrey] at (.2,.6) {\tiny $\Q^Z_3$};
      \node[dgrey] at (.2,.15) {\tiny $\Q^Z_5$};
      \node[dyellow] at (.3,.7) {\tiny $\bullet$};
      \node[blue] at (.3,.45) {\tiny $\bullet$};
      \node[blue] at (.3,.3) {\tiny $\bullet$};
      \node[dyellow] at (.3,.15) {\tiny $\bullet$};
      \node[red] at (.7,.9) {\tiny $\bullet$};
      \node[dgreen] at (.7,.75) {\tiny $\bullet$};
      \node[red] at (.7,.6) {\tiny $\bullet$};
      \node[red] at (.7,.45) {\tiny $\bullet$};
      \node[dgreen] at (.7,.3) {\tiny $\bullet$};
      %
      \node[dyellow] at (.42,.7) {\tiny $4,1$};
      \node[blue] at (.42,.45) {\tiny $2,2$};
      \node[blue] at (.42,.3) {\tiny $2,1$};
      \node[dyellow] at (.42,.15) {\tiny $4,2$};
      \node[red] at (.82,.9) {\tiny $1,3$};
      \node[dgreen] at (.82,.75) {\tiny $3,2$};
      \node[red] at (.82,.6) {\tiny $1,1$};
      \node[red] at (.82,.45) {\tiny $1,2$};
      \node[dgreen] at (.82,.3) {\tiny $3,1$};
    \end{tikzpicture}
    \caption{This is how our algorithm proceeds to cover all points.}\label{fig:algorithm}
  \end{figure}

  In the case $q=1$, we have $\Sigma(\Q^Z)=\Sigma$: let $\zeta\in\R^{d-1}$ and let
  $\smash{1\le \beta,\beta'\le l(\Q^Z)}$ be two indices such that
  for all $(i,j)\in \Q_\beta$ and all $\smash{(i',j')\in \Q_{\beta'}}$ we have $\smash{\pr_{\zeta}(z_i^{\smash{j}})=\pr_{\zeta}(z_{i'}^{\smash{j'}})=\zeta}$;
  by construction we have $\smash{\pr_t(z_{i'}^{\smash{j'}})<\pr_t(z_{i}^{\smash{j}})}$ if and only if $\beta'>\beta$
  or $\beta'=\beta$ and $(i',j')\prec_\beta (i,j)$. In particular, if $i=i'$, we have
  $\sigma_i(j)<\sigma_i(j')$ if and only if $\smash{\pr_t(z_{i'}^{\smash{j'}})<\pr_t(z_i^{\smash{j}})}$, and this holds
  if and only if $(i,j)\prec(i,j')$; this shows that $\Sigma(\Q^Z)=\Sigma$.

  Suppose that $\Q$ is another ray partition witnessed by $Z$, and let $l\coloneqq l(\Q)$.
  We want to show that $\omega(\Q)<\omega(\Q^Z)$ and we do this by showing
  for each $1\le\gamma\le\min(l,l^Z)$ that if $\smash{\Q_\beta=\Q_\beta^Z}$ for $\smash{1\le\beta <\gamma}$,
  then $\smash{\Q_\gamma\subseteq \Q^Z_\gamma}$, from which we can deduce inductively that since $\Q\ne\Q^Z$
  by assumption, there is a $\gamma$ such that $\Q_\beta=\Q^Z_\beta$ for $\smash{1\le \beta<\gamma}$
  and $\smash{\Q_\gamma\subsetneq \Q_\gamma}$, so by definition of the lexicographic ordering, we get
  $\smash{\omega(\Q)<\omega(\Q^Z)}$.

  To do so, assume $\Q_\beta=\Q_\beta^Z$ for $\smash{1\le\beta <\gamma}$ and let $(i_\gamma,j_\gamma)$ be the minimum of $\smash{\ind_\mathbf{k}\smallsetminus (\Q_1\sqcup\dotsb\sqcup \Q_{\gamma-1}) = \ind_\mathbf{k}\setminus (\Q_1^Z\sqcup\dotsb\sqcup\Q_{\gamma-1}^Z)}$ as before. By {\bfseries\small R1}, the pair $(i_\gamma,j_\gamma)$ has to lie inside $\Q_\gamma$, and by {\bfseries\small R2} it is the minimum with respect to $\prec_\gamma$. By {\bfseries\small W1}, all $(i,j)\in \Q_\gamma$ have to satisfy $\smash{z_{i}^{\smash j}=(\zeta_{i_\gamma}^{\smash{j_\gamma}},t)}$ for some $t\in\R$, and by {\bfseries\small W2}, we additionally require $\smash{t\ge t_{i_\gamma}^{\smash{j_\gamma}}}$. Hence $\Q_\gamma\subseteq \Q^Z_\gamma$ as desired.
\end{proof}

\begin{proof}[Proof of Lemma~\ref{lem:filtrationclosed}]
  We show that ${\tV}\cpt\setminus F_{\lambda}$ is open.
  Let $\mathring Z\in{\tV}\cpt\setminus F_{\lambda}$, then %
  %
  we have $\omega(\Q^{\smash{\mathring Z}})<\lambda$. %
  %
  %
  Let $\varepsilon>0$ be defined as follows: we consider all (Euclidean) distances in $\R^{d-1}$ between any two \emph{distinct} projections
  $\smash{\pr_{\zeta}(\mathring z_{i}^{\smash j})}$ and $\smash{\pr_{\zeta}(\mathring z_{i'}^{\smash{j'}})}$, and also all distances in $\R$ between any two \emph{distinct} projections
  $\pr_t(\mathring z_{i}^{\smash j})$ and $\pr_t(\mathring z_{i'}^{\smash{j'}})$, for $(i,j)\neq (i',j')$ in $\ind_{\mathbf{k}}$. We obtain a finite
  set of strictly positive real numbers, and $\varepsilon$ is defined as the minimum of all these numbers.

  Let $Z$ be
  %
  any configuration
  in $\tV$ such that, for all $(i,j)\in\ind_{\mathbf{k}}$,
  the distance in $\R^d$ between $\mathring z_{i}^{\smash{j}}$ and $z_{i}^{\smash j}$ is less than $\tfrac{\varepsilon}{2}$.
  We claim that $\omega(\Q^{Z})\le \omega(\Q^{\smash{\mathring Z}})$:
  %
  %
  %
  the claim implies that for $Z$ in a neighbourhood of $\mathring Z$ in ${\tV}$,
  also $Z\notin F_{\lambda}$. This would conclude the proof, as $\tV$ is open in ${\tV}\cpt$.

  To prove the claim, we use a method similar to the proof of \textsc{Lemma~\ref{lem:rayz}}, namely we show for each $1\le\gamma\le\min(l^{\smash{\mathring Z}},l^{Z})$ that if $\smash{\Q_\beta^Z=\Q_\beta^{z'}}$ for all $1\le \beta< \gamma$, then $\smash{\Q_\gamma^{Z}\subseteq\Q_\gamma^{\mathring Z}}$, which immediately implies $\smash{\omega(\Q^{Z})\le\omega(\Q^{\mathring Z})}$.

  The minimum $(i_\gamma,j_\gamma)$ of $\smash{\ind_\mathbf{k}\setminus(\Q^{\mathring Z}_1\sqcup\dotsb\sqcup \Q_{\gamma-1}^{\mathring Z})=\ind_\mathbf{k}\setminus(\Q^{Z}_1\sqcup\dotsb\sqcup \Q_{\gamma-1}^{Z})}$ has to lie in both $\Q_\gamma^{Z}$ and $\Q_\gamma^{\mathring Z}$. Now for $(i,j)\in\ind_{\mathbf{k}}\setminus (\Q_1^{Z}\sqcup\dotsb\sqcup \Q_{\gamma-1}^{Z})$, the following holds:
  if $\mathring z_{i}^{\smash j}$ does not lie on the ray starting at $\mathring z_{i_\gamma}^{\smash {j_\gamma}}$
  and running
  %
  in the positive $t$-direction,
  then either $\pr_{\zeta}(\mathring z_{i_\gamma}^{\smash{j_\gamma}})\neq \pr_{\zeta}(\mathring z_{i}^{\smash j})$, or the two
  projections are equal but $\pr_t(\mathring z_{i_\gamma}^{\smash {j_\gamma}})>\pr_t(\mathring z_{i}^{\smash j})$. In both cases,
  by the choice of $\varepsilon$, we would also have that
  $z_{i}^{\smash j}$ does not lie on the ray starting at $z_{i_\gamma}^{\smash {j_\gamma}}$
  and running
  %
  in positive $t$-direction. This shows in particular
  that $\smash{\Q^{Z}_\gamma\subseteq\Q^{\mathring Z}_\gamma}$ as desired.
\end{proof}

\pagebreak
\begin{proof}[Proof of Lemma~\ref{lem:stratacontractible}]
  Define $H_{\Q}\subseteq{\tV}$ as the subspace of configurations of the form $Z=(z_1^1,\dots,z_r^{\smash{k_r}})$
  such that the following condition holds:
  for each $1\leq \beta\leq l$ and $(i,j)\prec_{\beta}(i',j')$, we have
  $\pr_{\zeta}(z_i^{\smash j})=\pr_{\zeta}(z_{i'}^{\smash{j'}})$
  and $\pr_t(z_i^{\smash j})\leq\pr_t(z_{i'}^{\smash{j'}})$.
  Then $H_{\Q}$ is a closed subspace of ${\tV}$, as it is defined by imposing some equalities and some weak inequalities
  (using `$\leq$') between the coordinates.

  Note, however, that the \emph{same} space $H_\Q$ can be defined, as a subspace of $\tV$, by replacing the second condition
  '$\pr_t(z_{i}^{\smash j})\leq\pr_t(z_{i'}^{\smash {j'}})$' with '$\pr_t(z_i^{\smash j})<\pr_t(z_{i'}^{\smash{j'}})$'.
  As subspaces of $(\R^d)^{|\mathbf{k}|}$, we then have the following:
  \begin{itemize}
   \item There is a linear subspace of $(\R^d)^{|\mathbf{k}|}$ determined by $\pr_1(z_{i}^{\smash j})=\pr_1(z_{i}^{\smash{j'}})$ for all $1\le i\le r$
   and $1\le j,j'\le k_i$, and inside this linear subspace, $\tV$ is open, defined by the strict inequalities
   $z_{i}^{\smash j}\neq z_{i'}^{\smash{j'}}$ for each $(i,j)\neq (i',j')\in T_\mathbf{k}$;
   \item There is a linear subspace of $(\R^d)^{|\mathbf{k}|}$ determined by the linear equations
   \begin{itemize}
    \item $\pr_1(z_{i}^{\smash j})=\pr_1(z_{i}^{\smash {j'}})$ for all $1\le i\le r$ and $1\le j,j'\le k_i$;
    \item $\pr_{\zeta}(z_i^{\smash j})=\pr_{\zeta}(z_{i'}^{\smash{j'}})$ for all $1\le\beta\le l(\Q)$ and $(i,j),(i',j')\in\Q_\beta$;
   \end{itemize}
  inside this linear subspace, $H_\Q$ is open, defined by the strict inequalities
  \begin{itemize}
   \item $z_{i}^{\smash j}\neq z_{i'}^{\smash {j'}}$ for $(i,j)\neq (i',j')\in T_\mathbf{k}$;
   \item $\pr_t(z_i^{\smash j})<\pr_t(z_{i'}^{\smash {j'}})$ for all $1\le\beta\le l(\Q)$ and $(i,j)\prec_{\beta}(i',j')\in\Q_\beta$.
  \end{itemize}
  \end{itemize}
 It follows that $H_\Q$ is an orientable submanifold without boundary of $\tV$, which in turn is an orientable
  submanifold without boundary of $(\R^d)^{|\mathbf{k}|}$.

  The dimension of $H_\Q$
  is computed by noting that, locally, we have the following parameters describing a configuration $Z\in H_\Q$.
  \begin{itemize}
  \item For all $1\leq \beta\leq l$, we have a parameter $\zeta_\beta=(\zeta^1_\beta,\zeta^2_\beta)\in \R^p\times \R^{q-1}$ which corresponds
    to the (unique) value attained by $\pr_{\zeta}(z_{i}^{\smash j})$
    for all $(i,j)\in\Q_{\beta}$. However, if two rays $\Q_\beta$ and $\Q_{\beta'}$ share a cluster, their further projections $\zeta^1_{\smash \beta}$ and $\zeta^1_{\smash{\beta'}}$ have to coincide inside $\R^p$.
    Hence, we get for each equivalence class of rays a choice in $\R^p$, and for each ray a choice in $\R^{q-1}$. This yields $p\cdot a(\Q) + (q-1)\cdot l(\Q)$ parameters in $\R$.
  \item For each $(i,j)\in\ind_{\mathbf{k}}$ we have a parameter $t_i^{\smash j}=\pr_t(z_i^{\smash j})$ in $\R$.
  \end{itemize}
  We clearly have $W_\Q\subseteq H_\Q$, and ${W}_{\Q}$ can be characterised
  as the subspace of $H_\Q$ containing configurations $Z$ for which the following condition
  holds: for all $1\leq \beta<\beta'\leq l$ and for all $(i,j)\in\Q_{\beta}$ and $(i',j')\in\Q_{\beta'}$,
  either $\pr_{\zeta}(z_{i}^{\smash j})\neq \pr_{\zeta}(z_{i'}^{\smash {j'}})$ or $\pr_t(z_{i'}^{\smash {j'}})<\pr_t(z_{i}^{\smash j})$.
  Thus, $W_\Q\subseteq H_\Q$ is an open subspace (it is a finite intersection of open subspaces).

  To see that $W_\Q$
  is dense in $H_\Q$, note that one can slightly perturb all parameters of any configuration
  $z\in H_\Q$ of the forms
  \begin{itemize}
   \item $\zeta_\beta^2\in \R^{q-1}$, for $1\le \beta\le l(\Q)$;
   \item $\zeta_\beta^1\in\R^p$, for $\beta$ ranging in a set of representatives of the $a(\Q)$ equivalence classes of rays,
  \end{itemize}
   to ensure that they attain all different values, so that the new perturbed con\-figuration $Z'$ lies in $W_\Q$.
   Hence, $H_\Q$ is the closure $\ol{W}_\Q$ inside $\tV$.

  To prove that ${W}_\Q$ is contractible, we
  choose distinct numbers $\mathring t^{\, 1}_1,\dotsc \mathring t_{r}^{\, k_r}\in \R$ such that for $1\le \beta,\beta'\le l$ and $(i,j)\in\Q_\beta$ and $(i',j')\in\Q_{\beta'}$, we have $\mathring t_{i'}^{\smash {j'}}<\mathring t_{i}^{\smash j}$ if and only if $\beta'>\beta$ or $\beta'=\beta$ and $(i',j')\prec_\beta (i,j)$, i.e.\ the $\mathring t_{i}^{\smash j}$ are ordered exactly as the stacked order on $\Q_l\sqcup\dotsb\sqcup \Q_1$ from \textsc{Remark~\ref{rem:q1}} prescribes.

  We define $\mathring{z}_{i}^{\smash j}\coloneqq (0,\mathring{t}_{i}^{\smash j})\in\R^d$ for all $(i,j)\in\ind_{\mathbf{k}}$; note that the configuration $\mathring{Z}=(\mathring{z}_{1}^1,\dotsc,\mathring{z}_{r}^{k_r})$ lies in ${W}_\Q$. We can connect any configuration $Z\in W_\Q$ to $\mathring{Z}$ by \emph{linear interpolation} inside
  $(\R^d)^{|\mathbf{k}|}$: for all $0\leq s\leq 1$ we consider the configuration $s\cdot Z+(1-s)\cdot \mathring{Z}$,
  where we set
  \[
    (s\cdot Z+(1-s)\cdot \mathring{Z})_{i}^{\smash j}\coloneqq s\cdot z_{i}^{\smash j}+(1-s)\cdot \mathring{z}_{i}^{\smash j}\in\R^d.
  \]
    \begin{figure}
  \centering
  \begin{tikzpicture}[xscale=5,yscale=3.6]
    \node [thin,grey] at (2.35,1.2) {\tiny $\R^3$};
    \draw [very thin,bgrey] (0,0) -- (.6,.4) -- (2.3,.4);
    \draw [very thin,bgrey]  (.6,.4) -- (.6,1.2);
    \draw [blila] (.2,0) -- (.77,.4) -- (.77,1.2);
    \draw [blila] (.65,0) -- (1.1525,.4) -- (1.1525,1.2);
    \draw [blila] (1.6,0) -- (1.96,.4) -- (1.96,1.2);
    \draw [thin,grey] (2,0) -- (2.3,.4) -- (2.3,1.2) -- (2,1);
    \draw [thin,grey] (0,1) -- (.6,1.2) -- (2.3,1.2);
    \draw [dgrey,dgrey] (1.225,1.1) -- (1.225,.15);
    \draw [very thick, white] (.55,.7) -- (1.225,.25);
    \draw [thin, byellow] (.55,.7) -- (1.225,.25);
    \draw [very thick,white] (.3,.5) -- (1.225,.35);
    \draw [thin, byellow] (.3,.5) -- (1.225,.35);
    \draw [thin, bgreen] (1.7,.7) -- (1.225,.65);
    \draw [very thick, white] (1.7,.35) -- (1.225,.55);
    \draw [thin, bgreen] (1.7,.35) -- (1.225,.55);
    \draw [very thick,white] (1.05,.85) -- (1.225,.47);
    \draw [thin, bblue] (1.05,.85) -- (1.225,.47);
    \draw [very thick, white] (.8,.5) -- (1.225,.75);
    \draw [thin, bred] (.8,.5) -- (1.225,.75);
    \draw [very thick, white] (.8,.7) -- (1.225,.85);
    \draw [thin, bblue] (.8,.7) -- (1.225,.85);
    \draw [very thick,white] (.8,.8) -- (1.225,1.05);
    \draw [thin, bred] (.8,.8) -- (1.225,1.05);
    \draw [lila] (.65,0) -- (.65,1) -- (1.1525,1.2);
    \draw [lila] (1.6,0) -- (1.6,1) -- (1.96,1.2);
    \draw [lila] (.2,0) -- (.2,1) -- (.77,1.2);
    \draw [thin,grey] (0,0) rectangle (2,1);
    \draw [very thick,dgrey] (1.225,.65) -- (1.225,.55);
    \draw [very thick,dgrey] (1.225,1.05) -- (1.225,.75);
    \node[dgrey] at (1.32,1.145) {\tiny $(0,0)\times \R$};
    \node [dred] at (1.225,1.05) {\tiny $\bullet$};
    \node [blue] at (1.225,.85) {\tiny $\bullet$};
    \node [dred] at (1.225,.75) {\tiny $\bullet$};
    \node [dgreen] at (1.225,.65) {\tiny $\bullet$};
    \node [dgreen] at (1.225,.55) {\tiny $\bullet$};
    \node [blue] at (1.225,.47) {\tiny $\bullet$};
    \node [dyellow] at (1.225,.35) {\tiny $\bullet$};
    \node [dyellow] at (1.225,.25) {\tiny $\bullet$};
    \node [dred] at (.8,.5) {\tiny $\bullet$};
    \node [blue] at (.8,.7) {\tiny $\bullet$};
    \node [dred] at (.8,.8) {\tiny $\bullet$};
    \node [blue] at (1.05,.85) {\tiny $\bullet$};
    \node [dyellow] at (.3,.5) {\tiny $\bullet$};
    \node [dyellow] at (.55,.7) {\tiny $\bullet$};
    \node [dgreen] at (1.7,.7) {\tiny $\bullet$};
    \node [dgreen] at (1.7,.35) {\tiny $\bullet$};
    \node [dred] at (.752,.5) {\tiny $1,1$};
    \node [dyellow] at (.5,.7) {\tiny $4,2$};
    \node [dyellow] at (.25,.5) {\tiny $4,1$};
    \node [blue] at (1,.85) {\tiny $3,2$};
    \node [blue] at (.752,.7) {\tiny $3,1$};
    \node [dred] at (.752,.8) {\tiny $1,2$};
    \node [dgreen] at (1.75,.7) {\tiny $2,2$};
    \node [dgreen] at (1.75,.35) {\tiny $2,1$};
    \node[dgrey] at (1.27,.25) {\tiny $\Q_5$};
    \node[dgrey] at (1.27,.35) {\tiny $\Q_4$};
    \node[dgrey] at (1.27,.47) {\tiny $\Q_3$};
    \node[dgrey] at (1.27,.6) {\tiny $\Q_2$};
    \node[dgrey] at (1.27,.9) {\tiny $\Q_1$};
  \end{tikzpicture}
  \caption{The linear interpolation from $Z$ to the configuration $\mathring{Z}$ where all points are on the line $(0,0)\times\R$ and the components of the ray partition are ordered.}
\end{figure}
Since $\pr_{\zeta}$ is a linear map, for all $0\leq s\leq 1$ we have that, for fixed $1\leq \beta\leq l$,
  the map $\pr_{\zeta}$ attains the same value on all points of the form $s\cdot z_i^{\smash j}+(1-s)\cdot \mathring{z}_{i}^{\smash j}$,
  for $(i,j)$ ranging in $\Q_{\beta}$;
  similarly, for all $(i,j)\prec_{\beta}(i',j')\in\Q_{\beta}$ we have an inequality
  %
    $\pr_t((s\cdot Z+(1-s)\cdot \mathring{Z} \big)_{i,j})<\pr_t((s\cdot Z+(1-s)\cdot \mathring{Z} )_{i'}^{\smash {j'}})$.
  %
  This means\linebreak~\\[-14px] that the linear interpolation takes place inside the subspace $\ol{W}_{\Q}$.

  With some more effort one can show that the linear interpolation takes place inside ${W}_\Q$
  by using the characterisation
  of the points of ${W}_\Q$ inside $\ol{W}_\Q$:
  for all $0<s\leq 1$, $1\leq \beta<\beta'\leq l$, $(i,j)\in\Q_{\beta}$, and $(i',j')\in\Q_{\beta'}$,
  if $\pr_{\zeta}(z_{i}^{\smash j})\neq \pr_{\zeta}(z_{i'}^{\smash{j'}})$ then we
  also have the inequality
  \begin{align*}
    \pr_{\zeta}\pa{\big(s\cdot Z+(1-s)\cdot \mathring{Z} \big){}_{i}^{j}}%
    %
    \ne\pr_{\zeta}\pa{ \big(s\cdot Z+(1-s)\cdot \mathring{Z} \big){}_{i'}^{j'}},
  \end{align*}
  and if $\pr_t(z_{i'}^{\smash{j'}})<\pr_t(z_{i}^{\smash j})$ then also
  \[
    \pr_t\pa{\big(s\cdot Z+(1-s)\cdot \mathring{Z} \big){}_{i}^{j}} < \pr_t\pa{\big(s\cdot Z+(1-s)\cdot \mathring{Z} \big){}_{i'}^{j'}},
  \]
  by the same argument used above. At time $s=1$, we already know that $z$ lies in
  $W_\Q$. Thus, we have exhibited a contraction of $W_\Q$ onto its point $\mathring{Z}$.
\end{proof}

\pagebreak
\subsection{Growth of Betti numbers and cup product indecomposables}
We will not attempt to give a precise description of $H^*(\tV_{\mathbf{k}}(\R^{p,q}))$ as a ring. The aim
of this short subsection is to disprove a natural, yet naïve conjecture on multiplicative generators
of $H^*(\tV_{\mathbf{k}}(\R^{p,q}))$.

\begin{nota}
For all $1\le i<j\le r$ there is a Fadell–Neuwirth map of the form
$\pr_{i,j}\colon \tV_{\mathbf{k}}(\R^{p,q})\to\tV_{(k_i,k_{j})}(\R^{p,q})$ which forgets all clusters but
the $i$\textsuperscript{th} and $j$\textsuperscript{th} ones.
The map $\pr_{i,j}$ is in general not a fibration, though it is a fibration in the quite special case
in which $k_l=1$ for all $l\neq i,j$.
\end{nota}

In the case $k_1,\dotsc,k_r=1$, the space $\tV_{\mathbf{k}}(\R^{p,q})$ is homeomorphic to the classical
ordered configuration space $\tilde{C}_r(\R^d)$,
and the maps $\pr_{i,j}$ reduce to a version of\linebreak~\\[-14px] the classical Fadell–Neuwirth fibrations
$\pr_{i,j}\colon \tilde{C}_r(\R^d)\to \tilde{C}_2(\R^d)$. Denote by\linebreak~\\[-14px] $\theta\in H^{d-1}(\tilde{C}_2(\R^d))\cong \Z$ a generator,
and let $\theta_{i,j}\coloneqq\pr_{i,j}^*\theta\in H^{d-1}(\tilde{C}_r(\R^d))$ be the pulled
back cohomology class. It is then a classical result by Arnol'd \cite{arnold} that the classes
$\theta_{i,j}$ generate $H^*(\tilde{C}_r(\R^d))$ as a ring.
A natural conjecture would then be the following:
\begin{conj}[Naïve conjecture]
 \label{conj:naive}
The ring $H^*(\tV_{\mathbf{k}}(\R^{p,q}))$ is generated
in arity 2,\linebreak i.e.\ by all cohomology classes that can be obtained as a pullback along
$\smash{\pr_{i,j}}$, for some $1\le i<j\le r$, from a cohomology class in $\smash{H^*(\tV_{(k_i,k_{j})}(\R^{p,q}))}$.
\end{conj}
The following example shows that \textsc{Conjecture~\ref{conj:naive}} is wrong in general.

\newcommand{\mathrm{id}}{\mathrm{id}}
\newcommand{{\mathrm{Id}}}{{\mathrm{Id}}}
\begin{expl}
Consider the case $r=3$, $k\ge2$, and $p\ge1$ and select the com\-ponent
of $\smash{\tV_3^k(\R^{p,1})=\tV_{(k,k,k)}(\R^{p,1})}$ corresponding to ${\mathrm{Id}} \coloneqq (\mathrm{id},\mathrm{id},\mathrm{id}) \in (\mathfrak{S}_{k})^3$.

By \textsc{Theorem~\ref{theo:hom}}, $\smash{H^*(\tV^k_3(\R^{p,1})_{{\mathrm{Id}}})}$ is concentrated in degrees
$0$, $p$ and $2p$, with Betti numbers equal, respectively, to:
\begin{itemize}
 \item $1$ in degree $0$;\vspace*{-3px}
 \item $3\cdot (\tbinom{2k}k-1)$ in degree $p$;\vspace*{-3px}
 \item $\tbinom{3k}{k}\cdot \tbinom{2k}{k} -3\cdot\tbinom{2k}{k}+2$ in degree $2p$.
\end{itemize}
Similarly, $H^*(\tV^k_2(\R^{p,1})_{{\mathrm{Id}}}$ is concentrated
in degrees $0$ and $p$, with Betti numbers equal, respectively, to $1$ and $\binom{2k}k-1$.
The projections $\pr_{1,2},\pr_{1,3},\pr_{2,3}$ exhibit an isomorphism between
$H^p(\tV^k_3(\R^{p,1})_{{\mathrm{Id}}})$ and %
$H^p(\tV^k_2(\R^{p,1})_{{\mathrm{Id}}})^{\oplus 3}$.

If \textsc{Conjecture~\ref{conj:naive}} were true, the set of all cup products of pairs of classes in $H^p(\tV^k_3(\R^{p,1})_{{\mathrm{Id}}})$ would suffice to generate the entire cohomology group
$H^{2p}(\tV^k_3(\R^{p,1})_{{\mathrm{Id}}})$; in particular we would have
\[
 \tbinom{3k}{k}\cdot \tbinom{2k}{k} -3\cdot\tbinom{2k}{k}+2\le \pa{3\cdot \pa{\tbinom{2k}k-1}}^2.
\]
However, using Stirling's approximation,
the left hand side grows as fast as
$\smash{27^k\cdot \frac{\sqrt{3}}{2\pi k}}$ for $k\to\infty$, whereas the right hand side grows as fast as $\smash{9\cdot 16^k\cdot\frac{1}{\pi k}}$.
Hence for large $k$ the inequality does not hold; in particular the graded ring $H^*(\tV^k_3(\R^{p,1})_{{\mathrm{Id}}})$
has non-trivial indecomposable elements in degree $2p$.
\end{expl}
We do not expect that the situation becomes
better when considering more than three clusters, or taking a value of $q$ higher than $1$.
\pagebreak

\subsection{Setting and results}

We fix throughout the section a cluster size $k\ge 1$ and we will abbreviate $\tV_r(\R^{p,q})\coloneqq \tV_r^k(\R^{p,q})$ and $V_r(\R^{p,q})\coloneqq V_r^k(\R^{p,q})$. If $p$ and $q$ are fixed and clear from the context, we may also just write $\tV_r$ resp. $V_r$.

\newcommand{\mathrm{stab}}{\mathrm{stab}}
\begin{constr}
  For each $r\ge 0$, $p\ge 0$, and $q\ge 1$, we have \emph{stabilisation maps}
  \[\mathrm{stab}\colon V^k_r(\R^{p,q})\to V^k_{r+1}(\R^{p,q})\]
  by adding an extra cluster on the 'far right' with respect to the first coordinate of $\R^{p+q}$, as depicted in \textsc{Figure~\ref{fig:stabilisation}}.
\end{constr}

\begin{figure}[h]
  \centering
  \begin{tikzpicture}[scale=3]
    \draw[thin,dgrey] (0,0) rectangle (1,1);
    \draw[dgrey,thick] (.7,.7) -- (.7,.5);
    \draw[dgrey,thick] (.7,.3) -- (.7,.2);
    \draw[dgrey,thick] (.4,.87) -- (.4,.35);
    \draw[dgrey,thick] (.3,.8) -- (.26,.8) -- (.26,.4) -- (.3,.4);
    \draw[dgrey,thick] (.3,.6) -- (.34,.6) -- (.34,.3) -- (.3,.3);
    \node at (.3,.3) {\tiny $\bullet$};
    \node at (.3,.4) {\tiny $\bullet$};
    \node at (.3,.6) {\tiny $\bullet$};
    \node at (.3,.8) {\tiny $\bullet$};
    \node at (.7,.5) {\tiny $\bullet$};
    \node at (.7,.7) {\tiny $\bullet$};
    \node at (.7,.2) {\tiny $\bullet$};
    \node at (.7,.3) {\tiny $\bullet$};
    \node at (.4,.87) {\tiny $\bullet$};
    \node at (.4,.35) {\tiny $\bullet$};
  \end{tikzpicture}~~~\raisebox{38px}{$\mapsto$}~~
  \begin{tikzpicture}[scale=3]
    \draw[thin,bgrey] (1,0) -- (1,1);
    \draw[thin,dgrey] (0,0) rectangle (1.5,1);
    \draw[dgrey,thick] (.7,.7) -- (.7,.5);
    \draw[dgrey,thick] (.7,.3) -- (.7,.2);
    \draw[dgrey,thick] (.4,.87) -- (.4,.35);
    \draw[dgrey,thick] (.3,.8) -- (.26,.8) -- (.26,.4) -- (.3,.4);
    \draw[dgrey,thick] (.3,.6) -- (.34,.6) -- (.34,.3) -- (.3,.3);
    \draw[bblue,thick] (1.25,.4) -- (1.25,.6);
    \node at (.3,.3) {\tiny $\bullet$};
    \node at (.3,.4) {\tiny $\bullet$};
    \node at (.3,.6) {\tiny $\bullet$};
    \node at (.3,.8) {\tiny $\bullet$};
    \node at (.7,.5) {\tiny $\bullet$};
    \node at (.7,.7) {\tiny $\bullet$};
    \node at (.7,.2) {\tiny $\bullet$};
    \node at (.7,.3) {\tiny $\bullet$};
    \node at (.4,.87) {\tiny $\bullet$};
    \node at (.4,.35) {\tiny $\bullet$};
    \node[dblue] at (1.25,.4) {\tiny $\bullet$};
    \node[dblue] at (1.25,.6) {\tiny $\bullet$};
  \end{tikzpicture}
  \caption{The stabilisation map $\on{stab}\colon V^2_5(\R^{1,1})\to V^2_6(\R^{1,1})$, which adds a new cluster (blue) on the far right.}
  \label{fig:stabilisation}
\end{figure}

\definecolor{ddgreen}{rgb}{0,.5,0}
\begin{prop}
  The induced maps in homology
  \[\mathrm{stab}_*\colon H_m(V_r(\R^{p,q}))\to H_m(V_{r+1}(\R^{p,q})).\]
  are split monic for each $m,r,p\ge 0$ and $q\ge 1$.
\end{prop}
\begin{proof}
  This proof generalises the one from \cite[Lem.\,5.1]{palmerdis} which uses the classical idea of a 'power-set map': From \cite{dolddec}, we want to use the following
  \begin{quote}
    \mbox{\textbf{Lemma~2.}} \emph{Suppose we are given a sequence $(0=A_0\stackrel{s_0}{\to} A_1\stackrel{s_1}{\to} \dotsb)$ of abelian groups, and assume that there are $\tau_{j,r}:A_r\to A_j$ for $1\le j\le r$ such that $\tau_{r,r}=\on{id}$ and $\tau_{j,r}-\tau_{j,r+1}\circ s_r:A_r\to A_j$ lies in the image of $s_{j-1}$. Then every $s_r$ is split monic.}
  \end{quote}
  In order to do so, we first note that $V_0=*$, so all spaces $V_r=V_r(\R^{p,q})$ are canonically based by $\mathrm{stab}^{r}(*)\in V_r$, and the stabilisation maps are basepoint-preserving by definition.
  For a fixed $m\ge 0$, let $A_r\coloneqq \tilde{H}_m(V_r)$, so we have maps $s_r\colon A_r\to A_{r+1}$ induced by the stabilisation.

  Now recall for $\ell\ge 0$ the $\ell$-fold symmetric product $\on{SP}^\ell V_r\coloneqq (V_r)^\ell/\mathfrak{S}_\ell$ where $\mathfrak{S}_\ell$ acts by coordinate permutation. We will denote elements of $\on{SP}^\ell V_r$ as formal sums of elements
  of $V_r$, but we will use a \emph{green} sign $\color{ddgreen}{+}$ resp. $\color{ddgreen}{\sum}$ in order to distinguish the notation for the symmetric product from
  \textsc{Notation~\ref{nota:suggestivesum}}. For the binomial coefficient $\ell\coloneqq\tbinom{r}{j}$, consider the maps
  \begin{align*}
    \gamma_{j,r}\colon V_r\to \on{SP}^\ell V_j,\quad
    \sum_{i=1}^r[z_i] \mapsto {\color{ddgreen}{\sum_{\substack{S\subseteq\{1,\dotsc,r\}\\\# S=j}}}}\sum_{i\in S} [z_i].
  \end{align*}
  A priori, $\gamma_{j,r}$ is not based, but it can be homotoped to a based map since $V_r$ is well-based.
  Then $\gamma_{r,r}=\on{id}$ and we have a homotopy
  \[
    \gamma_{j,r+1}\circ \on{stab}_r\simeq \gamma_{j,r}{~\color{ddgreen}{+}~} {\textstyle\on{SP}^{\smash{\tbinom{r}{j-1}}}}\pa{\on{stab}_{j-1}}\circ \gamma_{j-1,r}.
 \]%
  of maps $V_r\to\on{SP}^{\binom{r+1}{j}}V_j$. Applying the functor $\pi_m\circ \on{SP}^\infty\cong \tilde{H}_m$ and using the 'flattening' map $\varphi_\ell\colon \on{SP}^\infty\on{SP}^\ell V_r\to \on{SP}^\infty V_r$, we obtain the desired system $(\tau_{j,r})$ of morphisms for Dold's lemma by
  \[
    \begin{tikzcd}[column sep=5em, row sep=1.2em]
      A_r\ar[d,equal]\ar[rr,dashed,"\tau_{j,r}"] && A_j\ar[d,equal]\\
      \pi_m\pa{\on{SP}^\infty V_r}\ar{r}[swap]{\pi_m\pa{\on{SP}^\infty \gamma_{i,r}}} & \pi_m\pa{\on{SP}^\infty\on{SP}^\ell V_j}\ar{r}[swap]{\pi_m(\varphi_\ell)} & \pi_m\pa{\on{SP}^\infty V_j}.
    \end{tikzcd}
  \]
  ~\\[-30px]
\end{proof}
~\\[-24px]

In contrast, surjectivity of $\mathrm{stab}_*\colon H_m(V^k_r(\R^{p,q}))\to H_m(V^k_{r+1}(\R^{p,q}))$ holds
only in a certain range. The rest of this section is devoted to the proof of the following stability theorem:

\begin{theo}\label{theo:stab}
  For all $p\ge 0$ and $q\ge 1$ with $(p,q)\ne (0,1)$, the induced maps
  \begin{align*}
    H_m(V^k_r(\R^{p,q}))\to H_m(V^k_{r+1}(\R^{p,q})).
  \end{align*}
  are isomorphisms for $m\le \tfrac{r}{2}$.
\end{theo}

\noindent Many cases of \textsc{Theorem~\ref{theo:stab}} have already been solved:
\begin{itemize}
\item {We know that $\pi_0V_r(\R^{0,1})\cong \mathfrak{S}_{rk}/(\mathfrak{S}_k)^r$, so there is no stability result to be expected in the case $p=0$ and $q=1$.}
\item {For $p=0$, we are in the case without any vertical coupling condition. This can alternatively be described by embeddings of (disconnected) $0$-dimensional manifolds into $\R^q$. For these cases, the theorem was proven for $q\ge 3$ in \cite{palmerdis} and for $q= 2$ in \cite{tran}.}
\item {In \cite{latifi}, the case $p+q\ge 3$ was considered and proven. Actually, Latifi writes down the proof only for $p=2$ and $q=1$, but her strategy works whenever $p+q\ge 3$.}
\end{itemize}
Hence we only have to prove the single remaining case $(p,q)=(1,1)$. However, since the method is the same, we will provide a proof for arbitrary $(p,1)$ with $p\ge 1$. Our proof uses different methods than Latifi's proof.

\subsection{The dexterity filtration}

\begin{nota}
  In the remainder of the section we assume $q=1$. In this case, a vertical cluster $[z]=\{z^1,\dotsc,z^k\}\subseteq\R^{p+1}$ is canonically ordered by the last coordinate $t^{\smash j}\coloneqq \pr_t(z^{\smash j})\in \R$, and $[z]$ is determined by their common projection $\zeta\coloneqq\smash{\pr_\zeta}(z)\in\R^p$ and the real numbers $t^1,\dotsc,t^k$. Hence, we can write
  \[\{z^1,\dotsc,z^k\} = (\zeta;t^1<\dotsb<t^k).\]
\end{nota}

\begin{defi}
Let $Z\coloneqq \pa{z_1,\dots,z_r}\in \tilde{V}_{r}$ be an ordered configuration, where $z_i=(z_{i}^1,\dots,z_{i}^k)$. %
We define an equivalence relation $\sim_Z$ on the set $\set{1,\dots,r}$. First, set $i\sim_Z i'$ whenever the two following
conditions hold:
\begin{itemize}
 \item $z_i$ and $z_{\smash{i'}}$ are \emph{aligned}, i.e.\ they are contained in the same $t$-line, or equivalently
 $\pr_{\zeta}(z_i)=\pr_{\zeta}(z_{i'})$ in $\R^{p}$, since $q=1$;
 \item $z_i$ and $z_{i'}$ are \emph{entangled}, i.e.\ their convex hulls (contained in the vertical line) intersect each other, see \mbox{\textsc{Figure~\ref{fig:entangle}}}.
\end{itemize}
Let $\sim_Z$ be the equivalence relation generated by the above basic relations $\sim_Z$.
We define the \emph{dexterity}
of $Z$, denoted $\delta(Z)$, to be the number $s$ of equivalence classes of $\sim_Z$. Since the notion of dexterity is invariant under the permutation action of the group $\mathfrak{S}_k\wr\mathfrak{S}_r$, we obtain a notion of dexterity
also for unordered configurations in $V_r$.
\end{defi}

\begin{figure}[h]
  \centering
  \begin{tikzpicture}[scale=3]
    \draw[thin] (0,0) rectangle (1,1);
    \draw[dgrey,thick] (.7,.7) -- (.7,.5);
    \draw[dgrey,thick] (.7,.3) -- (.7,.2);
    \draw[dgrey,thick] (.5,.87) -- (.5,.35);
    \draw[dgrey,thick] (.3,.8) -- (.26,.8) -- (.26,.4) -- (.3,.4);
    \draw[dgrey,thick] (.3,.1) -- (.3,.2);
    \draw[dgrey,thick] (.3,.6) -- (.34,.6) -- (.34,.3) -- (.3,.3);
    \node at (.3,.3) {\tiny $\bullet$};
    \node at (.3,.4) {\tiny $\bullet$};
    \node at (.3,.1) {\tiny $\bullet$};
    \node at (.3,.2) {\tiny $\bullet$};
    \node at (.3,.6) {\tiny $\bullet$};
    \node at (.3,.8) {\tiny $\bullet$};
    \node at (.7,.5) {\tiny $\bullet$};
    \node at (.7,.7) {\tiny $\bullet$};
    \node at (.7,.2) {\tiny $\bullet$};
    \node at (.7,.3) {\tiny $\bullet$};
    \node at (.5,.87) {\tiny $\bullet$};
    \node at (.5,.35) {\tiny $\bullet$};
    \draw[thin,bgrey] (.2,.85) rectangle (.4,.25);
  \end{tikzpicture}
  \caption{The leftmost two upper clusters are entangled and hence form an equivalence class. Therefore, the dexterity is $5$, while the number of clusters is $6$.}\label{fig:entangle}
\end{figure}

\begin{defi}[Dexterity filtration]
For $s\ge -1$ we let $F_s V_{r}$ be the subspace of all $[Z]\in V_{r}$ satisfying $\delta[Z]\ge r-s$.
We have inclusions
\[
 \emptyset=F_{-1} V_{r}\subseteq F_0 V_{r}\subseteq\dotsb\subseteq F_{r-1}V_{r}=V_{r}.
\]
We denote by $\mathfrak{F}_s V_r$ the $s$\textsuperscript{th} stratum of the filtration
\[
\mathfrak{F}_s V_r\coloneqq F_sV_r\setminus F_{s-1}V_r\subset V_r.
\]
\end{defi}
Note that each filtration level $F_sV_{r}$ is an open subspace of $V_r$, in particular it is a manifold
of the same dimension $p\cdot r+r\cdot k$; the stratum $\mathfrak{F}_sV_r$ is a closed subset of $F_sV_r$.

Additionally, the stabilisation map $\mathrm{stab}\colon V_{r}\to V_{r+1}$ is filtration-preserving, i.e.\ it restricts to maps $F_sV_r\to F_sV_{r+1}$ and even to maps of strata $\mathfrak{F}_sV_r\to \mathfrak{F}_sV_{r+1}$.
\begin{lem}
The stratum $\mathfrak{F}_sV_r\subset F_sV_r$ is a closed submanifold of codimension $s\cdot p$, i.e.\ of dimension $p\cdot (r-s)+r\cdot k$.
\end{lem}
\begin{proof}
 Let $[\mathring{Z}] =\sum_i [\mathring{z}_i]\in \mathfrak{F}_sV_r$ with $[\mathring{z}_i]=(\mathring \zeta_i;\mathring t_i^1,\dotsc,\mathring t_i^k)$, where $\mathring \zeta_i\in \R^p$ and $\mathring t_i^1<\dotsb< \mathring t_i^k\in \R$. A small neighbourhood of $[\mathring{Z}]$ in $V_r$ is described by the following local parameters constituting $[Z]=\sum_i(\zeta_i,t_i^1,\dotsc,t_i^{k})$:
 \begin{itemize}
  \item $\zeta_i$, ranging in a neighbourhood of $\mathring{\zeta}_i\in\R^p$, for all $1\le i\le r$;\vspace*{-3px}
  \item $t_i^{\smash j}$, ranging in a neighbourhood of $\mathring t_i^{\smash j}\in\R$, for all $1\le i\le r$ and $1\le j\le k$.
 \end{itemize}
 For $[Z]$ in a neighbourhood of $[\mathring{Z}]$, the condition $[Z]\in\mathfrak{F}_sV_r$ is, up to a per\-mutation of clusters of $Z$, equivalent to the equality $\zeta_i=\zeta_{i'}$ whenever $\smash{i\sim_{\smash{\mathring{Z}}}i'}$.

Let $1\le i_1<\dotsb <i_{r-s}\le r$ be the unique representatives of the $r-s$ equivalence classes of the
relation $\smash{\sim_{\smash{\mathring{Z}}}}$, satisfying for each $1\le c\le r-s$ and each $\smash{i\sim_{\smash{\mathring{Z}}}i_c}$ the inequality $\smash{\mathring t_{i}^1\ge \mathring t_{i_c}^1}$.
Thus, a neighbourhood of $\smash{[\mathring{Z}]}$ in $\mathfrak{F}_s$ is described by the following local parameters:
 \begin{itemize}
  \item $\zeta_{i_{\smash{c}}}$ for all $1\le c\le r-s$;\vspace*{-3px}
  \item $t_{i}^{\smash j}$ for all $1\le i\le r$ and $1\le j\le k$.\qedhere
 \end{itemize}
\end{proof}
\begin{rem}
\label{rem:normalfrF}
The argument in the previous proof can be pushed a bit further to describe the normal cotangent bundle
$N^*(\mathfrak{F}_sV_r,F_sV_r)$ of $\mathfrak{F}_sV_r$ in $F_sV_r$.
Recall that, for fixed $[\mathring{Z}]\in\mathfrak{F}_sV_r$, the normal cotangent space $\smash{N^{\vphantom{\underline{j}}*}_{\smash{[\mathring{Z}]}}(\mathfrak{F}_sV_r,F_sV_r)}$
is the \emph{subspace} of the cotangent space $\smash{T^{\vphantom{\underline{j}}*}_{\smash{[\mathring{Z}]}}F_sV_r}$ of all linear functionals
that vanish on the vector subspace $T_{\smash{[\mathring{Z}]}}\mathfrak{F}_sV_r$. The previous proof shows that
$N^{\vphantom{\underline{j}}*}_{\smash{[\mathring{Z}]}}(\mathfrak{F}_sV_r,F_sV_r)$ is 'spanned' by the parameters $\zeta_i$ for $i\in\set{1,\dots,r}\setminus\set{i_1,\dots,i_{r-s}}$. By this we formally mean the following:
\begin{itemize}
 \item for each index $i\in\set{1,\dots,r}\setminus\set{i_1,\dots,i_{r-s}}$ we consider the list of $p$ linear
 functionals $\mathrm{d}\zeta_i^1,\dots,\mathrm{d}\zeta_i^p$, where $\zeta_i^1,\dots,\zeta_i^p\in\R$ are the $p$ coordinates of the parameter
 $\zeta_i$, which takes values in $\R^p$;
 \item a basis for $N^{\vphantom{\underline{j}}*}_{\smash{[\mathring{Z}]}}(\mathfrak{F}_sV_r,F_sV_r)$ is given by the linear functionals $\mathrm{d}\zeta_i^{\smash j}$
 where $i$ ranges in $\set{1,\dots,r}\setminus\set{i_1,\dots,i_{r-s}}$ and $1\le j\le p$.
\end{itemize}
\end{rem}

\subsection{Coloured configuration spaces}

\begin{nota}[Distributions]
  Let $E$ be an index set. A \emph{distribution} is a map $\alpha\colon E\to \N$ with finite support. We write $\alpha_e\coloneqq \alpha(e)$ and $\alpha=\sum_e \alpha_e\cdot e$.

  In particular, for a fixed $e_0\in E$, we denote by $\alpha+e_0$ the distribution which coincides with $\alpha$, except for the fact that it increases $\alpha_{e_0}$ by $1$.
\end{nota}

\begin{defi}[Coloured labelled configuration spaces]
  Let $E$ be a set and $\alpha\colon E\to \N$ a distribution. Define $\lvert \alpha\rvert\coloneqq \sum_{e\in E}\alpha_e$ and $\mathfrak{S}({\alpha})\coloneqq \prod_{e\in E}\mathfrak{S}_{\alpha_e}\subseteq \mathfrak{S}_{|\alpha|}$. Moreover, let $X\coloneqq (X_e)_{e\in E}$ be a family of spaces. Then we define
  \[C_{\alpha}(\R^{p+1};X) \coloneqq \tilde{C}_{|\alpha|}(\R^{p+1})\times_{\mathfrak{S}(\alpha)} \prod_{e\in E} X^{\alpha_e}_e.\]
  In case $X_e=*$ for all $e$, we just write $C_{\alpha}(\R^{p+1})=\tilde{C}_{\lvert \alpha\rvert}(\R^{p+1})/\mathfrak{S}(\alpha)$.
\end{defi}
\pagebreak
Informally, we consider $C_\alpha(\R^{p+1};X)$ as the space of unordered configurations of $|\alpha|$ unordered
points, each equipped with a label in $\coprod_{e}X_e$, such that for all $e\in E$, there are precisely $\alpha_e$
points carrying a label in $X_e$.

\begin{nota}
  For unordered labelled configurations as before, we will use again the suggestive 'sum notation': For distinct points $y_1,\dotsc,y_{\lvert \alpha\rvert}\in \R^{p+1}$ and labels $x_1,\dotsc,x_{\lvert \alpha\rvert}\in X$, we will denote the unordered labelled configuration $\{y_1,\dotsc,y_n\}\subseteq \R^{p+1}$ in which the point $y_l$ carries the label $x_l$ by
  \[\Theta\coloneqq \sum_{l=1}^{\abs{\alpha}} y_l\otimes x_l\in C_\alpha(\R^{p+1};X).\]
  %
\end{nota}

\begin{defi}
  For $w\ge 1$, an unordered partition of $\{1,\dotsc,w\cdot k\}$ into subsets $S_1,\dots,S_w$
  of size $k$ is \emph{irreducible} if there is no $1\le i\le w-1$ for which the subset
  $\{1,\dots,i\cdot k\}$ is a union of some pieces $S_j$ of the partition.
  We denote by $\mathbb{E}_w$ the set of all irreducible, unordered partitions of $\{1,\dotsc,w\cdot k\}$.
\end{defi}

\begin{nota}
\label{nota:E}
  We denote $\mathbb{E}\coloneqq \coprod_{w\ge 1}\mathbb{E}_w$ the union of all $\mathbb{E}_w$. For $e\in \mathbb{E}_w\subset \mathbb{E}$, we will write
  $w(e)\coloneqq w\ge 1$ for the \emph{weight} of $e$. Note that there is precisely one partition $e_0\in \mathbb{E}$ with $w(e_0)=1$.
  For all $e=\set{S_1,\dots,S_w}\in \mathbb{E}_w$, we use the convention that $\min(S_b)<\min(S_{b'})$ for $1\le b<b'\le w(e)$.
\end{nota}
\begin{figure}[h]
  \centering
  \begin{tikzpicture}[xscale=.5,yscale=.6]
    \draw[thick,bred] (0,0) -- (0,.3) -- (2,.3) -- (2,0);
    \draw[thick,bred] (1,0) -- (1,-.3) -- (3,-.3) -- (3,0);
    \draw[thick,bred] (4,0) -- (4,.3) -- (5,.3) -- (5,0);
    \node[dred] at (0,0) {\tiny $\bullet$};
    \node[dred] at (1,0) {\tiny $\bullet$};
    \node[dred] at (2,0) {\tiny $\bullet$};
    \node[dred] at (3,0) {\tiny $\bullet$};
    \node[dred] at (4,0) {\tiny $\bullet$};
    \node[dred] at (5,0) {\tiny $\bullet$};
  \end{tikzpicture}
  ~~~~~~
  \begin{tikzpicture}[xscale=.5,yscale=.6]
    \draw[thick,bgreen] (0,0) -- (0,.3) -- (2,.3) -- (2,0);
    \draw[thick,bgreen] (1,0) -- (1,-.3) -- (4,-.3) -- (4,0);
    \draw[thick,bgreen] (3,0) -- (3,.3) -- (5,.3) -- (5,0);
    \node[dgreen] at (0,0) {\tiny $\bullet$};
    \node[dgreen] at (1,0) {\tiny $\bullet$};
    \node[dgreen] at (2,0) {\tiny $\bullet$};
    \node[dgreen] at (3,0) {\tiny $\bullet$};
    \node[dgreen] at (4,0) {\tiny $\bullet$};
    \node[dgreen] at (5,0) {\tiny $\bullet$};
  \end{tikzpicture}
  \caption{We can picture partitions of $\{1,\dotsc,w\cdot k\}$ as in this figure. Here we see a reducible partition (red) and an irreducible one (green) with $k=2$ and $w=3$.}
\end{figure}

\begin{constr}
  For each $e\in \mathbb{E}$ we denote by $D_e$ the product $(D^p)^{w(e)-1}$, where $D^p\subset\R^p$ denotes the standard,
  Euclidean unit disc; note that $D_e$ is, topologically, also a disc. We thus obtain a family of discs $D\coloneqq (D_e)_{e\in \mathbb{E}}$, and regard each $D_e$ as a subset of $\R^{p\cdot(w(e)-1)}$. For reasons that will become clearer later,
  we denote by $\xi_2,\dots,\xi_{w(e)}$ the $w(e)-1$ parameters of $D_e$, each taking values in $D^p$; each parameter
  $\xi_b$ consists of $p$ coordinates $\xi_b^1,\dots,\xi_b^p\in\R$.

  For a distribution $\alpha\colon \mathbb{E}\to \N$, we define the \emph{degree} $\deg(\alpha)$ as the pair of non-negative numbers
  \[\deg(\alpha)=(r(\alpha),s(\alpha))\coloneqq \pa{\sum_e \alpha_e\cdot w(e)\, ,\, \sum_e \alpha_e\cdot \pa{w(e)-1}}.\]
  We let $C_{r,s}\coloneqq C_{r,s}(\R^{p+1};D)$ be the union of all spaces $C_\alpha(\R^{p+1};D)$, where $\alpha$ ranges among distributions
  $\alpha\colon \mathbb{E}\to\N$ with $\deg(\alpha)=(r,s)$. %
 %

  Each disc $D_e$ contains a centre $0_e\in D_e$, and we denote by $0\coloneqq (0_e)_{e\in \mathbb{E}}$ the family of centres; we obtain an inclusion %
  $C_\alpha(\R^{p+1};0)\subseteq C_\alpha(\R^{p+1};D)$ which is a closed embedding of a submanifold of codimension $s(\alpha)\cdot p$ for each distribution $\alpha\colon \mathbb{E}\to \N$.  We write $C_\alpha^*(\R^{p+1};D)\coloneqq C_\alpha(\R^{p+1};D)\setminus C_\alpha(\R^{p+1};0)$.

  Moreover, we define $C_{r,s}^0\subseteq C_{r,s}$ to be the union of all  $C_\alpha(\R^{p+1};0)$ with $\deg(\alpha)=(r,s)$ and define its complement $C^*_{r,s}\coloneqq C_{r,s}\setminus C_{r,s}^0$.
\end{constr}
\begin{rem}
  \label{rem:normalCa}
  A generic point in $C_{r,s}$ is of the form
  \[\Theta=\sum_{l=1}^{r-s} y_l\otimes (e_l,\bm{\xi}_l),\]
  where the points $y_1,\dotsc,y_{r-s}\in \R^{p+1}$ are distinct, $e_l\in \mathbb{E}$, and $\bm{\xi}_l\in D_{e_l}$ is expanded as $\bm{\xi}_l=(\xi_{l,2},\dotsc,\xi_{\smash{l,w(e_l)}})$ with $\xi_{l,b}\in D^p$. Consider a distribution $\alpha$ of degree $\deg(\alpha)=(r,s)$, and
 let $\mathring{\Theta}\coloneqq \sum_{l=1}^{r-s} \mathring{y}_l\otimes (e_l,0_{e_l})\in C_{\alpha}(\R^{p+1};0)\subseteq C_{r,s}^0$. In order to describe a small neighbourhood of $\mathring{\Theta}$
 in $C_{r,s}^0$, we can use the local parameters $y_l\in\R^{p+1}$, each ranging in a small neighbourhood of $\mathring{y}_l$;
 to describe a small neighbourhood of $\smash{\mathring{\Theta}}$ in $C_{r,s}$ we can additionally use the parameters
 $\bm{\xi}_l=(\xi_{l,2},\dotsc,\xi_{l,w(e)})\in D_{e_l}$, each ranging in a small neighbourhood of $0_{e_l}$. It follows that the
 $N^*_{\mathring{\Theta}}(C_{r,s}^0,C_{r,s})$ is 'spanned by the parameters $\bm{\xi}_l$ for $1\le l\le r-s$'. By this we mean the following:
\begin{itemize}
 \item for each $1\le l\le r-s$ we consider the list of $p\cdot(w(e_l)-1)$ linear
 functionals $\mathrm{d}\xi_{l,b}^\tau$, for $2\le b\le w(e_l)$ and $1\le \tau\le p$.
 Here $\xi_{l,b}^\tau$, for $2\le b\le w(e_l)$ and $1\le \tau\le p$, are the $p\cdot (w(e_l)-1)$ coordinates of
 the parameter $\bm{\xi}_l$, which takes values in $D_{e_l}\subset \R^{p\cdot (w(e_l)-1)}$;
 \item a basis for $N^*_{\mathring{\Theta}}(C_{r,s}^0,C_{r,s})$ is given by all linear functionals $\mathrm{d}\xi_{l,b}^\tau$ indexed by $1\le l\le r-s$, $2\le b\le w(e_l)$, and $1\le \tau\le p$.
\end{itemize}
\end{rem}
\let\del\partial

There are stabilisation maps $C_{r,s}\to C_{r+1,s}$ given by placing a new point with label in $D_{e_0}=*$ on the 'far right' with respect to the first coordinate of $\R^{p+1}$.\linebreak
The stabilisation increases the parameter $r$ by $1$, but leaves $s$ constant. Moreover, $C^0_{r,s}$ is sent to $C^0_{r+1,s}$ under the stabilisation map, and $C^*_{r,s}$ is sent to $C^*_{r+1,s}$

\subsection{The insertion map}

We will now connect the filtration pairs $(F_sV_r,F_{s-1}V_r)$ to the
pairs $(C_{r,s},C^*_{r,s})$
of labelled configurations via an 'insertion map'.

\begin{constr}
  For each $1\le s\le r$, we have a map of pairs
  \[\varphi_{r,s}\colon (C_{r,s},C^*_{r,s}) \to (F_sV_r,F_{s-1}V_r),\]
  which pictorially does the following, see \mbox{\textsc{Figure~\ref{fig:insert}}}: given a labelled configuration in $C_{r,s}$, we draw pairwise disjoint cylinders around each point in $\R^{p+1}$ and place inside each cylinder a small 'standard configuration' which corresponds to the given indecomposable partition, and is 'perturbed' by the $w(e)-1$ disc parameters from the label. More formally, it is constructed as follows:
  \begin{itemize}
  \item For each $w\ge 1$, each subset $S\subseteq \{1,\dotsc,w\cdot k\}$ of cardinality $k$, and each $\xi\in D^p$, we define the unordered 'standard cluster'
    \[
    T_S(\xi) \coloneqq \pa{\xi,\pa{-1+\tfrac{2}{k\cdot w+1}\cdot h}_{h\in S}}.
    \]
    Pictorially, $T_S(\xi)$ is the unordered vertical cluster of $k$ points which projects to $\xi\in D^p$ and whose $t$-coordinate takes the values
    corresponding to $S$, among all values arising from a uniform distribution of $w\cdot k$ points in the interior of the interval $[-1\until 1]$.
  \item For a partition $e\in \mathbb{E}$ write $S_1,\dotsc,S_{w(e)}\subseteq\{1,\dotsc,w\cdot k\}$ for the partition components; since $e$ is an unordered partition, we assume without loss of generality that $\min(S_i)<\min(S_{i+1})$. For all $\xi_2,\dotsc,\xi_{w(e)}\in D^p$, we set $\xi_{1}\coloneqq 0\in D^p$ and define
    \[T_e(\xi_2,\dotsc,\xi_{w(e)}) \coloneqq \sum_{b=1}^{w(e)} T_{S_b}(\xi_{b})\in V_{w(e)}.\]
    Note that the 'lowest cluster' (the one attaining the lowest value of the $t$-coordinate) is 'in the middle' (projects to the centre of $D^p$). %
    Note also that $S_b$ and $S_{\smash{b'}}$ are disjoint for $b\neq b'$, hence the clusters $T_{S_b}(\xi_b)$ and $\smash{T_{S_{\smash{b'}}}(\xi_{\smash{b'}})}$ are also disjoint, and the sum defining $\smash{T_e(\xi_2,\dotsc,\xi_{w(e)})}$ is well-defined.
      \item Consider on $\R^p\times \R$ the 'product distance'
    \[d((\zeta,t),(\zeta',t')) \coloneqq \max(d(\zeta,\zeta'),d(t,t'))\]
    with respect to the Euclidean distances on $\R^p$ resp. $\R$. This means that for a radius $\rho>0$, the closed $\rho$-ball around $(\zeta,t)$ is given by the cylinder $B_{\rho}(\zeta,t)=(\zeta+\rho\cdot D^p)\times [t-\rho\until t+\rho]$.
  \end{itemize}
  \begin{figure}[t]
  \centering
  \begin{tikzpicture}[xscale=3.9,yscale=3.9]
    \draw [very thin,bgrey] (0,0) -- (.6,.4) -- (1.4,.4);
    \draw [very thin,bgrey]  (.6,.4) -- (.6,1.2);
    \draw[very thick,white] (.3,.8) -- (.3,.1);
    \draw[very thick,white] (.8,1) -- (.8,.3);
    \draw[very thick,white] (1.023,.55) -- (1.023,0);
    \draw [dgrey,thin] (.3,1.033) -- (.3,0.072);
    %
    %
    %
    \draw [dgrey,thin] (.8,1.043) -- (.8,0.082);
    \draw [dgrey,thin] (1.023,1.14) -- (1.023,0.283);
    %
    \draw[very thick,white] (1,1) -- (1.4,1.2);
    \draw[very thick,white] (1,.55) -- (1,0);
    \draw[very thick,white] (0.1,1) -- (.9,1);
    \draw [grey] (0,0) rectangle (1,1);
    \draw [grey] (1,0) -- (1.4,.4) -- (1.4,1.2) -- (1,1);
    \draw [grey] (0,1) -- (.6,1.2) -- (1.4,1.2);
    \node at (.3,.3) {\tiny $\bullet$};
    \node at (.3,.7) {\tiny $\bullet$};
    \node at (1.023,.8) {\tiny $\bullet$};
    \node at (.8,.5) {\tiny $\bullet$};
    \draw[dred] (.14,.745) -- (.12,.745) -- (.12,.685) -- (.14,.685);
    \draw[dblue] (.14,.715) -- (.16,.715) -- (.16,.655) -- (.14,.655);
    \node[dred] at (.14,.743) {$\cdot$};
    \node[dblue] at (.14,.713) {$\cdot$};
    \node[dred] at (.14,.683) {$\cdot$};
    \node[dblue] at (.14,.653) {$\cdot$};
    \draw[dred] (.225,.7) circle (1.4pt);
    \node[dred] at (.21,.681) {$\cdot$};
    \draw[dblue] (.26,.315) -- (.26,.285);
    \node[dblue] at (.26,.313) {$\cdot$};
    \node[dblue] at (.26,.283) {$\cdot$};
    \draw[dblue] (1.063,.815) -- (1.063,.785);
    \node[dblue] at (1.063,.813) {$\cdot$};
    \node[dblue] at (1.063,.783) {$\cdot$};
    \draw[dgreen] (.725,.5) circle (1.4pt);
    \draw[dred] (.615,.5) circle (1.4pt);
    \node[dred] at (.59,.5) {$\cdot$};
    \node[dgreen] at (.725,.498) {$\cdot$};
    \draw[dred] (.53,.575) -- (.51,.575) -- (.51,.515) -- (.53,.515);
    \draw[dgreen] (.53,.545) -- (.55,.545) -- (.55,.455) -- (.53,.455);
    \draw[dblue] (.53,.485) -- (.51,.485) -- (.51,.425) -- (.53,.425);
    \node[dred] at (.53,.573) {$\cdot$};
    \node[dgreen] at (.53,.543) {$\cdot$};
    \node[dred] at (.53,.513) {$\cdot$};
    \node[dblue] at (.53,.483) {$\cdot$};
    \node[dgreen] at (.53,.453) {$\cdot$};
    \node[dblue] at (.53,.423) {$\cdot$};
    %
    %
    %
  \end{tikzpicture}
  ~~~~\raisebox{42px}{$\mapsto$}~~~
  \begin{tikzpicture}[xscale=3.9,yscale=3.9]
    \draw [very thin,bgrey] (0,0) -- (.6,.4) -- (1.4,.4);
    \draw [very thin,bgrey]  (.6,.4) -- (.6,1.2);
    \draw[very thick,white] (.3,.8) -- (.3,.1);
    \draw[very thick,white] (.8,1) -- (.8,.3);
    \draw[very thick,white] (1.023,.55) -- (1.023,0);
    \draw [dgrey,thin] (.3,1.033) -- (.3,0.072);
    %
    %
    %
    \draw [dgrey,thin] (.8,1.043) -- (.8,0.082);
    \draw [dgrey,thin] (1.023,1.14) -- (1.023,0.283);
    %
    \draw[very thick,white] (1,1) -- (1.4,1.2);
    \draw[very thick,white] (1,.55) -- (1,0);
    \draw[very thick,white] (0.1,1) -- (.9,1);
    \draw [grey] (1,0) -- (1.4,.4) -- (1.4,1.2) -- (1,1);
    \draw [grey] (0,1) -- (.6,1.2) -- (1.4,1.2);
    %
    \draw[very thick,white] (.3,.38) ellipse (2pt and .8pt);
    \draw[very thick,white] (.3,.22) ellipse (2pt and .8pt);
    \draw[thin,dyellow] (.3,.38) ellipse (2pt and .8pt);
    \draw[thin,dyellow] (.3,.22) ellipse (2pt and .8pt);
    \draw[very thick, white] (.3,.38) -- (.3,.45);
    \draw[dgrey,thin] (.3,.38) -- (.3,.45);
    \draw[very thick, white] (.3,.22) -- (.3,.28);
    \draw[dgrey,thin] (.3,.22) -- (.3,.28);
    \draw[dyellow] (.23,.22) -- (.23,.38);
    \draw[dyellow] (.37,.22) -- (.37,.38);
    %
    \draw[very thick,white] (1.023,.88) ellipse (2pt and .8pt);
    \draw[very thick,white] (1.023,.72) ellipse (2pt and .8pt);
    \draw[thin,dyellow] (1.023,.88) ellipse (2pt and .8pt);
    \draw[thin,dyellow] (1.023,.72) ellipse (2pt and .8pt);
    \draw[very thick, white] (1.023,.88) -- (1.023,.95);
    \draw[dgrey,thin] (1.023,.88) -- (1.023,.95);
    \draw[very thick, white] (1.023,.72) -- (1.023,.78);
    \draw[dgrey,thin] (1.023,.72) -- (1.023,.78);
    \draw[dyellow] (.953,.72) -- (.953,.88);
    \draw[dyellow] (1.093,.72) -- (1.093,.88);
    %
    \node[dred] at (.76,.554) {$\cdot$};
    \draw[dred,semithick] (.76,.557) -- (.76,.511);
    \draw[very thick,white] (.3,.78) ellipse (2pt and .8pt);
    \draw[very thick,white] (.3,.62) ellipse (2pt and .8pt);
    \draw[thin,dyellow] (.3,.78) ellipse (2pt and .8pt);
    \draw[thin,dyellow] (.3,.62) ellipse (2pt and .8pt);
    %
    \draw[very thick, white] (.3,.78) -- (.3,.85);
    \draw[dgrey,thin] (.3,.78) -- (.3,.85);
    \draw[very thick, white] (.3,.62) -- (.3,.68);
    \draw[dgrey,thin] (.3,.62) -- (.3,.68);
    \draw[dyellow] (.23,.62) -- (.23,.78);
    \draw[dyellow] (.37,.62) -- (.37,.78);
    \draw[very thick,white] (.8,.58) ellipse (2pt and .8pt);
    \draw[very thick,white] (.8,.42) ellipse (2pt and .8pt);
    \draw[thin,dyellow] (.8,.58) ellipse (2pt and .8pt);
    \draw[thin,dyellow] (.8,.42) ellipse (2pt and .8pt);
    \draw[very thick, white] (.8,.58) -- (.8,.65);
    \draw[dgrey,thin] (.8,.58) -- (.8,.65);
    \draw[very thick, white] (.8,.42) -- (.8,.48);
    \draw[dgrey,thin] (.8,.42) -- (.8,.48);
    \draw[dyellow] (.73,.42) -- (.73,.58);
    \draw[dyellow] (.87,.42) -- (.87,.58);
    %
    %
    %
    \node[dred] at (.28,.676) {\Large $\cdot$};
    \node[dblue] at (.3,.71) {\Large$\cdot$};
    \node[dred] at (.28,.739) {\Large$\cdot$};
    \draw[dblue,semithick] (.3,.3267) -- (.3,.2733);
    \node[dblue] at (.3,.3267) {\Large $\cdot$};
    \node[dblue] at (.3,.2733) {\Large $\cdot$};
    \node[dblue] at (.3,.647) {\Large $\cdot$};
    \draw[dred,semithick] (.28,.68) -- (.28,.744);
    \draw[dblue,semithick] (.3,.716) -- (.3,.652);
    \node[dgreen] at (.8,.531) {\Large $\cdot$};
    \node[dred] at (.76,.508) {\Large $\cdot$};
    \node[dblue] at (.8,.485) {\Large $\cdot$};
    \node[dgreen] at (.8,.463) {\Large $\cdot$};
    \node[dblue] at (.8,.439) {\Large $\cdot$};
    \draw[dgreen,semithick] (.8,.534) -- (.82,.534) -- (.82,.466) -- (.8,.466);
    \draw[dblue,semithick] (.8,.442) -- (.78,.442) -- (.78,.488) -- (.8,.488);
    \draw[dblue,semithick] (1.023,.8267) -- (1.023,.7733);
    \node[dblue] at (1.023,.8267) {\Large $\cdot$};
    \node[dblue] at (1.023,.7733) {\Large $\cdot$};
    \draw[very thick, white] (1,.5) -- (1,.95);
    \draw [grey] (0,0) rectangle (1,1);
  \end{tikzpicture}
  \caption{An instance of the insertion map
  $\varphi_{7,3}\colon (C_{7,3},C^*_{7,3})\to (F_3V_7,F_2V_7)$.
  The result even lies in $F_1V_7$.}\label{fig:insert}
\end{figure}
\noindent Now we have everything together to define the desired insertion map: Given an element $\Theta=\sum_{l} y_l\otimes (e_l,\bm{\xi}_l)$ in $C_{r,s}$, we define
    \[\rho=\rho(\Theta) \coloneqq \begin{cases}\tfrac{1}{5}\cdot \min_{l\ne l'}d(y_l,y_{l'}) & \text{for $r-s\ge 2$},\\1 & \text{for $r-s=1$.}\end{cases}\]
and accordingly, the map
  \[\varphi_{r,s}\colon C_{r,s}\to F_sV_r,\quad \Theta=\sum_{l=1}^{r-s} y_l\otimes (e_l,\bm{\xi}_l)\mapsto \sum_{l=1}^{r-s} \pa{y_l + \rho(\Theta)\cdot T_{e_l}(\bm{\xi}_l)}.\]
  First of all, note that the signs '$+$' and '$\cdot$' in the expression  '$y_l + \rho(\Theta)\cdot T_{e_l}(\bm{\xi}_l)$' denote a translation and a dilation in $\R^{p+1}$; %
  the sum sign always describes an unordered collection (of points, or of vertical clusters). Note that the second sum is well-defined as the configurations $y_l+\rho(\Theta)\cdot T_{e_l}(\bm{\xi}_l)$ lie inside the cylinders $B_{\rho}(y_l)$ and are hence disjoint. Finally, we have $\delta(\varphi_{r,s}(\Theta))\ge r-s$, so the image of $\varphi_{r,s}$ is actually contained in the filtration level $F_sV_r\subset V_r$.

  We see that $\varphi_{r,s}(\Theta)$ lies in the stratum $\mathfrak{F}_sV_r$ if and only if $\bm{\xi}_l=0_{e_l}$ for all $1\le l\le r-s$. In particular
  $\varphi_{r,s}$ restricts to maps
  $C^*_{r,s}\to F_{s-1}V_r$ and
  $C^0_{r,s}\to \mathfrak{F}_sV_r$.
\end{constr}

\begin{rem}
  \begin{enumerate}
  \item [{\small\bfseries 1.}] In fancier language, and up to homotopy, we constructed the following: Let $D\coloneqq \coprod_e D_e$, and define a map $\varphi\colon D\to V:=\coprod_{r\ge 0}V_r$, restricting for all $e\in \mathbb{E}$ to
  a map $D_e\to V_{w(e)}$ similar to the map $T_e$ above. Then we use that $V$ is an $\mathscr{C}_{p+1}$-algebra and consider the adjoint map of $\mathscr{C}_{p+1}$-algebras, with source the free $\mathscr{C}_{p+1}$-algebra on $D$:
    \[\varphi\colon C(\R^{p+1};D)\simeq F^{\mathscr{C}_{p+1}}(D)\to V.\]
    The left hand side decomposes as a disjoint union of the spaces $C_{r,s}$ as before, while the right hand side decomposes into the spaces $V_r$, which are filtered by the spaces $F_sV_r$, and the map $\varphi$ is compatible with this decomposition and filtration. We spelled out the insertion map directly for two reasons: firstly, we need the explicit choice of $\rho$ later in the proof, and secondly, this adjoint description gives a definition of $\varphi$
    which is sensible only up to homotopy, and in particular we could not immediately make sense of a statement like: \emph{$\varphi$ is compatible with the filtrations and their strata.}
  \item [{\small\bfseries 2.}] The insertion maps respect the stabilisation maps on both sides up to homotopy:
  the stabilisation maps in the following diagram of maps of pairs can be chosen so that the diagram commutes on the nose
    \[
      \begin{tikzcd}
        (C_{r,s},C^*_{r,s}) \ar[r,"\varphi_{r,s}"]\ar[d,swap,"\mathrm{stab}"] & (F_sV_r,F_{s-1}V_r)\ar[d,"\mathrm{stab}"]\\
        (C_{r+1,s},C^*_{r+1,s})\ar[r,swap,"\varphi_{r+1,s}"] & (F_sV_{r+1},F_{s-1}V_{r+1})
      \end{tikzcd}
    \]
  \item[{\small\bfseries 3.}] In order to ensure that $\varphi_{r,s}$ is well-defined, it would have been enough to choose $\rho$ slightly smaller than $\frac{1}{2}\cdot\min_{l\ne l'}d(y_l,y_{l'})$. However, we wanted to ensure that $\varphi_{r,s}\colon C_{r,s}\to F_sV_r$ is even an embedding, so we need to ensure that $\varphi_{r,s}(\Theta)$ still 'knows' which clusters come from the same label.
  \end{enumerate}
\end{rem}

\begin{prop}
\label{prop:varphiiso}
  For each $1\le s\le r$, the map
  $\varphi_{r,s}\colon(C_{r,s}, C^*_{r,s})\to (F_sV_r,F_{s-1}V_r) $
  induces an isomorphism in relative homology.
\end{prop}
The proof of \mbox{\textsc{Proposition \ref{prop:varphiiso}}} relies on the following lemma.
\begin{lem}
 \label{lem:varphiiso}
  For $1\le s\le r$, the map
  $\varphi_{r,s}\colon C^0_{r,s} \to \mathfrak{F}_sV_r$ is a homotopy equivalence.
\end{lem}
We first finish the proof of \mbox{\textsc{Proposition \ref{prop:varphiiso}}} assuming \mbox{\textsc{Lemma \ref{lem:varphiiso}}},
and then prove \mbox{\textsc{Lemma \ref{lem:varphiiso}}}.
\begin{proof}[Proof of Proposition~\ref{prop:varphiiso}]
Since $\mathfrak{F}_sV_r\subseteq F_sV_r$ is an embedded, closed submanifold, the relative homology groups of
$(F_sV_r,F_{s-1}V_r)$ can be computed using excision and the Thom isomorphism for
the normal bundle of $\mathfrak{F}_sV_r$ in $F_sV_r$,
\[
 H_*(F_sV_r,F_{s-1}V_r)=H_*(F_sV_r, F_sV_r\setminus \mathfrak{F}_s V_r)\cong H_{*-s\cdot p}(\mathfrak{F}_sV_r;\mathcal{O}_{r,s}),
\]
for a suitable choice of local coefficients $\mathcal{O}_{r,s}$ on $\mathfrak{F}_sV_r$.
Similarly, the relative homology groups of $(C_{r,s}, C^*_{r,s})$ are given by
\[
 H_*(C_{r,s}, C^*_{r,s})=H_*(C_{r,s}, C_{r,s}\setminus C^{\smash 0}_{\smash{r,s}})\cong H_{*-s\cdot p}(C^0_{r,s};\mathcal{O}'_{r,s}),
\]
for a suitable choice of local coefficients $\mathcal{O}'_{r,s}$ on $C^0_{r,s}$.

Recall \textsc{Remark~\ref{rem:normalfrF}} and \textsc{Remark~\ref{rem:normalCa}}:
the key observation is that the insertion map $\varphi_{r,s}$ induces
a map of normal bundles $N(C^0_{r,s},C_{r,s})\to N(\mathfrak{F}_sV_r,F_sV_r)$, which is an isomorphism on fibres.
To see this, fix $\mathring{\Theta}\in C^0_{r,s}$ as in \textsc{Remark~\ref{rem:normalCa}}, denote
$[\mathring{Z}]=\varphi_{r,s}(\mathring{\Theta})$, and use the notation from \textsc{Remark~\ref{rem:normalfrF}}.
By construction $\phi_{r,s}$ is a smooth map, and our aim is to check that
\[
 \phi_{r,s}^*\colon N^*_{[\mathring Z]}(\mathfrak{F}_sV_r,F_sV_r)\to N^*_{\mathring{\Theta}}(C^0_{r,s},C_{r,s})
\]
is an isomorphism of vector spaces. This follows directly from the observation that $\phi_{r,s}^*$
sends the
linear functional $\mathrm{d}\zeta_i^{\smash j}$
to the linear functional
$\smash{\frac{1}{\rho(\mathring{\Theta})}\cdot \mathrm{d}\xi_{l,b}^\tau}$, whenever the cluster $[\mathring{z}_i]$ of the collection $[\mathring{Z}]$
is obtained from the labelled point $\mathring{y}_l\otimes \bm{\xi}_l$, using the $b$\textsuperscript{th}
component $S_b$ of the partition $e_l$.%
Hence the local coefficient system $\mathcal{O}'_{r,s}$ coincides with $\varphi_{r,s}^*\mathcal{O}_{r,s}$, and so,
\[
 \begin{tikzcd}[column sep=4em]
  H_*(C_{r,s}, C^*_{r,s})\ar[r,"\cong","\text{Thom}"'] \ar[d,"(\varphi_{r,s})_*"'] & H_{*-s\cdot p}(C^0_{r,s};\mathcal{O}'_{r,s})\ar[d,"(\varphi_{r,s})_*"]\\
  H_*(F_sV_r, F_{s-1}V_r) \ar[r,"\cong","\text{Thom}"'] & H_{*-s\cdot p}(\mathfrak{F}_sV_r;\mathcal{O}_{r,s}).
 \end{tikzcd}
\]
commutes. By \mbox{\textsc{Lemma \ref{lem:varphiiso}}}, the right vertical map is an isomorphism; hence also the left vertical map is an isomorphism.
\end{proof}

\begin{proof}[Proof of Lemma \ref{lem:varphiiso}]

The insertion map $\varphi_{r,s}\colon C^0_{r,s}\hookrightarrow \mathfrak{F}_sV_r$ is a closed injection, and we define
a deformation retraction of $\mathfrak{F}_sV_r$ onto $C^0_{r,s}$:
for each \mbox{$[Z]\in \mathfrak{F}_sV_r$} there is, up to permutation, a unique sequence $e_1,\dotsc,e_{r-s}\in \mathbb{E}$ such that %
\[
  [Z]=\sum_{l=1}^{r-s}\sum_{S\in e_l}\pa{\zeta_l,(u_{\smash l}^{\smash h})_{h\in S}},
\]
where we assume $u_{l}^{\smash h}<u_{l}^{h+1}$ for all $1\le h< w(e_l)\cdot k$. %
We %
define
\[u_l\coloneqq \tfrac{1}{k\cdot w(e_l)}\cdot (u_{l}^1+\dotsb+u_{l}^{\smash{k\cdot w(e_l)}})\]
Moreover, let $y_l \coloneqq (\zeta_{l},u_l)\in \R^{p+1}$
and $\smash{\rho\coloneqq \rho(y_1,\dotsc,y_{r-s})}$. %
The homotopy $H\colon \mathfrak{F}_sV_r\times[0\until 1]\to \mathfrak{F}_sV_r$ is given by linear interpolation of the local parameters:
we move $u_{l}^h$ to $\smash{u_l+\rho\cdot\big({-1} + \frac{2}{k\cdot w(e_l)+1}\cdot h\big)}$, and keep the values $\zeta_l$ fixed.
For each $1\le l\le r-s$, the interpolation takes place
in the convex hull of the points $(\zeta_l,u_{l}^h)\in\R^{p+1}$; these are $r-s$ disjoint vertical segments in $\R^{p+1}$, and at each time the original vertical order of the points lying on each of these segments is preserved.
Therefore no collision between distinct points of the vertical configuration occurs.

We note that $H([Z],1)$ is in the subspace $\varphi_{r,s}(C^0_{r,s})$, and by construction, $H$ fixes pointwise at all times
the subspace $\varphi_{r,s}(C^0_{r,s})$.
\end{proof}

\subsection{The stability proof}

\newcommand{\on{sg}}{\on{sg}}

\begin{lem}\label{lem:colstab}
  The stabilisation map $C_{r,s}\to C_{r+1,s}$ induces isomorphisms
  \[H_m\pa{C_{r,s},C^*_{r,s}} \to H_m(C_{r+1,s},C_{r+1,s}^*)\]
  in homology for $m\le \tfrac{r}{2}$ and all $1\le s\le r$.
\end{lem}
\begin{proof}
  For each distribution $\alpha$ of degree $(r,s)$, we have a (not always orientable) disc bundle $C_\alpha(\R^{p+1};D)\to C_\alpha(\R^{p+1})$ of dimension $p\cdot s$
  and structure group $\mathfrak{S}(\alpha)$, which gives us a Thom isomorphism
  \[H_{m}\pa{C_\alpha(\R^{p+1};D),C^*_\alpha(\R^{p+1};D)}\cong H_{m-p\cdot s}\pa{C_{\alpha}(\R^{p+1});\hspace*{1px}\pr_\alpha^*\mathcal{O}_\alpha}=:M_{m,\alpha},\]
  where $\pr_\alpha\colon \pi_1(C_{\alpha}(\R^{p+1}))\to \mathfrak{S}(\alpha)$ is the projection and $\mathcal{O}_\alpha$ is of the form
  \[\mathcal{O}_\alpha\colon \mathfrak{S}(\alpha)\to \{\pm 1\}, \quad(\sigma_e)_{e\in \mathbb{E}}\mapsto \prod_e \on{sg}(\sigma_e)^{p\cdot (w(e)-1)}.\]
  In particular, we have a natural isomorphism $\pr^*_\alpha\mathcal{O}_{\alpha}\cong \mathrm{stab}^*\pr^*_{\alpha+e_0}\mathcal{O}_{\alpha+e_0}$ for the canonical stabilisation $C_\alpha(\R^{p+1}) \to C_{\alpha+e_0}(\R^{p+1})$, which gives us induced stabilisation morphisms $M_{m,\alpha}\to M_{m,\alpha+e_0}$. Now we have a commutative square
  \[
    \begin{tikzcd}
      H_m(C_{r,s},C^*_{r,s})\ar[d,swap]\ar[r,"\cong"] & \displaystyle\bigoplus_{\deg(\alpha)=(r,s)}M_{m,\alpha}\ar[d]\\
      H_m(C_{r+1,s},C^*_{r+1,s})\ar[r,swap,"\cong"] & \displaystyle\bigoplus_{\deg(\alpha)=(r,s)}M_{m,\alpha+e_0}\oplus \bigoplus_{\substack{\deg(\alpha)=(r+1,s)\\\alpha_{e_0}=0}} M_{m,\alpha}
    \end{tikzcd}
  \]
  where the left vertical arrow is the desired stabilising map and the right side is a sum of maps $M_{m,\alpha}\to M_{m,\alpha+e_0}$. Therefore, we can prove the statement by showing that for $m\le \frac{r}{2}$, we have $M_{m,\alpha}=0$ for each distribution $\alpha$ of degree $(r+1,s)$ with $\alpha_{e_0}=0$, and secondly, that the stabilising map $M_{m,\alpha}\to M_{m,\alpha+e_0}$ is an isomorphism for each distribution $\alpha$ of degree $(r,s)$.
  For the first part, we use $w(e)\ge 2$ for all $e\in \mathbb{E}$ with $\alpha_e\neq 0$ to obtain
  \[
    p\cdot s\ge p\cdot \sum_e \alpha_e\cdot \pa{w(e)-\tfrac 12\cdot  w(e)}=p\cdot \tfrac{r+1}2\ge \tfrac{r+1}2,
  \]
  so $m-p\cdot s<0$, whence $M_{m,\alpha}=0$.
  For the second part, have to check
  that $H_{m-p\cdot s}(C_\alpha(\R^{p+1});\mathcal{O}_\alpha)\to H_{m-p\cdot s}(C_{\alpha+e_0}(\R^{p+1});\mathcal{O}_{\alpha+e_0})$ is an isomorphism for $m\le \frac{r}{2}$.
  To do so, we first observe that $\tfrac{r}{2} \ge \tfrac{1}{2}\cdot\alpha_{e_0} + \sum_{e\ne e_0}\alpha_e$ since $w(e)\ge 2$ for $e\ne e_0$; and as $p\ge 1$, we obtain
  \begin{align*}
    m-p\cdot s &\le \tfrac{r}{2}-p\cdot \sum_e \alpha_e\cdot (w(e)-1)
    \le -\tfrac{r}{2} + \sum_{e}\alpha_e
    \le \tfrac 12\cdot \alpha_{e_0}.
  \end{align*}
  Now we want to use a technique from \cite{palmertwist}, so we adapt his notation by writing $\lambda\coloneqq (\alpha_e)_{e\ne e_0}$, so $\abs{\lambda} = \sum_{e\ne e_0}\alpha_e$ as well as $\lambda[n]$ for the distribution with $\lambda[n](e_0)=n-\abs{\lambda}$ and $\lambda[n](e)=\alpha_e$ for $e\ne e_0$, so we have  $\lambda[r-s]=\alpha$ and\linebreak $\lambda[r-s+1] = \alpha+e_0$.
  This notation has the advantage that $n=\sum_e\lambda[n](e)$. We have a stabilisation map $C_{\lambda[n]}(\R^{p+1})\to C_{\lambda[n+1]}(\R^{p+1})$ by placing an additional point with label $e_0$, which for $n=r-s$ is our map from before.

  The orientation system $\mathcal{O}_{\lambda[n]}\colon \mathfrak{S}(\lambda[n])\to \{\pm 1\}$ extends over the inclusion $\mathfrak{S}(\lambda[n])\hookrightarrow \mathfrak{S}_n$ to a weighted sign function $\mathcal{O}_n\colon \mathfrak{S}_n\to \{\pm 1\}$. Now we construct a signed version of \cite[Ex.\,4.6]{palmertwist}: Let $\mathbf{PInj}$ be the category whose objects are non-negative integers and whose morphisms $n\to n'$ are partially defined injections $\eta\colon \{1,\dotsc,n\}\dashrightarrow \{1,\dotsc, n'\}$. We define a functor $\mathcal{P}_\lambda\colon \mathbf{PInj}\to \mathbf{Ab}$ to the category of abelian groups as follows: We set
  \[\mathcal{P}_\lambda(n) \coloneqq \Z\mathopen{}\left\langle (P_e)_{e\ne e_0};\,P_e\subseteq \{1,\dotsc,n\},~ P_e\cap P_{e'}=\emptyset~\text{ and }~\# P_e=\lambda_e\right\rangle\mathclose{},\]
  and for each partially-defined injection $\eta\colon n\to n'$ and $P\coloneqq (P_e)_{e\ne e_0}$, we define
  \[\eta_*(P) \coloneqq \begin{cases}\mathcal{O}_{\abs{\lambda}}(\eta|_{\bigcup_e\hspace*{-1px} P_e})\cdot (\eta(P_e))_{e\ne e_0} & \text{if $\eta$ is defined on $\bigcup_e P_e$},\\0 & \text{else,}\end{cases}\]
  where in the first case, the restriction $\eta|_{\bigcup_e\hspace*{-1px}P_e}\colon \bigcup_e P_e \to \bigcup_e \eta(P_e)$ is canonically identified with a permutation in $\mathfrak{S}_{\abs{\lambda}}$ since $\bigcup_e P_e$ and $\bigcup_e\eta(P_e)$ are totally or\-dered as subsets of $\{1<\dotsb<n\}$ resp.\ $\{1<\dotsb<n'\}$. By the same in\-ductive argument as in \cite[Lem.\,4.7]{palmertwist}, $\mathcal{P}_\lambda$ is a polynomial coefficient system with $\deg(\mathcal{P}_\lambda)=\abs{\lambda}=r-s-\alpha_{e_0}$ and we have natural isomorphisms
  \[
    \begin{tikzcd}
      H_{m'}\pa{C_{\lambda[n]}(\R^{p+1});\hspace*{1px}\pr^*_{\lambda[n]}\mathcal{O}_{\lambda[n]}}\ar[d,swap,"\cong"]\ar[r]& H_{m'}\pa{C_{\lambda[n+1]}(\R^{p+1});\hspace*{1px}\pr^*_{\lambda[n+1]}\mathcal{O}_{\lambda[n+1]}}\ar[d,"\cong"]\\
      H_{m'}\pa{C_n(\R^{p+1});\hspace*{1px}\pr^*_n\mathcal{P}_\lambda(n)}\ar[r] & H_{m'}\pa{C_{n+1}(\R^{p+1});\hspace*{1px}\pr^*_{n+1}\mathcal{P}_\lambda(n+1)},
    \end{tikzcd}
  \]
  where $\pr_n\colon \pi_1\pa{C_n(\R^{p+1})}\to \mathfrak{S}_n$ is the projection. Since $p+1\ge 2$, the bottom map is %
  an isomorphism for $m'\le \frac{1}{2}\cdot (n-r+s+\alpha_{e_0})$ by \cite[Thm.\,A]{palmertwist}. For us, $n=r-s$, we get the isomorphism for $m'\le \frac{1}{2}\cdot \alpha_{e_0}$ as desired.
    %
    %
    %
    %
    %
    %
    %
    %
\end{proof}
\pagebreak
Now we have all tools to prove \textsc{Theorem~\ref{theo:stab}}.

\begin{proof}[Proof of Theorem~\ref{theo:stab}]
  This is now a standard argument: Let $E(r)$ denote the Leray homology spectral associated with the filtered space
  $V_r(\R^{p,1})$;
  the fil\-tration-preserving stabilisation $V_r(\R^{p,1})\to V_{r+1}(\R^{p,1})$ induces a morphism $f\colon E(r)\to E(r+1)$ of spectral sequences, and on the first page we have exactly the morphisms
  \[f^1_{s,t}\colon E(r)^1_{s,t}=H_{s+t}(F_sV_r,F_{s-1}V_r)\to E(r+1)^1_{s,t}=H_{s+t}(F_sV_{r+1},F_{s-1}V_{r+1}).\]
  By the \mbox{\textsc{Lemma \ref{lem:colstab}}}, we know that $f^1_{s,t}$ is an isomorphism for $s+t\le \tfrac{r}{2}$, so by a standard comparison argument between spectral sequences, see \cite{Zeeman}, we obtain that $H_m(V_r)\to H_m(V_{r+1})$ is an isomorphism for $m\le \frac r2$.
\end{proof}

\begin{outl}
  \begin{enumerate}
  \item [{\small\bfseries 1.}]
    %
    As already remarked, the space $V^k(\R^{p,q})=\coprod_{r\ge0}V_r^k(\R^{p,q})$ is a $\mathscr{C}_{p+q}$-algebra with $\N$ as monoid of path components, hence the stable homology $H_*(V^k_{\infty}(\R^{p,q})):=\mathrm{colim}_{r\to \infty}H_*(V_r^k(\R^{p,q}))$ agrees with the homology of (a component of) some $\Omega^{p+q}$-space. %
  The second author \cite{kranhold2} provides a geometric model for the $p$-fold delooping of $V^k(\R^{p,q})$,
  and in the case $q=1$ even for the $(p+1)$-fold delooping. We still lack a geometric description of the $(p+q)$-fold
  delooping of $V^k(\R^{p,q})$ for $q\ge2$, even in the seemingly innocent case $p=0$.
  \item [{\small \bfseries 2.}]
    We believe that the Leray spectral sequence for the filtration $F_\bullet V_r$ collapses on its first page and that the extension problem is trivial. This would then imply that, using the notation from the proof of \mbox{\textsc{Lemma \ref{lem:colstab}},}
    \[H_m(V_r(\R^{p,1}))\cong\bigoplus_{s=0}^{r-1} H_m(C_{r,s},C^*_{r,s}) \cong \bigoplus_{s=0}^{r-1}\bigoplus_{\deg(\alpha)=(r,s)}M_{m,\alpha}.\]
    Our motivation is again the description of the stable homology
    \[
    H_m(V_{\infty}(\R^{p,1}))\cong \bigoplus_{s=0}^{\infty}\bigoplus_{\alpha}M_{m,\alpha},
    \]
    given in \cite{kranhold2}, where the last direct sum is extended over all distribution $\alpha\colon \mathbb{E}\to\N$
    with $\alpha_{e_0}=0$ and $s(\alpha)=s$.

  \item [{\small\bfseries 3.}]
  The strategy of proof of \mbox{\textsc{Theorem \ref{theo:stab}}} generalises to the following case: Let $\mathbf{k}=(k_1,\dots,k_r)$, and for
  $k\ge1$ let $r(k)\ge0$ be the number of indices $1\le i\le r$ with $k_i=k$; define a stabilisation
  map $V_{\mathbf{k}}(\R^{p,1})\to V_{(\mathbf{k},k)}(\R^{p,1})$ by inserting a new vertical cluster; then the induced map in homology
  \[
   \mathrm{stab}_*\colon H_m(V_{\mathbf{k}}(\R^{p,1}))\to H_m(V_{(\mathbf{k},k)}(\R^{p,1}))
  \]
  is an isomorphism for $m\le \frac{r(k)}{2}$. We leave to the reader the details of the generalisation of the proof.
  \end{enumerate}
\end{outl}

\section{Introduction}
\input{01-introduction.tex}

\section{Preliminaries}
\label{sec:preliminaries}
\input{02-preliminaries.tex}
\section{The cohomology of \texorpdfstring{$\tilde{V}_\mathbf{k}(\R^{p,q})$}{VK(Rpq)}}
\label{sec:orderedHomology}
\input{03-ray_partitions.tex}

\section{Homological stability}
\label{sec:stability}
\input{04-homological_stability.tex}

\printbibliography[heading=bibintoc]

\begin{addr}
  \auth
  {6cm}
  {Andrea Bianchi}
  {Department of Mathematical Sciences,\\
    University of Copenhagen,\\
    Universitetsparken 5,\\
    DK-2100 Copenhagen, Denmark}
  {anbi@math.ku.dk}
  \hspace*{8mm}
  \auth
  {5.2cm}
  {Florian Kranhold}
  {Mathematical Institute,\\
    University of Bonn,\\
    Endenicher Allee 60,\\
    53111 Bonn, Germany}
  {kranhold@math.uni-bonn.de}
\end{addr}

\end{document}

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GDL Racing clinched last weekend the title of the GT Open in the last round of Targa Tricolore Porsche at Imola with the Italian-Swiss duo Vittorio Bagnasco-Nicola Bravetti, who were joined by Dubai's Bashar Mardini.
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A: If you have installed the Code Runner extension, the play button in the upper right corner will have three options to run the code.

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From 2004 to 2008 Yamaha was on the cutting edge of our industry, coming out with three exciting vehicles - YFZ450, Raptor 700, and Rhino 660 and 700. The YFZ led the way for 450cc sport ATV's. The Raptor 700 overhauled the entire industry with its trail stomping power and fuel injected motor.
The Rhino was the first popular sport side x side. I owned a YFZ450 and a Rhino 700 and I loved both of them. I loved taking my YFZ to the track and the dunes. What a great bike. But, I was getting older and my daughter was getting older so it was easy to fall in love with riding the dunes out at Glamis while strapped in to with a seat belt, not having to change gears, with my little trooper strapped in beside me (I am aware that taking a small child for a ride in the front seat of a Rhino is not the smartest idea. We'll just add it to the list). I know that's lazy but hey, why do you think side x sides are so popular? But 2008 was it. Polaris introduced the RZR in 2008 and Yamaha just kind of stepped aside and let them take over the industry.
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Why not come out with a really cool 1000cc sxs that has a motor based on one of Yamaha's killer snowmobile or street bike motors? Was that not feasible? Is the Viking really all they have as a comeback for the XP900 and XP1000? What do you think?

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