\begin{document}

\title{Cellular approximations of fusion systems}

\begin{abstract}
 In this paper we study the cellularization of classifying spaces of saturated fusion systems over finite $p$-groups with respect to classifying spaces of finite $p$-groups. We give explicit criteria to decide when a classifying space is cellular and we explicitly compute the cellularization for a family of exotic examples.
\end{abstract}

\section{Introduction}

The transfer plays an important role both in stable homotopy theory of classifying spaces and group cohomology. Given a finite group $G$ and a prime $p$, if $S$ is a Sylow $p$-subgroup, the properties of the transfer imply that the mod $p$ cohomology of $G$ injects into the mod $p$ cohomology of $S$. In stable homotopy theory, the spectrum of $\p{BG}{p}$ is a retract of the spectrum of $BS$ and the splitting is constructed by an idempotent in stable selfmaps of the spectrum of $BS$.

In 1990s, E. Dror-Farjoun and W. Chach\'olski generalized the concept of $CW$-complex, spaces build from spheres by means pointed homotopy colimits. Let $A$ be a pointed space and 
 let $\mathcal{C}(A)$ denote the smallest collection of pointed spaces that contains $A$ and it is closed by weak equivalences and pointed homotopy colimits. A pointed space $X$ is
 \textit{$A$-cellular} if $X \in \mathcal{C}(A)$. These concepts are also defined in stable homotopy category.
 
 In the stable context, the fact that $\p{BG}{p}$ is a stable retract of $BS$ implies that that $\Sigma^\infty_+\p{BG}{p}$ belongs to $\mathcal C(\Sigma^\infty_+BS)$.
In unstable homotopy theory of classifying spaces, $\p{BG}{p}$ is not a retract of $BS$, but we can ask ourselves whether $\p{BG}{p}$ is in the cellular class $\mathcal C(BS)$, or more generally, given a finite $p$-group $P$, $\p{BG}{p}\in \mathcal C(BP)$?

 The homotopy type of $\p{BG}{p}$ is determined by the $p$-local structure of $G$, the fusion system associated to $G$. Given a finite $p$-group $S$, $p$ a prime, a \textit{fusion system} over $S$ is a subcategory of the category of groups whose objects are the subgroup of $S$ and morphisms are given a set of injective
 homomorphisms, containing those which are induced by conjugation by elements of $S$. A fusion system $\mathcal{F}$ is \textit{saturated} if it verifies certain axioms such as would be 
 holded if $S$ were a Sylow $p$-subgroup of a finite group. These ideas were developed by L. Puig in an unpublished notes. Afterwards, D. Benson suggested the idea of associating a 
 ``classifying space'' to each saturated fusion system (see \cite{Benson}). The notion of classifying space was formulated by C. Broto, R. Levi and B. Oliver
 in \cite{MR1992826}, where the notion of ``centric linking system'' (or ``$p$-local finite group'') associated to saturated fusion systems appears.  At that time, it was not known if every saturated fusion has an associated linking system.

K. Ragnarsson \cite{MR2199459} constructed a classifying space spectrum $\mathbb B\mathcal{F}$ associated to $\mathcal{F}$ by splitting the spectrum of $BS$ via an idempotent stable selfmap. Analogously to the situation for finite groups, we have then $\mathbb B\mathcal{F}\in \mathcal C(BS)$. 
 
 
Recently, A. Chermak 
 \cite{MR3118305} has proved the existence and uniqueness of centric linking systems, that means, each saturated fusion system $\mathcal{F}$ has a unique (up to 
 isomorphism) centric linking system associated to $\mathcal{F}$, and so a unique (up to homotopy equivalence) classifying space $B\mathcal{F}$. See Section~\ref{s:fus sys} for 
 specific definitions, details and main results which we will use in the rest of the paper about fusion systems. 
 
Previous works in finite groups (see \cite{MR2272149}, \cite{MR2351607} and \cite{MR2823972}) suggest the strong relationship between the cellularity properties of $\p{BG}{p}$ with respect to classifying spaces of finite $p$-groups and the fusion structure of $G$ at the prime $p$.
 
 Let $P$ be a finite $p$-group, we will denote by $Cl_{\mathcal{F}}(P)$ the smallest strongly $\mathcal{F}$-closed subgroup in $S$ which contains all the images of homomorphisms $P \rightarrow S$ (see Section~\ref{s:ker f} below). The main result of this paper is the following theorem.

 \noindent{\bf Theorem~\ref{t:BP << Lp}.}
  \noindent {\it  Let $\mathcal{F}$ be a saturated fusion system over a finite $p$-group $S$ and let $P$ be a finite $p$-group. Then $\mathcal{L}p$ is $BP$-cellular if and only if $S = Cl_{\mathcal{F}}(P)$.
 }
 

 \noindent{\bf Corollary~\ref{c:BS << Lp}.}
 \noindent {\it 
  Let $(S,\mathcal{F})$ be a saturated fusion system and $P$ a finite $p$-group.
  \begin{enumerate}[(a)]
   \item The classifying space $\mathcal{L}p$ is $BS$-cellular.
   \item Let $(S,\mathcal{F}')$ be a saturated fusion system with $\mathcal{F}\subset \mathcal{F}'$. If $B\mathcal{F}$ is $BP$-cellular then $B\mathcal{F}'$ is also $BP$-cellular.
   \item Let $A$ be a pointed connected space. If $Cl_\mathcal{F}((\pi_1 A)_{ab})=S$, then $\mathcal{L}p$ is $A$-cellular.
   \item Let $\Omega_{p^m}(S)$ be the (normal) subgroup of $S$ generated by its elements of order $p^i$, which $i \leq m$. Then $\mathcal{L}p$ is $\bz{p^m}$-cellular if and only if 
         $S = Cl_{\mathcal{F}}(\Omega_{p^m}(S))$. In particular, there is a non-negative integer $m_0 \geq 0$ such that $\mathcal{L}p$ is $\bz{p^m}$-cellular for all $m \geq m_0$.
  \end{enumerate}
 }
 

There exists an augmented idempotent endofunctor $CW_A \colon Spaces_{\ast} \rightarrow Spaces_{\ast}$ such that for all
 pointed space $X$, the space $CW_A X$ is $A$-cellular and the augmention map $c_X \colon CW_A X \rightarrow X$ is an $A$-equivalence, that means, it is induced a weak equivalence in
 pointed mapping space $\xymatrix@1@C=1.25em{(c_X)_{\ast} \colon \mbox{map}p(A, CW_A X) \ar[r]^-{\simeq} & \mbox{map}p(A, X)}$. Roughly speaking, $CW_A X$ is the best $A$-cellular approximation 
 of $X$.  We will say that $CW_A X$ is the \textit{$A$-cellularization} of $X$ and the map $c_X \colon CW_A X \rightarrow X$ is the \textit{$A$-cellular approximation} of $X$. See 
 \cite{MR1392221} for more details about the construction and main properties of the functor $CW_A$.
 
 
 The strategy of proof of Theorem \ref{t:BP << Lp} is the analysis of the Chach\'olski fibration to compute $CW_{BP}(B\mathcal{F})$ which is described in \cite{MR1408539}. Let $C$ be the homotopy cofibre of the 
 evaluation map $ev \colon \bigvee_{[BP,\mathcal{L}p]_{\ast}} BP \rightarrow \mathcal{L}p$, then $CW_{BP}(B\mathcal{F})$ is the homotopy fibre of the composite 
 $r \colon B\mathcal{F} \rightarrow C \rightarrow P_{\Sigma BP} C$, where $P_{\Sigma BP}$ denotes the $\Sigma BP$-nullification functor defined by A. K. Bousfield in \cite{MR1257059}. We 
 prove that $CW_{BP}(B\mathcal{F}) \simeq B\mathcal{F}$ if and only if the map $\p{r}{p}$ is null-homotopic (see the proof of Theorem~\ref{t:BP << Lp}). Therefore, the $BP$-cellularity of $B\mathcal{F}$ is 
 equivalent to the homotopy nullity of the map $\p{r}{p}$. 
 
 To approach this question, we study the kernel of $\p{r}{p}$ in the sense of D. Notbohm introduced in \cite{MR1286829} for maps from 
 classifying spaces of compact Lie groups (Section~\ref{s:ker f}). Given a map $f \colon B\mathcal{F} \rightarrow Z$, where $Z$ is a connected $p$-complete and 
 $\sbz{p}$-null space (that is, $\p{Z}{p} \simeq Z$ and $\mbox{map}p(\sbz{p}, Z) \simeq \ast$), then $\ker(f) := \{x \in S \mid f|_{B\langle x \rangle} \simeq \ast \}$. We show that 
 $\ker(f)$ is a strongly $\mathcal{F}$-closed subgroup of $S$ (Proposition~\ref{p:kerf strongly}) with the following  main property.
 

 \noindent{\bf Theorem~\ref{t:Dwyer p local}.}
  \noindent {\it Let $(S,\mathcal{F})$ be a saturated fusion system. Let $Z$ be a connected $p$-complete and $\sbz{p}$-null space. A map 
            $f \colon \mathcal{L}p \rightarrow Z$ is null-homotopic if and only if $\ker(f) = S$.
 }

 
 Furthermore, any strongly $\mathcal{F}$-closed subgroup $K\leq S$ is the kernel of a map from $B\mathcal{F}$ to $\p{B(\Sigma_{|S/K|} \wr \Sigma_{p^m})}{p}$ for certain $m \geq 0$ 
 (Proposition~\ref{p:exist m and f}). 
 
 In Section~\ref{s:cw BP} we study homotopy properties of $CW_{BP}(B\mathcal{F})$ needed for the proof of the main theorem. Section~\ref{s:cw BF} contains the proof of Theorem~\ref{t:BP << Lp}. A key step is the computation of  the kernel of $\p{r}{p}$.
 
 

 \noindent{\bf Proposition~\ref{p:Kerf = Cl}.}
  \noindent {\it Let $(S,\mathcal{F})$ be a saturated fusion system. Then $\ker(\p{r}{p}) = Cl_{\mathcal{F}}(P)$.
 }

 The last two sections are devoted to give explicit examples. Concretely, in Section~\ref{s:examples} we describe a strategy to compute the $BP$-cellularization of $B\mathcal{F}$ when $S \neq Cl_{\mathcal{F}}(P)$. This is the case when we have a homotopy factorization of $\p{r}{p} \colon B\mathcal{F} \rightarrow \p{(P_{\Sigma BP} C)}{p}$ by a map 
 $\p{\tilde{r}}{p} \colon B\mathcal{F}' \rightarrow \p{(P_{\Sigma BP} C)}{p}$ with trivial kernel (verifying certain technical conditions in Proposition~\ref{p:factor}). This is the case when $Cl_{\mathcal{F}}(S)$ is normal in $\mathcal{F}$. 
 

 \noindent{\bf Corollary~\ref{c:normal}.}
  \noindent {\it Let $(S,\mathcal{F})$ be a fusion system and let $P$ be a finite $p$-group. If $Cl_{\mathcal{F}}(P) \vartriangleleft \mathcal{F}$, then $CW_{BP} (\mathcal{L}p)$ is homotopy equivalent to the homotopy
                 fibre of $\mathcal{L}p \rightarrow B(\mathcal F/Cl_{\mathcal{F}}(P))$.
 }

 This result allow us compute, for all $r \geq 1$, the $\bz{p^r}$-cellularation of the classifying space of $\z{/p^n} \wr \z{/q}$, with $p \neq q$, and of the Suzuki group 
 $\operatorname{Sz}(2^n)$, with $n$ an odd integer at least $3$ (Example~\ref{e:NG(S) controls fus}).
 
The last section contains an explicit description of the $\bz{3^l}$-cellularation of the classifying space of a family of exotic fusion systems over a finite $3$-group given in \cite{MR2272147}. 

 \noindent{\bf Corollary~\ref{c:exotic example}.}
  \noindent {\it Let $\mathcal{F}$ be an exotic fusion system over $B(3,r;0,\gamma,0)$ such that $\mathcal{F}$ has at least one $\mathcal{F}$-Alperin rank two elementary abelian $3$-subgroup given in \cite[Theorem 5.10]{MR2272147}.
  Then 
  \begin{enumerate}[(i)]
   \item If $\gamma = 0$, then $\mathcal{L}p$ is $\bz{3^l}$-cellular for all $l \geq 1$.
   \item Assume $\gamma \neq 0$. Then $\mathcal{L}p$ is $\bz{3^l}$-cellular if and only if $l \geq 2$. If $l=1$, $Cl_{\mathcal{F}}(\z{/3})=\langle s,s_2\rangle$.
   \end{enumerate}
 }

Moreover, when $\gamma \neq 0$ and $l=1$, we show that $CW_{\bz3}(B\mathcal{F})$ is the homotpy fibre of a map $B\mathcal{F}\rightarrow \p{(B\Sigma_3)}{3}$.

\textbf{Acknowledgements.}
 We would like to thank Ram\'on J. Flores and J\'er\^ome Scherer for many suggestions throughout the course of this work. We appreciate the help of Antonio D\'iaz with the computations for the examples in Section~\ref{s:exotic example}. The second author would also like to thank the Departamento de \'Algebra, Geometr\'ia y Topolog\'ia of the Universidad de M\'alaga and the Universidade da CoruÃ±a, in special, Antonio Viruel and Cristina Costoya for their kind hospitality. 

\section{Preliminaries on the homotopy theory of fusion systems}\label{s:fus sys}

 A saturated fusion system is a small subcategory of the category of groups which encodes fusion/conjugacy data between subgroups of a fixed finite $p$-group $S$, as formalized by L. Puig (see \cite{MR2253665} and also 
 \cite{MR2848834}). Such objects have classifying spaces which satisfy many of the rigid homotopy theoretic properties of $p$-completed classifying spaces of finite groups, as generalized  by C. Broto, R. Levi and B. Oliver (see \cite{MR1992826}).
  
 \begin{defi}
  Let $S$ be a finite $p$-group. A \textit{saturated fusion system on $S$} is a subcategory $\mathcal{F}$ of the category of groups with $\operatorname{Ob}(\mathcal{F})$  the set of all subgroups of $S$ and 
  such that it satisfies the following properties. For all $P,Q \leq S$:
   \begin{enumerate}[($f.1$)]
    \item $\operatorname{Hom}_S (P,Q) \subset \operatorname{Hom}_{\mathcal{F}} (P,Q) \subset \operatorname{Inj}(P,Q)$; and
    \item each $\varphi \in \operatorname{Hom}_{\mathcal{F}} (P,Q)$ is the composite of an isomorphism in $\mathcal{F}$ followed by an inclusion.
   \end{enumerate}
     
  For all $P \leq S$ and all $P' \leq S$ which is \textit{$\mathcal{F}$-conjugate to $P$} ($P$ and $P'$ are isomorphic as objects in $\mathcal{F}$):
   \begin{enumerate}[($s.1)$]
    \item For all $P \leq S$ which is \textit{fully normalized in $\mathcal{F}$} (i.e. $|N_S(P)| \geq |N_S(P')|$ for all $P'$ $\mathcal{F}$-conjugate to $P$), $P$ is also \textit{fully centralized in $\mathcal{F}$} ($|C_S(P)| \geq |C_S(P')|$ for all $P'$ $\mathcal{F}$-conjugate to $P$), and 
          $\operatorname{Out}_S(P) \in \operatorname{Syl}_p(\operatorname{Out}_{\mathcal{F}}(P))$.
    \item Let $P \leq S$ and $\varphi \in \operatorname{Hom}_{\mathcal{F}}(P,S)$ be such that $\varphi(P)$ is fully centralized. If we set
          $$
           N_{\varphi} = \{g \in N_S(P) \mid \varphi c_g \varphi^{-1} \in \operatorname{Aut}_S(\varphi(P))\},
          $$
          then there is $\bar{\varphi} \in \operatorname{Hom}_{\mathcal{F}} (N_{\varphi},S)$ such that $\bar{\varphi}|_P = \varphi$.
    \end{enumerate}
 \end{defi}
 
 The standard example is given by the fusion category of a finite group $G$. Given a finite group $G$ with a fixed Sylow $p$-subgroup $S$, let $\mathcal{F}_S(G)$ be the category with 
 $\operatorname{Mor}_{\mathcal{F}_S(G)} (P,Q) = \operatorname{Hom}_G (P,Q)$ for all $P,Q \leq S$, where $\operatorname{Hom}_G (P,Q) = \{\varphi \in \operatorname{Hom}(P,Q) \mid \varphi = c_g \mbox{ for some } g \in G\}$.  This category $\mathcal{F}_S(G)$ satisfies the saturation axioms (see \cite[Proposition 1.3]{MR1992826}).  
 
 In order to recover the homotopy type of the Bousfiel-Kan $p$-completion of the classifying space $BG^\wedge_p$, Broto-Levi-Oliver \cite{MR1961340} introduce a new category defined using the group $G$. A $p$-subgroup $P\leq G$ is $p$-centric if $Z(P')$ is a Sylow $p$-subgroup of $C_G(P')$ for all $P'$ $G$-conjugate to $P$. Let $\mathcal{L}_S(G)$ be the category whose objects are $p$-centric subgroups of $G$
 with $\operatorname{Mor}_{\mathcal{L}_S(G)} (P,Q) = \{ x \in G \mid xP^{-1}x \leq Q\}/O^p(C_G(P))$ for all $P,Q \leq S$. In \cite{MR1961340} the authors proved that the classifying space of this $p$-local structure is $\p{|\mathcal{L}_S(G)|}{p} \simeq \p{BG}{p}$.  This new structure one can associate to a finite group can also be generalized in the context of abstract fusion systems.

 
 \begin{defi}\cite{MR1992826}
  Let $\mathcal{F}$ be a saturated fusion system over a finite $p$-group $S$. A \textit{centric linking system associated to $\mathcal{F}$} is a category $\mathcal{L}$ whose objects are the 
  \textit{$\mathcal{F}$-centric subgroups of $S$} (i.e., the subgroups $P \leq S$ such that $C_S (P) = Z(P)$), together with a functor $\pi \colon \mathcal{L} \rightarrow \mathcal{F}^c$, and 
  ``distinguished'' monomorphisms $\delta_P \colon P \rightarrow \operatorname{Aut}_{\mathcal{L}}(P)$ for each $\mathcal{F}$-centric subgroup $P \leq S$, which satisfies the following conditions:
  \begin{enumerate}[($l.1)$]
   \item $\pi$ is the identity on objects and surjective on morphisms. For each pair of objects $P,Q \leq \mathcal{L}$, $Z(P)$ acts freely on $\operatorname{Mor}_{\mathcal{L}}(P,Q)$ by 
         composition (upon identifying $Z(P)$ with $\delta_P(Z(P)) \leq \operatorname{Aut}_{\mathcal{L}}(P)$), and $\pi$ induces a bijection
         $$
          \xymatrix@1@C=1.25em{\operatorname{Mor}_{\mathcal{L}}(P,Q)/Z(P) \ar[r]^-{\cong} & \operatorname{Hom}_{\mathcal{F}}(P,Q).}
         $$
   \item For each $\mathcal{F}$-centric subgroup $P \leq S$ and each $g \in P$, $\pi$ sends $\delta_P(g)\in \operatorname{Aut}_{\mathcal{L}}(P)$ to $c_g \in \operatorname{Aut}_{\mathcal{F}}(P)$.
   \item For each $f \in \operatorname{Mor}_{\mathcal{L}}(P,Q)$ and each $g \in P$, $f \circ \delta_P(g) = \delta_Q(\pi f(g)) \circ f$.
  \end{enumerate}
 \end{defi}
 
 Recently, in 2013, A. Chermak \cite{MR3118305} proved the existence and uniqueness of centric linking systems associated to saturated fusion systems (see also \cite{MR3118306}).
 
 \begin{theorem}{(\cite{MR3118305},\cite{MR3118306})}
  Let $\mathcal{F}$ be a saturated fusion system on a finite $p$-group $S$. Then there exists a centric linking system $\mathcal{L}$ associated to $\mathcal{F}$. Moreover, $\mathcal{L}$ is uniquely determined by $\mathcal{F}$ up to isomorphism.
 \end{theorem}
 
 \begin{defi}
  The \textit{classifying space} $B\mathcal{F}$ of a saturated fusion system $(S,\mathcal{F})$ is the Bousfield-Kan $p$-completion of the nerve of the associated centric linking system $\p{|\mathcal{L}|}{p}$. We denote by $\Theta\colon BS\rightarrow \mathcal{L}p$ the map induced by the distinguished monomorphism $\delta_S$.
 \end{defi}
 

 Given a map $f\colon \mathcal{L}p\rightarrow X$ and $P\leq S$, we denote by $f|_{BP}$ the composite $f\circ \Theta\circ Bi$ where $i\colon P\leq S$. 

We describe some results concerning the homotopy type of the classifying space $B\mathcal{F}$ and mapping spaces $\mbox{map}(BP,\mathcal{L}p)$ which will be used in the rest of the paper.
   
 \begin{prop}[{\cite{MR1992826},\cite{MR2302515},\cite{MR2518640}}]\label{l:hmtpy gps}
  For any satured fusion system $(S,\mathcal{F})$, $\mathcal{L}p$ is a $p$-complete space and $\pi_i(\mathcal{L}p)$ are finite $p$-groups for all $i\geq 1$. The fundamental group 
  $\pi_1(\mathcal{L}p) \cong S/\mathcal{O(F)}p$, where 
  $
   \mathcal{O(F)}p := \langle [Q,\mathcal{O}^p(\operatorname{Aut}_{\mathcal{F}}(Q))] \mid Q \leq S \rangle.
  $
 \end{prop}

 \begin{proof}
  The description of the fundamental group $\pi_1(\mathcal{L}p)$ is given in \cite[Theorem B]{MR2302515}. The fact that $\mathcal{L}p$ is $p$-complete follows from 
  \cite[Proposition I.5.2]{MR0365573} since the nerve of the associated linking system $|\mathcal{L}|$ is $p$-good by \cite[Proposition 1.12]{MR1992826}. Finally, $\pi_i(\mathcal{L}p)$ are finite 
  $p$-groups for all $i\geq 1$ by \cite[Lemma 7.6]{MR2518640}.
 \end{proof}
 
 \begin{prop}\cite[Theorem 6.3]{MR1992826}\label{p:mapping space}
 Let $(S,\mathcal{F})$ be a saturated fusion system. If $P$ is a finite $p$-group and $\rho\colon P\rightarrow S$ a group homomorphism such that $\rho(P)$ is fully $\mathcal{F}$-centralized, there is a saturated fusion system $(C_S(\rho(P)),C_\mathcal{F}(\rho(P)))$ and a homotopy equivalence  $BC_\mathcal{F}(\rho(P))\stackrel{\simeq}{\rightarrow} \mbox{map}(BP,B\mathcal{F})_\rho$. In particular, the evaluation $\mbox{map}(BP,B\mathcal{F})_c\rightarrow \mathcal{L}p$ is a homotopy equivalence.
 \end{prop}
 
 Moreover, any finite covering of the classifying space of a saturated fusion system is again the classifying space of a saturated fusion system.
 
 \begin{theorem}[{\cite[Theorem A]{MR2302515}}]\label{t:coverings}
  Let $(S,\mathcal{F})$ be a saturated fusion system and $\mathcal{L}$ be the associated linking system. Then there is a normal subgroup $H \vartriangleleft \pi_1 |\mathcal{L}|$ which is minimal among all 
  those whose quotient is finite and $p$-solvable. Any covering space of the geometric realization $|\mathcal{L}|$ whose fundamental group contains $H$ is homotopy equivalent to $|\mathcal{L}'|$ 
  for some linking system $\mathcal{L}'$ associated to a saturated fusion system $(S',\mathcal{F}')$, where $S' \leq S$ and $\mathcal{F}' \leq \mathcal{F}$.
 \end{theorem}

In this work it is important to understand the homotopy properties of mapping spaces between classifying paces. This question was already studied in \cite{MR1992826}.
 One of the standard techniques used when studying maps between $p$-completed classifying spaces of finite groups is to replace them by the $p$-completion of a homotopy colimit of 
 classifying spaces of subgroups.
 
 \begin{defi}
  Let $\mathcal{F}$ be a saturated fusion system over a finite $p$-group $S$. The \textit{orbit category} of $\mathcal{F}$ is the category $\mathcal{O(F)}$ whose objects are the subgroups of $S$, and whose 
  morphisms are defined by
  $$
   \operatorname{Mor}_{\mathcal{O(F)}}(P,Q) = \operatorname{Rep}_{\mathcal{F}}(P,Q) := \operatorname{Inn}(Q)/\operatorname{Hom}_{\mathcal{F}}(P,Q).
  $$
  We let $\mathcal{O(F)}c$ denote the full subcategory of $\mathcal{O(F)}$ whose objects are the $\mathcal{F}$-centric subgroups of $S$. If $\mathcal{L}$ is a centric linking system associated to $\mathcal{F}$, then 
  $\tilde{\pi}$ denotes the composite functor 
  $$
   \xymatrix@1@C=1.25em{\tilde{\pi} \colon \mathcal{L} \ar[r]^-{\pi} & \mathcal{F}^c \ar@{->>}[r] & \mathcal{O(F)}c}.
  $$
 \end{defi}

 The homotopy type of the nerve of a centric linking system can be described as a homotopy colimit over the orbit category.

 \begin{prop}[{\cite[Proposition 2.2]{MR1992826}}]\label{p:L = hocolim}
  Fix a saturated fusion system $\mathcal{F}$ over a finite $p$-group $S$ and an associated centric linking system $\mathcal{L}$, and let $\tilde{\pi} \rightarrow \mathcal{O(F)}c$ be the projection functor. 
  Let 
  $\tilde{B} \colon \mathcal{O(F)}c \rightarrow \operatorname{Top}$
  be the left homotopy Kan extension over $\tilde{\pi}$ of the constant functor $\xymatrix@1@C=1.25em{\mathcal{L} \ar[r]^-{\ast} & \operatorname{Top}}$. Then $\tilde{B}$ is a lift of the classifying space functor $P \mbox{map}sto BP$ to the category of topological spaces, and
  $$
   |\mathcal{L}| \simeq \operatorname{hocolim}_{\mathcal{O(F)}c} (\tilde{B}).
  $$
 \end{prop}  
 
 Given fusion systems $\mathcal{F}$ and $\mathcal{F}'$ on $S$ and $S'$ respectively, a homomorphism $\psi \colon  S \to S'$ is called \emph{fusion preserving} if for every $\varphi \in \mathcal{F}(P,Q)$ there 
 exists some $\varphi' \in \mathcal{F}'(\psi(P),\psi(Q))$ such that $\psi \circ \varphi = \varphi' \circ \psi$.
   
 \begin{theorem}[{\cite[Theorem 1.3]{MR2518640}}]
  Let $(S,\mathcal{F})$ and $(S',\mathcal{F}')$ be saturated fusion systems. Suppose that $\rho \colon S \to S'$ is a fusion preserving homomorphism. Then there exists some $m \geq 0$ and a map $\tilde{f}\colon B\mathcal{F} \to B(\mathcal{F}'\wr \Sigma_{p^m})$ such that the diagram below commutes up to homotopy
  $$
   \xymatrix{
             BS  \ar[r]^{ \Theta} \ar[d]_{B\rho} & B\mathcal{F}  \ar[dr]^{\tilde{f}} \\
             BS' \ar[r]_{ \Theta'}               & B\mathcal{F}' \ar[r]_(0.4){{\Delta}^\wedge_p} & B(\mathcal{F}'\wr \Sigma_{p^m}).
            }
  $$ where $\mathcal{F}'\wr \Sigma_{n}$ is the saturated fusion system whose classifying space is $\p{((B\mathcal{F}')^n_{h\Sigma_n})}{p}$.
 \end{theorem} 
    
\section{The kernel of a map from a classifying space}\label{s:ker f}
  
Given a connected space $A$, we say that a space $X$ is $A$-null if the evaluation map $ev\colon \mbox{map}(A,X)\rightarrow X$ is a weak equivalence (see \cite{MR1392221}). If $X$ is connected, $X$ is $A$-null iff $\mbox{map}_*(A,X)$ is weakly contractible. There is a nullification functor $P_A\colon Spaces\rightarrow Spaces$ with a natural transformation $\eta_X\colon X\rightarrow P_A(X)$ which is initial with respect to maps into $A$-null spaces. We say that $X$ is $A$-acyclic if $P_AX\simeq *$.
 
 The kernel of a map $f \colon \p{BG}{p} \rightarrow Z$, where $G$ is a compact Lie group and $Z$ is a connected $p$-complete $\sbz{p}$-null space, is defined by D. Notbohm in 
 \cite{MR1286829}. We adapt his definition to our context. 
  
 \begin{defi}\label{d:kerf}
  Let $(S,\mathcal{F})$ be a saturated fusion system and let $Z$ be a connected $p$-complete $\sbz{p}$-null space. If $f \colon \mathcal{L}p \rightarrow Z$ is a pointed map, we define the kernel of $f$ 
  $$
   \ker(f) := \{g \in S \mid f|_{B\langle g \rangle} \simeq \ast \}.
  $$
 \end{defi}
 
 \begin{remark} \label{r:bf sigmabzp-null}
 By Proposition \ref{p:mapping space}, we have $\mbox{map}(\bz p,\mathcal{L}p)_c\simeq \mathcal{L}p$. It follows then by looping that $\Omega \mathcal{L}p$ is $\bz p$-null, or equivalently, that $\mathcal{L}p$ is $\Sigma \bz p$-null (\cite[3.A.1]{MR1392221}).
 \end{remark}
 
 \begin{remark}\label{r:BZp null hence BP}
  If $X$ is a $\bz{p}$-null space, then $X$ is $BP$-null for any finite $p$-group $P$. There are weak equivalences $\mbox{map}p(BP,X) \simeq \mbox{map}p(P_{\bz{p}}(BP), X) \simeq \ast$, where
  the last equivalence follows from Lemma 6.13 in \cite{MR1397728} which states that $BP$ is $\bz{p}$-acyclic. A direct proof can be obtained by induction using the central 
  extension of a $p$-group and Zabrodsky's Lemma \cite[Proposition 3.4]{MR1397728}.
 \end{remark}
 
 We will show that $\ker(f)$ is a subgroup of $S$ with some important properties.
 
 \begin{defi}
  Let $\mathcal{F}$ be a fusion system over a finite $p$-group $S$. Then a subgroup $K \leq S$ is \textit{strongly $\mathcal{F}$-closed} if for all $P \leq K$ and all morphism 
  $\varphi \colon P \rightarrow S$ in $\mathcal{F}$ we have $\varphi(P) \leq K$.
 \end{defi}
 
 If $G$ is a finite group and $S \in \operatorname{Syl}_p(G)$,  $K \leq S$ is strongly $\mathcal{F}_S(G)$-closed if and only if $K$ is \textit{strongly closed in $G$}, i.e., if for all $k \in K$ and 
 $g \in G$ such that $c_g(s) \in S$, then $c_g(s) \in K$.
 
 Since the intersection of strongly $\mathcal{F}$-closed subgroups is again strongly $\mathcal{F}$-closed, given a finite $p$-group $P$, we  define $Cl_{\mathcal{F}}(P)$ to be the smallest strongly 
 $\mathcal{F}$-closed subgroup of $S$ that contains $f(P)$ for all $f \in \operatorname{Hom}(P,S)$.
 
 \begin{prop}\label{p:kerf strongly}
  Let $f \colon \mathcal{L}p \rightarrow Z$ be a pointed map as in Definition \ref{d:kerf}. The kernel $\ker(f)$ is a strongly $\mathcal{F}$-closed subgroup of $S$.
 \end{prop}
  
 \begin{proof}
  Since  $\langle x \rangle = \langle x^{-1} \rangle$, in order to show that $\ker(f)$ is a subgroup of $S$ it is sufficient to prove that if $x,y \in \ker(f)$, then 
  $xy \in \ker(f)$. The composite $B\langle x,y \rangle \rightarrow BS \rightarrow \mathcal{L}p \rightarrow Z$ is null-homotopic by \cite[Proposition 2.4]{MR1286829}. Since 
  $\langle xy \rangle \hookrightarrow \langle x,y \rangle$, $f|_{B\langle xy \rangle} \simeq \ast$.
   
  Let $P \leq \ker(f)$ and $\varphi \in \operatorname{Hom}_{\mathcal{F}}(P, S)$. We have the following homotopy commutative diagram
  $$
   \xymatrix@R=.5em{ 
              BP \ar[dr]^-{B\delta_P} \ar[dd]_-{B\varphi}^-{\simeq} &                 &   \\
                                                                   & B\mathcal{F} \ar[r]^-{f} & Z \\
              B\varphi(P) \ar[ru]_-{B\delta_{\varphi(P)}}           &                 &
             }
  $$
  which shows that $f|_{BP}$ is null-homotopic if and only if $f|_{B\varphi(P)}$ is null-homotopic.
 \end{proof}
  
 W. Dwyer shows in \cite[Theorem 5.1]{MR1397728} that if we have a finite group $G$, a map $f \colon \p{BG}{p} \rightarrow Z$ as in Definition \ref{d:kerf} is null-homotopic if
 and only if $\ker(f) = S$. This statement is also true for classifying spaces of saturated fusion systems.

 \begin{theorem}\label{t:Dwyer p local}
  Let $(S,\mathcal{F})$ be a saturated fusion system. Let $Z$ be a connected $p$-complete $\sbz{p}$-null space. Then a map $f \colon \mathcal{L}p \rightarrow Z$ is 
  null-homotopic if and only if $\ker(f) = S$.
 \end{theorem}

 \begin{proof}
  If $f \simeq \ast$, then $f|_{BS} \simeq \ast$ and therefore $\ker(f) = S$. Now assume that $f|_{BS} \simeq \ast$, we will show that $f \simeq \ast$. 
  
  

  
  \noindent \textbf{Step 1:} Assume that $\pi_1(Z)$ is abelian. By Proposition \ref{p:L = hocolim}, $\mathcal{L}p \simeq \p{(\operatorname{hocolim}_{\mathcal{O(F)}c} \tilde{B}P)}{p}$, where $\tilde{B}P \simeq BP$ for $P \in \mathcal{F}^c$. Since any map $BP \rightarrow \mathcal{L}p$ 
  factors through 
  $\Theta \colon BS\rightarrow \mathcal{L}p$ by \cite[Theorem 4.4]{MR1992826}, $f|_{BP} \simeq \ast$ for all $P \in \mathcal{F}^c$. Therefore we have two maps
  $$
   \xymatrix@C=1.25em{\operatorname{hocolim}_{\mathcal{O(F)}c} (\tilde{B}P) \ar[r] & \p{(\operatorname{hocolim}_{\mathcal{O(F)}c} (\tilde{B}P))}{p} \ar@<.75ex>[r]^-f \ar@<-.25ex>[r]_-{\ast} & Z},
  $$
  such that both are nullhomotopic when restricted to $BP$ for all $P \in \mathcal{F}^c$. 
  
  The obstructions for these maps to be homotopic are in $\lim_{\mathcal{O(F)}c}^i \pi_{i}(\mbox{map}(BP, Z)_c)$, for $i \geq 1$ (see \cite{MR928836}). Since a $\bz{p}$-null space is $BQ$-null for 
  any finite $p$-group $Q$ (Remark \ref{r:BZp null hence BP}), $Z$ is $\Sigma BP$-null and hence $\mbox{map}p(BP,Z)$ is homotopically discrete, therefore $\mbox{map}p(BP,Z)_c \simeq \ast$ 
  and, from the fibration $\mbox{map}p(BP,Z)_c \rightarrow \mbox{map}(BP,Z)_c \rightarrow Z$, we obtain $\mbox{map}(BP,Z)_c \simeq Z$. 
  
  Given a morphism $\varphi \colon P \rightarrow Q$ in $\mathcal{F}^c$, $B\varphi$ is a pointed map and induces a commutative diagram 
  $$
   \xymatrix{\mbox{map}(BQ,Z)_c \ar[r]^-{B\varphi^{\ast}} \ar[d]_-{\simeq}^-{ev} & \mbox{map}(BP,Z)_c \ar[d]^-{\simeq}_-{ev} \\
                     Z    \ar[r]_-{id}                               & Z
             }
  $$
  which shows that the obstructions are in $\lim_{\mathcal{O(F)}c}^i \pi_i Z$, where $\pi_{\ast} Z$ is a constant functor in $\mathcal{O(F)}c$. Since $Z$ is $p$-complete and $\pi_1(Z)$ is abelian, the constant functors $\pi_i(Z)$ all take values in 
  $\z{}_{(p)}-\operatorname{Mod}$. 
  
  Let $F := \pi_{\ast}Z$ be the constant functor. Fix $P$ in $\mathcal{O(F)}c^{\operatorname{op}}$ and consider the atomic functors 
  $F_P \colon \mathcal{O(F)}c^{\operatorname{op}} \rightarrow \z{}_{(p)}-\operatorname{Mod}$
  $$
   F_P(Q) := \left\{
                   \begin{array}{ll}
                     \pi_{\ast} Z & \mbox{, if $Q = P$,} \\
                           0      & \mbox{, if $Q \neq P$.}
                    \end{array}
              \right.
  $$
  and $\tilde{F}_P(Q) := F(Q)/F_P(Q)$. Filtering $F$ as a series of extensions of functors $F_P$, one for each object $P$, and using the long exact sequence for higher limits 
  (see proof of Theorem 1.10 of \cite{MR1320996}), if $\lim^i F_P = 0$ for all $P$ and $i > 0$, then $\lim^i F = 0$ for $i > 0$. By \cite[Proposition 6.1 (i),(ii)]{MR1154593}, 
  if $p \nmid |\operatorname{Out}_{\mathcal{F}} (P)|$, then 
  $$
   {\lim}^i F_P = \left\{
              			             \begin{array}{ll}
                                              (\pi_{\ast} Z)^{\operatorname{Out}_{\mathcal{F}} (P)} = \pi_{\ast} Z & \mbox{, if $i = 0$,} \\
                                              0 & \mbox{, if $i > 0$.}
                                             \end{array}
                                            \right.
  $$
  and if $p \mid | \operatorname{Out}_{\mathcal{F}} (P)|$, then $ \lim^i F_P = 0$ for $i \geq 0$. Therefore $\lim_{\mathcal{O(F)}c}^i \pi_i Z= 0$ for $i > 0$, and finally $f \simeq \ast$.
  
  

  
  \noindent\textbf{Step 2}: We will prove that a map $f \colon B\mathcal{F} \rightarrow BG$ is 
  null-homotopic if $f|_{BS}\colon BS\rightarrow BG$ is null-homotopic where $G$ is a discrete group.
  
  
 We will apply Zabrodsky's lemma \cite[Proposition 3.5]{MR1397728} to the map $f|_{BS}$ and the fibre sequence $F\rightarrow BS\rightarrow B\mathcal{F}$ where $F$ is connected since $\pi_1(\Theta)$ is an epimorphism (Lemma \ref{l:hmtpy gps}). Note that $\mbox{map}_*(F,BG)$ is homotopically discrete, then Zabrodsky's lemma shows that there is a homotopy equivalence $\mbox{map}(B\mathcal{F},BG)\simeq \mbox{map}(BS,BG)_{[c]}$ where that last mapping space corresponds to the components which are null-homotopic when restricted to $F$. The bijection between components implies that $f\simeq *$.
 
 

 
 Finally, we can prove the theorem by looking at the universal cover $\tilde{Z}$ of $Z$. Let $p\colon Z\rightarrow B\pi_1(Z)$. The composite $p\circ f$ is null-homotopic by Step 2. Therefore there is a lift $\tilde{f}\colon B\mathcal{F}\rightarrow \tilde{Z}$ to the universal cover of $Z$. In order to apply Step 1, we need to check that $\tilde{f}|_{BS} \colon BS\rightarrow \tilde{Z}$ is null-homotopic. Since both $Z$ and $B\pi_1(Z)$ are $\Sigma B\mathbb Z/p$-null, applying mapping spaces from $BS$ to the universal cover fibration shows that $\mbox{map}(BS,\tilde{Z})_{\{c\}}\simeq \tilde{Z}$ where $\{c\}$ is the set of maps which are nullhomotopic when postcomposed with $\tilde{Z}\rightarrow Z$. Note that $[\tilde{f}|_{BS}]\in \{c\}$. Since $\tilde{Z}$ is connected, the set $\{c\}$ only consists on the constant map and then $\tilde{f}|_{BS}$ is nullhomotopic.
 \end{proof}
  
 Given a strongly $\mathcal{F}$-closed subgroup $K \leq S$ in a saturated fusion system $\mathcal{F}$, there exists a map between classifying spaces $f \colon \mathcal{L}p \rightarrow \p{BG}{p}$ for a finite group $G$ such that $\ker(f) = K$.
 
 \begin{prop}\label{p:exist m and f}
  Let $(S,\mathcal{F})$ be a saturated fusion system and $K$ be a strongly $\mathcal{F}$-closed subgroup and $\mathcal{L}$ be the associated centric linking system. Let $\rho$ be the composite 
  $\xymatrix@1{S \ar[r]^-{\pi} & S/K \ar[r]^-{reg} & \Sigma_{|S/K|}}$, where $\pi$ is the quotient homomorphism and $reg$ is the regular representation of $S/K$. Then there is a 
  non-negative integer $m \geq 0$ and a map $f \colon |\mathcal{L}| \rightarrow \p{B(\Sigma_{|S/K|} \wr \Sigma_{p^m})}{p}$ such 
  that the following diagram
  $$
   \xymatrix{ BS     \ar[r]^-{\p{(\Delta B \rho)}{p}} \ar[d]_-{\iota_S} & \p{(B(\Sigma_{|S/K|})^{p^m})}{p} \ar[d]^{\p{\Delta}{p}}\\
              |\mathcal{L}| \ar[r]_-f                                          & \p{B(\Sigma_{|S/K|} \wr \Sigma_{p^m})}{p}
             }
  $$
  is commutative up to homotopy. Moreover, $\ker(\p{f}{p}) = K$. 
 \end{prop}

 \begin{proof}
  Let $n = |S/K|$. According to \cite[Theorem 1.2]{MR2518640}, if $\rho$ is fusion invariant then there is a non-negative integer $m \geq 0$ and a map
  $f \colon |\mathcal{L}| \rightarrow \p{B(\Sigma_{|S/K|} \wr \Sigma_{p^m})}{p}$ such that $f|_{BS}$ is homotopic to the composite 
  $\xymatrix@1{BS \ar[r]^-{\p{(\Delta B \rho)}{p}} & \p{(B(\Sigma_n)^{p^m})}{p} \ar[r]^-{\p{\Delta}{p}} & \p{B(\Sigma_n \wr \Sigma_{p^m})}{p}}$. 
  Therefore, it is sufficient to show that $\rho$ is fusion invariant: for all $P \leq S$ and $\varphi \colon P \rightarrow S$ in $\mathcal{F}$ there 
  is $\omega \in \Sigma_n$ such that $\rho|_{\varphi(P)} \circ \varphi = c_{\omega} \circ \rho|_P$.
  
  The homomorphisms $\rho|_P$ and $c_{\omega}\rho|_P$ equip $S/K$ with a structure of a $P$-set which are isomorphic. Hence, to prove the above equality, we only need to show that 
  $(S/K, \leq)$ and $(S/K, \leq_{\varphi})$ are equivalent as $P$-sets.
 
  Note that for any $\varphi \colon P \rightarrow S \in \mathcal{F}$, 
  $$
   (S/K, \leq_{\varphi}) \cong \operatorname{Iso}^{\ast}(\varphi) \operatorname{Res}_{\varphi(P)}^S (S/K) \cong \operatorname{Iso}^{\ast}(\varphi) \operatorname{Res}_{\varphi(P)}^S\operatorname{Ind}_K^S(\ast).
  $$
  Applying the Mackey formula to $\operatorname{Res}_{\varphi(P)}^S\operatorname{Ind}_K^S$, we get
  \begin{equation*}\begin{split}
   (S/K, \leq_{\varphi}) &\cong \coprod_{[x] \in \varphi(P) \backslash S/K} \operatorname{Iso}^{\ast}(\varphi) \operatorname{Ind}_{\varphi(P) \cap K^x}^{\varphi(P)} 
                                \operatorname{Iso}^{\ast}(c_x) \operatorname{Res}_{(\varphi(P) \cap K)^x}^K (\ast) \\ 
                         &=     \coprod_{[x] \in \varphi(P) \backslash S/K} \operatorname{Ind}_{\varphi^{-1}(\varphi(P) \cap K)}^P \operatorname{Iso}^{\ast}(\varphi) 
                                \operatorname{Iso}^{\ast}(c_x) \operatorname{Res}_{(\varphi(P) \cap K)^x}^K (\ast).
  \end{split}\end{equation*}
  where the second equality comes from the commutativity of isogation and induction and, where $K^x = K$ because $K$ is strongly $\mathcal{F}$-closed and
  $c_x \colon K \rightarrow S$ is in $\mathcal{F}$, hence $c_x(K)=K^x \leq K$ and since $c_x$ is an isomorphism, $K^x = K$. 
  
  Next we prove that $\varphi^{-1}(\varphi(P) \cap K) = P \cap K$. On the one hand, $\varphi^{-1}(\varphi(P) \cap K) \leq P \cap K$ because 
  $\varphi^{-1}|_{\varphi(P) \cap K} \colon \varphi(P) \cap K \rightarrow S$ is in $\mathcal{F}$,
  $\varphi(P) \cap K \leq K$, $K$ is strongly $\mathcal{F}$-closed and $\varphi^{-1}(\varphi(P) \cap K) \leq P$. The equality will follow if 
  $|\varphi^{-1}(\varphi(P) \cap K)| = |P \cap K|$. Since $\varphi$ is an isomorphism, it is enough to check $|\varphi(P) \cap K| = |\varphi(P \cap K)|$. We already know that 
  $|\varphi(P) \cap K| = |\varphi^{-1}(\varphi(P) \cap K)| \leq |P \cap K| = |\varphi(P \cap K)|$. Since $\varphi|_{P \cap K} \colon P \cap K \rightarrow S$ is in $\mathcal{F}$, 
  $P \cap K \leq K$ and  $K$ is strongly $\mathcal{F}$-closed, $\varphi(P \cap K) \leq K$ but also $\varphi(P \cap K) \leq \varphi(P)$, hence $\varphi(P \cap K) \leq \varphi(P) \cap K$ and 
  therefore $|\varphi(P \cap K)| \leq |\varphi(P) \cap K|$. 
  
  Therefore in the above formula, since $\operatorname{Iso}^{\ast}(\varphi) \operatorname{Iso}^{\ast}(c_x) \operatorname{Res}_{(\varphi(P) \cap K)^x}^K (\ast) = \ast$ as $(P \cap K)$-set and  
  $\varphi^{-1}(\varphi(P) \cap K) = P \cap K$, we get for all $\varphi \colon P \rightarrow S \in \mathcal{F}$
  $$
   (S/K, \leq_{\varphi}) \cong \coprod_{[x] \in \varphi(P) \backslash S/K} \operatorname{Ind}_{P \cap K}^P (\ast) \cong \coprod_{l_{\varphi}} P/P \cap K,
  $$
  where $l_{\varphi} = |\varphi(P) \backslash S/K| = |S/\varphi(P)\cdot K| = |S|/|\varphi(P)\cdot K|$ since $K \lhd S$. In particular, if $\varphi = id_P$, then
  $$
   (S/K, \leq) \cong \coprod_{[x] \in P \backslash S/K} \operatorname{Ind}_{P \cap K}^P (\ast) \cong \coprod_{l} P/P \cap K,
  $$
  where $l = |S|/|P\cdot K|$.
 
  Therefore, $\rho$ will be fusion invariant if $l = l_{\varphi}$. It is enough to show $|\varphi(P)\cap K| = |P \cap K|$ and, in fact, this equality has been already proved in a paragraph above.
  
  Finally, $f|_{BS}$ is the induced map on classifying spaces of the homomorphism 
  $$
   \xymatrix@1{S \ar[r]^-{\pi} & S/K \ar[r]^-{reg} & \Sigma_{|S/K|} \ar[r] & \Sigma_{|S/K|} \wr \Sigma_{p^m}}
  $$ 
  whose kernel is $K$ by construction.
 \end{proof}
 
 \noindent \textbf{Question:} Given a saturated fusion system $(S,\mathcal{F})$ and a map $f \colon \mathcal{L}p \rightarrow Z$ where $Z$ is a connected $\Sigma B\mathbb Z/p$-null $p$-complete space. Does $f$ 
 factors, up to homotopy, through $\tilde{f} \colon B\mathcal{F}' \rightarrow Z$ with trivial kernel, where $\mathcal{F}'$ is a saturated fusion system?
 
 

 
 Under some hypothesis we can give a positive answer to the previous question.
 
 \begin{prop}\label{p:K normal}
  Let $(S,\mathcal{F})$ be a saturated fusion system with associated linking system $\mathcal{L}$. Let $K$ be the kernel of $f \colon \mathcal{L}p \rightarrow Z$, where $Z$ is $\Sigma B\mathbb Z/p$-null $p$-complete space. 
  
  If $K$ is normal in $\mathcal{F}$, then there exist a saturated fusion system $(S/K,\mathcal{F}/K)$ with associated linking system $\mathcal{L} /K$ and a map 
  $pr \colon |\mathcal{L}| \rightarrow |\mathcal{L} /K|$, whose homotopy fibre is $BK$, such that $f$ factors via $\tilde{f} \colon B(\mathcal{F}/K) \rightarrow Z$ with trivial kernel.
 \end{prop}
 
 \begin{proof} 
  In \cite[Section 2]{MR2317758}, the authors prove that if $K$ is normal in $\mathcal{F}$, then there is a saturated fusion system $(S/K, \mathcal{F}/K)$ with linking system $\mathcal{L}/K$ and a map $pr \colon |\mathcal{L}| \rightarrow |\mathcal{L}/K|$ whose homotopy fibre is $BK$. 
  
  Since $Z$ is $p$-complete, one can consider the composite $g \colon |\mathcal{L}| \rightarrow \mathcal{L}p \rightarrow Z$ such that $\p{g}{p} \simeq f$. By assumption $Z$ is $\sbz{p}$-null, and also $BK$ is $\bz{p}$-acyclic, then we get a unique factorization, up to homotopy, $\tilde{g} \colon |\mathcal{L}/K| \rightarrow Z$ of $g$ by applying Zabrodsky lemma (see 
  \cite[Proposition 3.4]{MR1397728}) to the homotopy fibre sequence $BK\rightarrow |\mathcal{L}|\rightarrow |\mathcal{L}/K|$ and the map $g$. Take $\tilde{f} = \p{\tilde{g}}{p}$. 
  The same argument applied to the fibration $BK\rightarrow BS\stackrel{\pi}{\rightarrow} B(S/K)$ and $g|_{BS}$ shows that $\tilde{f}|_{B(S/K)} \circ B\pi \simeq f|_{BS}$, by uniqueness.
  
   Let $[x] \in \ker(\tilde{f}) \leq S/K$, then $\tilde{f}|_{B\langle [x] \rangle} \simeq \ast$ implies that $f|_{B\langle x \rangle} \simeq \ast$ and
  finally $x \in K$.
 \end{proof}
 
 \begin{coro}\label{c:normal resistant}
  Let $(S,\mathcal{F})$ be a saturated fusion system with associated linking system $\mathcal{L}$. Let $K$ be the kernel of $f \colon \mathcal{L}p \rightarrow Z$, where $Z$ is a connected $\Sigma B\mathbb Z/p$-null $p$-complete space. 
  
  If $K$ is abelian or $S$ is resistant, then there exist a saturated fusion system $(S/K,\mathcal{F}/K)$ with associated linking system $\mathcal{L}/K$ and a map 
  $pr \colon |\mathcal{L}| \rightarrow |\mathcal{L} /K|$ whose homotopy fibre is $BK$ and a homotopy factorization $\tilde{f} \colon \mathcal{L}p/K \rightarrow Z$ with trivial kernel.
 \end{coro}
 
 \begin{proof}
  Recall that $K$ is strongly $\mathcal{F}$-closed by Proposition \ref{p:kerf strongly}. If $K$ is abelian and strongly $\mathcal{F}$-closed, then it is normal in $\mathcal{F}$ by 
  \cite[Proposition 3.14]{MR2602388}. If $S$ is resistant, each strongly $\mathcal{F}$-closed subgroup is also normal in $\mathcal{F}$. Finally, apply Proposition \ref{p:K normal}.
 \end{proof}

\section{Homotopy properties of $BP$-cellular approximations}\label{s:cw BP}

 In this context, it is natural to study homotopy properties of $CW_{BP} X$ which are inherited by those of $X$. We will show in this section that $CW_{BP}(\mathcal{L}p)$ is a $p$-good nilpotent space whose fundamental group is a finite $p$-group. 
 
 \begin{prop}\label{p:CW Lin nilp}
  Let $(S,\mathcal{F})$ be a saturated fusion system. Then both $\mathcal{L}p$ and $CW_{BP}(\mathcal{L}p)$ are nilpotent. Furthermore, there exists a homotopy fibre sequence
  $$
   CW_{BP}(\mathcal{L}p) \rightarrow \p{CW_{BP}(\mathcal{L}p)}{p} \rightarrow (\p{CW_{BP}(\mathcal{L}p)}{p})_{\mathbb{Q}}.
  $$
 \end{prop}
 
 \begin{proof}
The classifying space $\mathcal{L}p$ is $p$-complete and $\pi_i(\mathcal{L}p)$ are finite $p$-groups by Lemma \ref{l:hmtpy gps}, hence $\mathcal{L}p$ is nilpotent according to 
  \cite[Proposition VII.4.3(ii)]{MR0365573}. Tthen $CW_{BP}(\mathcal{L}p)$ is also nilpotent  \cite[Corollary 3.2]{CASTFLOR}. 
  
  Moreover, it follows from \cite[Lemma 2.8]{CASTFLOR} that 
  $R_{\infty} CW_{BP}(\mathcal{L}p) \simeq \ast$ for $R = \mathbb{Q}$ and $R = \mathbb{F}_q$, $q \neq p$, since $\tilde{H}_{\ast}(BP; R) =0$. Finally, the Sullivan's arithmetic square applied 
  to the nilpotent space $CW_{BP}(\mathcal{L}p)$ gives the desired result.
 \end{proof}
 
 The $R$-completion functor is a homological localization functor when we restrict to $R$-good spaces (see \cite[p. 205]{MR0365573}). By \cite[Proposition VII.5.1]{MR0365573}, if the 
 fundamental group of a pointed space $X$ is finite, then $X$ is $p$-good for any prime $p$. Therefore, in order to show that $CW_{BP}(\mathcal{L}p)$ is a $p$-good space, we will prove that its fundamental group is a finite $p$-group. 
 
 \begin{prop}\label{p:pi1finite}
  Let $(S,\mathcal{F})$ be a saturated fusion system and let $P$ be a finite $p$-group. Then $\pi_1 CW_{BP}(\mathcal{L}p)$ is a finite $p$-group. Therefore, $CW_{BP}(\mathcal{L}p)$ is a $p$-good space.
 \end{prop}
 
 \begin{proof}
 The proof will be divided in several steps.  Let  $C$ be the homotopy cofibre of the evaluation map $ev \colon \bigvee_{[BP,\mathcal{L}p]_{\ast}} BP \rightarrow \mathcal{L}p$.
 
 

 
 \noindent \textbf{Step 1:} Assume  $C$ is simply connected.  The Chacholski's homotopy fibre sequence $CW_{BP}(\mathcal{L}p) \rightarrow \mathcal{L}p \rightarrow P_{\Sigma BP} C$ induces a long  exact sequence of homotopy groups
  $$
   \ldots \rightarrow \pi_2 (P_{\Sigma BP} C) \rightarrow \pi_1 CW_{BP}(\mathcal{L}p) \rightarrow \pi_1 (\mathcal{L}p) \rightarrow \ldots 
  $$
  where $\pi_1 (\mathcal{L}p)$ is a finite $p$-group by Lemma \ref{l:hmtpy gps}. Therefore, we are reduced to prove that $\pi_2 (P_{\Sigma BP} C)$ is a finite $p$-group.
 
  Hurewicz's theorem shows that $H_2 (P_{\Sigma BP} C; \z{}) \cong \pi_2(P_{\Sigma BP} C)$, since $P_{\Sigma BP} C$ is simply connected (\cite[2.9]{MR1257059}). Moreover, since $\Sigma BP$ is $1$-connected,
  we obtain an epimorphism $H_2 (C ; \z{}) \twoheadrightarrow H_2(P_{\Sigma BP} C; \z{})$ by \cite[Proposition 3.2]{CellInfEsp}. Then it is enough to prove that $H_2 (C ; \z{})$ is a finite
  $p$-group.
  
The cofibration sequence  
  $
   \bigvee_{[BP, \mathcal{L}p]_{\ast}} BP \rightarrow \mathcal{L}p \rightarrow C
  $ 
  induces a long exact sequence of homology groups 
  $$
   \xymatrix@1@C=1.25em{\ldots \ar[r] & H_2(\mathcal{L}p; \z{}) \ar[r]^-{f_1} & H_2(C; \z{}) \ar[r]^-{f_2} & 
               H_1 (\bigvee_{[BP, \mathcal{L}p]_{\ast}} BP; \z{}) \ar[r] & \ldots }
  $$
  which allows to describe $H_2 (C;\z{})$ by a short exact sequence
  $$
   0 \rightarrow \ker f_2 \rightarrow H_2 (C;\z{}) \rightarrow \operatorname{Im} f_2 \rightarrow 0.
  $$
  By exactness, $\ker f_2=\operatorname{Im} f_1$ is a quotient of $H_2(\mathcal{L}p; \z{})$, where $H_2(\mathcal{L}p; \z{})$ is a finite $p$-group since $H^{\ast}(\mathcal{L}p;\z{}) \hookrightarrow H^{\ast}(BS;\z{})$ by \cite[Theorem B]{MR1992826}.
  Hence $\ker f_2$ is a finite $p$-group. 
  
 Now note that 
  $$
   \operatorname{Im} f_2 \subset H_1 (\bigvee_{[BP, \mathcal{L}p]_{\ast}} BP; \z{}) \cong \pi_1(\bigvee_{[BP, \mathcal{L}p]_{\ast}} BP)_{ab} \cong \oplus_{[BP, \mathcal{L}p]_{\ast}} P_{ab},
  $$ 
  where $[BP,\mathcal{L}p]_{\ast}$ is a finite set because there is an epimorphism of sets $[BP,BS]_{\ast} \twoheadrightarrow [BP,\mathcal{L}p]_{\ast}$ by \cite[Theorem 4.4]{MR1992826}, and 
  $[BP, BS]_{\ast} \cong \operatorname{Hom}(P, S)$ is finite. Finally, $H_2 (C;\z{})$ is a finite $p$-group. 

 

 
 \noindent \textbf{Step 2:} We will show that  there exists a saturated fusion system $(S', \mathcal{F}')$ and a $BP$-equivalence $f \colon B\mathcal{F}' \rightarrow \mathcal{L}p$ such that the homotopy cofibre of $ev \colon \bigvee_{[BP,B\mathcal{F}']_{\ast}} BP \rightarrow B\mathcal{F}'$ is $1$-connected.
  
  Let $N$ be the normal subgroup of $\pi_1 \mathcal{L}p$ generated by the image of homomorphisms 
  $P \rightarrow \pi_1 \mathcal{L}p$ such that the induced pointed map $BP \rightarrow B\pi_1 \mathcal{L}p$ lifts to $\mathcal{L}p$. Let $X$ be the pullback of the universal cover 
  $
   \widetilde{\mathcal{L}p} \rightarrow \mathcal{L}p \rightarrow B \pi_1 \mathcal{L}p
  $
  along $BN \rightarrow B \pi_1 \mathcal{L}p$. Then we have a homotopy commutative diagram
  $$
   \xymatrix@C=1em@R=1em{\widetilde{\mathcal{L}p} \ar@{=}[r] \ar[d] & \widetilde{\mathcal{L}p} \ar[d] & \\
                                      X    \ar[r]^-f  \ar[d] &      \mathcal{L}p \ar[r] \ar[r] \ar[d] & B(\pi_1 \mathcal{L}p /N) \ar@{=}[d] \\
                                      BN   \ar[r]            &    B\pi_1 \mathcal{L}p   \ar[r]        & B(\pi_1 \mathcal{L}p /N)
                         }
  $$
   where the vertical arrows are homotopy fibrations and the horizontal arrows are principal homotopy fibrations. By Theorem \ref{t:coverings}, there is a satured fusion system $(S', \mathcal{F}')$ such that $S' \leq S$, $\mathcal{F}' \leq \mathcal{F}$ and $X \simeq \mathcal{L}p'$. Furthermore, $f \colon \mathcal{L}p'\simeq X \rightarrow \mathcal{L}p$ is
  a $BP$-equivalence by Proposition \ref{p:CCS corregit}.
  
  Let $C$ be the homotopy cofibre of $ev \colon \bigvee_{[BP, \mathcal{L}p']_{\ast}} BP \rightarrow \mathcal{L}p'$. By Seifert-Van Kampen's theorem, $C$ will be $1$-connected if $\pi_1(ev)\colon \ast_{[BP, \mathcal{L}p']_{\ast}} P\rightarrow \pi_1(\mathcal{L}p')\cong N$ is an epimorphism.
  
  Since $N$ is generated by the image of homomorphisms $f\colon P \rightarrow \pi_1 \mathcal{L}p$ such that the induced map $Bf\colon BP \rightarrow B\pi_1 \mathcal{L}p$ lifts to $\tilde{f}\colon BP\rightarrow \mathcal{L}p$, it is enough to show that for each of those $f$, $\operatorname{Im}(f)\leq \operatorname{Im}(\pi_1(ev))$. 
  
  By definition, there is a factorization $f\colon P \rightarrow N\leq \pi_1 \mathcal{L}p$ which lifts to $\mathcal{L}p$. There exists a unique $\beta \colon BP\rightarrow \mathcal{L}p'$ (up to homotopy) such that the diagram
  $$
   \xymatrix@C=1.5em@R=1.5em{ BP \ar@/_1em/[ddr]_-{f} \ar@/^1em/[drr]^-{\tilde{f}} \ar@{-->}[dr]_-{\beta} &                  &              \\
                                                                                                             & \mathcal{L}p'  \ar[d] \ar[r] & \mathcal{L}p \ar[d] \\
                                                                                                             & BN        \ar[r] & B \pi_1 \mathcal{L}p
                     }
  $$
 is homotopy commutative. Therefore $\operatorname{Im}(\pi_1(\beta))=\operatorname{Im}(f)\leq \operatorname{Im}(\pi_1(ev))$.

 Now we are ready to complete the proof.  By Step 2 there exists a saturated fusion system $(S', \mathcal{F}')$ and a $BP$-equivalence $f \colon B\mathcal{F}' \rightarrow \mathcal{L}p$ such that the homotopy cofibre of 
  $ev \colon \bigvee_{[BP,B\mathcal{F}']_{\ast}} BP \rightarrow B\mathcal{F}'$ is $1$-connected. Hence $CW_{BP}(\mathcal{L}p) \simeq CW_{BP}(B\mathcal{F}')$ and $\pi_1 CW_{BP}(\mathcal{L}p) \cong \pi_1 CW_{BP}(\mathcal{L}p')$ is a
  finite $p$-group by Step 1.

 \end{proof}
 
 For the proof of  Proposition \ref{p:pi1finite} we need the next version of a result of 
 N. Castellana, J.A. Crespo and J. Scherer.
 
 \begin{prop}[{\cite[Proposition 2.1]{MR2262843}}]\label{p:CCS corregit}
  Let $P$ be a finite $p$-group and let $\xymatrix@1@C=1em{F \ar[r] & E \ar[r]^-{\pi} & BG}$ be a fibration, where $G$ is a discrete group. Let $N$ be the normal 
  subgroup of $G$ generated by the image of homomorphisms $P \rightarrow G$ such that the induced pointed map $BP \rightarrow BG$ lifts to $E$. Then the pullback of the fibration along 
  $BN \rightarrow BG$
  $$
   \xymatrix@C=1em@R=1em{E' \ar[r]^-f \ar[d] & E \ar[d]^-{\pi} \ar[r]^-{pr}	 & B(G/N) \ar@{=}[d] \\
                         BN \ar[r]           & BG \ar[r]^-{pr'}           & B(G/N)  
                         }
  $$
  induces a $BP$-equivalence $f \colon E' \rightarrow E$ on the total space level.
 \end{prop}
 
 \begin{proof}
  We want to show that $f$ induces a $BP$-equivalence. The top fibration in the diagram yields a fibration
  $$
   \xymatrix@1@C=1.5em{\mbox{map}p(BP,E') \ar[r]^-{f_{\ast}} & \mbox{map}p(BP,E)_{\{c\}} \ar[r]^-{pr_{\ast}} & \mbox{map}p(BP,B(G/N))_c,}.
  $$
  Since the base space is homotopically discrete, we only need to check that all components of $\mbox{map}p(BP,E)$ 
  are sent by $pr_{\ast}$ to $\mbox{map}p(BP,B(G/N))_c$. Thus consider a pointed map $h \colon BP \rightarrow E$. The composite $pr \circ h$ is homotopy
  equivalent to a map induced by a group homomorphism $\alpha = \pi_1(pr \circ h) \colon P \rightarrow G$ whose image is in $N$ by construction. Therefore
  $pr \circ h \simeq pr' \circ \pi \circ h$ is null-homotopic.
 \end{proof}

 \begin{remark}
  In the original version the authors consider $P = \bz{p^m}$ for $m > 1$ and $\bar{N}$ to be the (normal) subgroup generated by all elements $g \in G$ of order $p^i$ for some 
  $i \leq r$ such that the inclusion $f_g \colon B\langle g \rangle \rightarrow BG$ lifts to $E$, but this subgroup does not have the desired properties. Consider the fibration 
  $\xymatrix@1@C=1em{\bz{2} \ar[r]^-{\iota} & \bz{4} \ar[r]^-{pr} & \bz{2}}$, this fibration has no section and hence $\bar{N} = \{ 0 \}$. Then $E' \simeq \bz{2}$ and 
  $CW_{\bz{4}}(\bz{2}) \simeq \bz{2} \nsimeq \bz{4} = CW_{\bz{4}}(\bz{4})$, contradicting the proposition. However, $N \cong \z{/2}\cong \langle g \rangle$, 
  since $f_{g} = pr$. Hence $E' \simeq \bz{4}$ and $f: E' \rightarrow \bz{4}$ is an equivalence, in particular it is a $\bz{4}$-equivalence.
 \end{remark}
 

\section{Cellular properties of the classifying space of a saturated fusion system}\label{s:cw BF}

 
 The goal of this section is to prove the main theorem of the paper. Given a finite $p$-group $P$, this result characterizes the property of being $BP$-cellular for classifying spaces of saturated fusion systems in terms of the fusion data.

 \begin{theorem}\label{t:BP << Lp}
  Let $(S,\mathcal{F})$ be a saturated fusion system and let $P$ be a finite $p$-group. Then $\mathcal{L}p$ is $BP$-cellular if and only if $S = Cl_{\mathcal{F}}(P)$.
 \end{theorem} 

 \begin{coro}\label{c:BS << Lp}
  Let $(S,\mathcal{F})$ be a saturated fusion system and let $P$ be a finite $p$-group.
  \begin{enumerate}[(a)]
   \item The classifying space $\mathcal{L}p$ is $BS$-cellular.
   \item Let $(S,\mathcal{F}')$ be a saturated fusion system with $\mathcal{F}\subset \mathcal{F}'$. If $B\mathcal{F}$ is $BP$-cellular then $B\mathcal{F}'$ is also $BP$-cellular.
   \item Let $A$ be a pointed connected space. If $Cl_\mathcal{F}((\pi_1 A)_{ab})=S$, then $\mathcal{L}p$ is $A$-cellular.
   \item Let $\Omega_{p^m}(S)$ be the (normal) subgroup of $S$ generated by its elements of order $p^i$, with $i \leq m$. Then $\mathcal{L}p$ is $\bz{p^m}$-cellular if and only if 
         $S = Cl_{\mathcal{F}}(\Omega_{p^m}(S))$. In particular, there is a non-negative integer $m_0 \geq 0$ such that $\mathcal{L}p$ is $\bz{p^m}$-cellular for all $m \geq m_0$.
  \end{enumerate}
 \end{coro}
 
 \begin{proof}
  \begin{enumerate}[(a)]
   \item Direct from Theorem~\ref{t:BP << Lp} since $Cl_{\mathcal{F}}(S) = S$.
   \item It follows from the the inclusions $Cl_\mathcal{F}(P)\leq Cl_{\mathcal{F}'}(P)\leq S$. 
   \item Notice that $SP^\infty A \simeq \prod_{i \geq 1} K(H_i(A;\z{}),i)$ is $A$-cellular by \cite[Corollary 4.A.2.1]{MR1392221}, so $B(\pi_1 A)_{ab} \simeq K(H_1(A;\z{}),1)$ is 
         also $A$-cellular from \cite[2.D]{MR1392221}. Then, $B\mathcal{F}$ is $B(\pi_1 A)_{ab}$-cellular using (a).
   \item We have $Cl_\mathcal{F}(\Omega_{p^m}(S))\cong Cl_\mathcal{F}(\mathbb Z/p^m)$. Note that $B\Omega_{p^m}(S)$ is $B\mathbb Z/p^m$-cellular and that there exists an $m_0 \geq 0$ such that $S$ is generated by elements of order a power of $p$ less than 
         $p^{m_0}$.
  \end{enumerate}
 \end{proof}

 The strategy of proof for Theorem  \ref{t:BP << Lp} goes by analyzing  the fibre sequence given in \cite[Theorem 20.5]{MR1408539}
 $$
  \xymatrix@1@C=1.25em{CW_{BP}(\mathcal{L}p) \ar[r]^-c & \mathcal{L}p \ar[r]^-r & P_{\Sigma BP} C},
 $$
 where $C$ is the homotopy cofibre of the evaluation map $ev \colon \bigvee_{[BP, \mathcal{L}p]_{\ast}} BP \rightarrow \mathcal{L}p$ and $r$ is the composite 
 $\mathcal{L}p \rightarrow C \rightarrow P_{\Sigma BP} C$.  
 
 The first goal is to compute the kernel of $r^\wedge_p$. 
 In order to apply the theory of kernels, developed in Section \ref{s:ker f}, the target of the map needs to be a connected $p$-complete $\sbz{p}$-null space. 
 
 Since $\pi_1 \mathcal{L}p$ is a finite $p$-group, the same holds for $P_{\Sigma BP} C$ \cite[2.9]{MR1257059}, therefore Bousfield-Kan $p$-completion of the previous homotopy fibration is a homotopy fibre sequence (\cite[II.5.1]{MR0365573})
 $$
  \xymatrix@1@C=1.25em{CW_{BP}(\mathcal{L}p)^\wedge_p \ar[r]^-{c^\wedge_p} & \mathcal{L}p \ar[r]^-{r^\wedge_p} & \p{(P_{\Sigma BP}C)}{p}}.
 $$

\begin{lemma}\label{l:Sigma null}
If $X$ is a $1$-connected space and $P$ is a finite $p$-group, then $\p{(P_{\Sigma BP}X)}{p}$ is $\Sigma B\mathbb Z/p$-null.
\end{lemma}

\begin{proof} We have the following weak homotopy equivalences
$$\mbox{map}_*(\Sigma B\mathbb Z/p, \p{(P_{\Sigma BP}X)}{p})\simeq \mbox{map}_*( B\mathbb Z/p, \Omega \p{(P_{\Sigma BP}X)}{p})\simeq \mbox{map}_*( B\mathbb Z/p, \p{(\Omega P_{\Sigma BP}X)}{p})$$
where the last equivalence holds by \cite[V.4.6 (ii)]{MR0365573} since $X$ is $1$-connected (and so is $P_{\Sigma BP}X$ by \cite[2.9]{MR1257059}). 

The commutation rules between nullification functors and loops in \cite[3.A.1]{MR1392221} show that $\Omega P_{\Sigma BP}X\simeq P_{BP}(\Omega X)$. Finally,
$\mbox{map}_*( B\mathbb Z/p, \p{(P_{BP}\Omega X)}{p})\simeq \mbox{map}_*( B\mathbb Z/p, (P_{BP}\Omega X))\simeq *$ where the first equivalence follows from Miller's theorem \cite[Thm 1.5]{MR750716} and the second from the fact that $BP$ is $B\mathbb Z/p$-acyclic (\cite[Lemma 6.13]{MR1397728}).
\end{proof}

Next we describe a criteria for detecting when a map from an $A$-cellular space is null-homotopic which will be useful later.

 \begin{prop}\label{p:map(CW X, Z) = *}
  Let $X$ and $Y$ be pointed connected spaces. Assume that $X$ is $A$-cellular and $Y$ is $\Sigma A$-null. Then a pointed map $f \colon X \rightarrow Y$ is null-homotopic 
  if and only if for any map $g \colon A \rightarrow X$ the composite $f \circ g$ is null-homotopic.
 \end{prop}

 \begin{proof}
  If $f$ is null-homotopic then for any map $g \colon A \rightarrow X$ the composite $f \circ g$ is null-homotopic.  A
  
  Assume that for any
  map $g \colon A \rightarrow X$, the composite $f \circ g$ is null-homotopic. Let $C$ be the homotopy cofibre of 
  $ev \colon \bigvee_{[A,X]_{\ast}} A \rightarrow X$. Since $f \circ ev \simeq \ast$ by assumption, there is a map 
  $\tilde{f} \colon C \rightarrow Y$ such that the following diagram is homotopy commutative
  $$
   \xymatrix{\bigvee_{[A,X]_{\ast}} A \ar[d]_-{ev} \ar[dr]^-{\simeq \ast} & \\
                                    X \ar[d]       \ar[r]^-{f}            & Y. \\
                                    C \ar@{-->}[ur]^-{\tilde{f}}          &
             }
  $$
Then $f \simeq \ast$ if and only if $\tilde{f} \simeq \ast$. 

Since $X$ is $A$-cellular, $P_{\Sigma A} C \simeq \ast$. Finally $\tilde{f} \simeq \ast$ because 
  $
   \mbox{map}p(C,Y) \simeq \mbox{map}p(P_{\Sigma A} C, Y) \simeq \mbox{map}p(\ast, Y) \simeq \ast,
  $ where the first equivalence follows from the fact that $Y$ is $\Sigma A$-null.
 \end{proof}
  
 A key step in the proof of Theorem \ref{t:BP << Lp} is the following computation of the kernel of the map $r^\wedge_p$.

 \begin{prop}\label{p:Kerf = Cl}
  Let $(S,\mathcal{F})$ be a saturated fusion system. Then $\ker(\p{r}{p}) = Cl_{\mathcal{F}}(P)$.
 \end{prop}
 
 \begin{proof}
 We will show that $\ker(\p{r}{p}) \leq Cl_{\mathcal{F}}(P)$ and that $f(P)\leq \ker(\p{r}{p})$ for all $f\colon P\rightarrow S$, then since $\ker(\p{r}{p})$ is strongly $\mathcal{F}$-closed, the definition of $Cl_{\mathcal{F}}(P)$ implies the equality.
 
 By universal properties of cellularization and $p$-completion, any map $ BP\rightarrow B\mathcal{F}$ lifts to $\p{CW_{BP}(\mathcal{L}p)}{p}$, then $f(P)\leq \ker(\p{r}{p})$ for any homomorphism $f\colon P\rightarrow S$.
 
 
  According to Proposition \ref{p:exist m and f}, there exist $m \geq 0$ and a pointed map between classifying spaces $k \colon \mathcal{L}p \rightarrow \p{B(\Sigma_{|S/K|} \wr \Sigma_{p^m})}{p}$ such that 
  $\ker(k) = Cl_{\mathcal{F}}(P)$. Let $\iota\colon B\ker(\p{r}{p})\rightarrow B\mathcal F$ be the composite $B\ker(\p{r}{p})\rightarrow BS \rightarrow B\mathcal{F}$. It is enough to show that $k\circ \iota$ is nullhomotopic.
  
  There is a lift $\tilde{\iota}\colon B\ker(\p{r}{p})\rightarrow CW_{BP}(\mathcal{L}p)$ such that following diagram is homotopy commutative
  $$
   \xymatrix@C=1em{  & \p{CW_{BP}(\mathcal{L}p)}{p} \ar[d]^-c \ar[dr]^-{k\circ c} & \\
   B\ker(\p{r}{p}) \ar[dr]_-{\ast \backsimeq } \ar[r]^-{\iota} \ar@{-->}[ur]^-{\tilde{\iota}} & \mathcal{L}p \ar[d]^-{\p{r}{p}} \ar[r]^-k & \p{B(\Sigma_{|S/K|} \wr \Sigma_{p^m})}{p} \\
                     & \p{(P_{\Sigma BP} C)}{p}. &
             }
  $$
  We will show that $k\circ c$ is nullhomotopic by applying Proposition \ref{p:map(CW X, Z) = *}. Recall that any map $f\colon BP\rightarrow \mathcal{L}p$ is homotopic to $\Theta \circ B\rho\colon BP\rightarrow BS\rightarrow \mathcal{L}p$ where $\rho \in \operatorname{Hom}(P,S)$ (see \cite[Theorem 4.4]{MR1992826}). Then, the map $k$ has the property that for any $f \colon BP \rightarrow \mathcal{L}p$, the composite $k \circ f$ is nullhomotopic. 
  
  
 In particular, if $c \colon CW_{BP}(\mathcal{L}p) \rightarrow \mathcal{L}p$ is the augmentation, for any $f\colon BP\rightarrow CW_{BP}(\mathcal{L}p)$ we also have $(k \circ c)\circ f \simeq \ast$. Therefore $k \circ c \simeq \ast$ by Proposition \ref{p:map(CW X, Z) = *} 
  (since  $\p{B(\Sigma_{|S/K|} \wr \Sigma_{p^m})}{p}$ is $\sbz{p}$-null and hence $\Sigma BP$-null by Remark \ref{r:BZp null hence BP}). 
 \end{proof}
 
 An immediate consequence of Proposition \ref{p:Kerf = Cl} is one implication in Theorem \ref{t:BP << Lp}, that is, if $\mathcal{L}p$ is $BP$-cellular then $S=Cl_{\mathcal{F}}(P)$. In order to prove the prove the other implication, we will need a couple of technical results. 
 
 \begin{lemma}\label{l:s=Cl}
  Let $(S,\mathcal{F})$ be a saturated fusion system. If $S = Cl_{\mathcal{F}}(P)$, then homotopy cofibre $C$ of the evaluation $ev \colon \bigvee_{[BP,\mathcal{L}p]_{\ast}} BP \rightarrow \mathcal{L}p$ is $1$-connected.
 \end{lemma}
 
 \begin{proof}
 From the cofibration sequence and Seifert-Van Kampen theorem, we have that  $\pi_1 C \cong \pi_1 \mathcal{L}p/N$, where $N$ is the minimal normal subgroup of $\pi_1 \mathcal{L}p$ containing $\operatorname{Im}(\pi_1(ev))$. Given $f\colon BP\rightarrow B\mathcal{F}$, there is group homomorphism $g\colon P\rightarrow S$ such that $f\simeq \Theta\circ Bg$ (see \cite[Theorem 4.4]{MR1992826}). Let $\bar{N}$ be normal subgroup of $S$ generated by all $g(P)$, where $g\in \operatorname{Hom}(P,S)$. Then the fundamental group can be described $\pi_1(C)\cong S/\bar{N}\mathcal{O(F)}p$.
 
 First, $\bar{N}\mathcal{O(F)}p$ is strongly $\mathcal{F}$-closed by \cite[Proposition A.9]{MR2835340}. Morever it contains  $g(P)$ for all $g\in \operatorname{Hom}(P,S)$. Therefore we have inclusions $Cl_{\mathcal{F}}(P)\leq \bar{N}\mathcal{O(F)}p \leq S$. Since we are assuming $S = Cl_{\mathcal{F}}(P)$, the previous inclusions are all equalities and $C$ is then $1$-connected.
 \end{proof}
 
 \begin{prop}\label{p:pcom-null}
  Let $Z$ be a $\Sigma^i BP$-null space where $P$ is a finite $p$-group and $i \geq 0$. If $\pi_1 Z$ is a finite $p$-group then $\p{Z}{p}$ is also $\Sigma^i BP$-null.
 \end{prop}
 
 \begin{proof}
  Since $\pi_1 Z$ is a finite $p$-group, there is a homotopy commutative diagram of fibre sequences
  $$
   \xymatrix@C=1.5em@R=1.5em{ F     \ar[r] \ar@{=}[d] & Z\langle 1 \rangle \ar[r] \ar[d] & \p{Z}{p}\langle 1 \rangle \ar[d] \\
              F     \ar[r] \ar[d]     & Z  \ar[r] \ar[d]                 & \p{Z}{p} \ar[d] \\
            \ast    \ar[r]            & B\pi_1(Z) \ar[r]^-{\simeq}       & B\pi_1(\p{Z}{p})      
             }
  $$
 From the proof of Proposition 3.1 in \cite{MR928826}, it is easy to see that $[\Sigma^i BP,\p{Z}{p}]_{\ast} \cong [\Sigma^i BP,Z]_{\ast} \cong \ast$. Then it is enough to check that 
  $\mbox{map}p(\Sigma^i BP, \Omega \p{Z}{p}) \simeq \ast$. Consider the homotopy fibre sequence $\Omega\p{Z}{p} \rightarrow F \rightarrow Z$. Since $Z$ is $\Sigma^i BP$-null, 
  $\mbox{map}p(\Sigma^i BP, \Omega \p{Z}{p})\simeq \mbox{map}p(\Sigma^i BP, F)$. This last mapping space is homotopy equivalent to $\mbox{map}p(\Sigma^i BP, \p{F}{p})$ by Miller's theorem \cite[Thm 1.5]{MR750716} since $F$ is nilpotent. Finally $\p{F}{p} \simeq \ast$.
 \end{proof}
 
 We are now ready to prove the main theorem of this paper.
 
 \begin{proof}[Proof of Theorem \ref{t:BP << Lp}]
  First assume that $\mathcal{L}p$ is $BP$-cellular. Then $P_{\Sigma BP} C$ is contractible and $r \simeq *$. This implies that $\ker(r^\wedge_p) = S$, but also $\ker(\p{r}{p}) = Cl_{\mathcal{F}}(P)$ 
  by Proposition \ref{p:Kerf = Cl}. Therefore $S = Cl_{\mathcal{F}}(P)$.
 
  Now assume that $S = Cl_{\mathcal{F}}(P)$. Since $C$ is $1$-connected by Lemma \ref{l:s=Cl} (and therefore $P_{\Sigma BP}C$ is so by \cite[2.9]{MR1257059}), consider the $p$-completed homotopy fibre sequence (\cite[II.5.1]{MR0365573})
  $$
   \xymatrix@1@C=1.25em{CW_{BP}(\mathcal{L}p)^\wedge_p \ar[r]^-{c^\wedge_p} & \mathcal{L}p \ar[r]^-{r^\wedge_p} & \p{(P_{\Sigma BP}C)}{p}}.
  $$
 
 We will show that $\p{(P_{\Sigma BP}C)}{p}$ is weakly contractible.
 For this we apply first the theory of kernels developed in Section \ref{s:ker f} to $r^\wedge_p$ (see Lemma \ref{l:Sigma null} and Lemma \ref{l:s=Cl}).
  By assumption and Proposition \ref{p:Kerf = Cl}, $\ker(r^\wedge_p)=S$ and therefore $r^\wedge_p$ is null-homotopic by Theorem \ref{t:Dwyer p local}. Then there is a splitting 
  $\p{CW_{BP}(\mathcal{L}p)}{p} \simeq \mathcal{L}p \times \Omega \p{(P_{\Sigma BP} C)}{p}$. 
  
 Let $X=\p{CW_{BP}(\mathcal{L}p)}{p}$ and $Y=\Omega \p{(P_{\Sigma BP}C)}{p}$ for simplicity. We will show that $\p{P_{BP}(X)}{p}\simeq *$ and that $\p{P_{BP}(Y)}{p}\simeq Y$. Since both nullification and $p$-completion functors commute with products, the previous splitting $X\simeq B\mathcal{F}\times Y$ shows that $Y\simeq *$. Since $\p{(P_{\Sigma BP}C)}{p}$ is $1$-connected, then it follows that it is contractible.
  
 First, we have that $P_{BP}(X)^\wedge_p\simeq P_{BP}(CW_{BP}(\mathcal{L}p))^\wedge_p\simeq \ast$ by checking the hypothesis 
  of \cite[Lemma 3.9]{CASTFLOR}. One only needs to check that $P_{BP}(X)$ is $p$-good and that $P_{BP}(X)^\wedge_p$ is $BP$-null. Notice that $\pi_1 X$ is a finite $p$-group since $C$ is $1$-connected by Lemma \ref{l:s=Cl}. Therefore the space $\p{P_{BP}({X})}{p}$ is $BP$-null by 
  Proposition~\ref{p:pcom-null} and the space $P_{BP}(X)$ is $p$-good by Proposition \ref{p:pi1finite}.  Finally, recall that $BP$-cellular spaces are $BP$-acyclic (\cite[3.B.1]{MR1392221}).
  
  Now, $$\p{P_{BP}(Y)}{p}\simeq \p{P_{BP}(\Omega \p{(P_{\Sigma BP}C)}{p})}{p}\simeq \p{(\Omega P_{\Sigma BP}(\p{(P_{\Sigma BP}C)}{p}))}{p}\simeq \p{(\Omega \p{(P_{\Sigma BP}C)}{p})}{p}\simeq \Omega \p{(P_{\Sigma BP}C)}{p}=Y$$ where the second equivalence follows from commutation rules \cite[3.A.1]{MR1392221}, the third holds because $\p{(P_{\Sigma BP}C)}{p}$ is $\Sigma BP$-null by Millers's theorem \cite[Thm 1.5]{MR750716}, and the forth is a commutation of taking loops and $p$-completion \cite[V.4.6 (ii)]{MR0365573}.
  
  
  
Summarizing, we have proved that  
  $c \colon CW_{BP}(\mathcal{L}p) \rightarrow \mathcal{L}p$ is a mod $p$ equivalence.  Finally, using Proposition \ref{p:CW Lin nilp} we get that $c$ is a weak equivalence since $\p{CW_{BP}(\mathcal{L}p)}{p}\simeq \mathcal{L}p$ and $\mathcal{L}p_\mathbb Q\simeq *$.
 \end{proof}

\section{Examples}\label{s:examples}

 Let $G$ be a finite group. The situation when $G$ is generated by elements of order $p$ is well studied by R. Flores and R. Foote in \cite{MR2823972}. We start by giving a simple example where $G$ is not generated by elements of order $p^i$.
 
 \begin{example}
  Let $G = \Sigma_3$, the permutation group of $3$ elements. $\Sigma_3$ is generated by transpositions, i.e, by elements of order $2$, but the Sylow $3$-subgroup of $\Sigma_3$ is $S=\z{/3}$. Therefore, $BS$ is $\bz{3^r}$-cellular for all $r \geq 1$ and hence $\p{(B \Sigma_3)}{3}$ is $\bz{3^r}$-cellular for all $r \geq 1$ by Corollary~\ref{c:BS << Lp}. 
  
  Notice that $B\Sigma_3$ is not $\bz{3^r}$-cellular for any $r \geq 1$: applying $\mbox{map}_*(\bz{3^r},-)$ to the homotopy fibre sequence 
  $
   \bz{3} \stackrel{Bi}{\rightarrow} B\Sigma_3 \rightarrow \bz{2}
  $, we see that $Bi$ is a $\bz{3^r}$-equivalence for any $r$, since $\mbox{map}_*(\bz{3^r},\bz 2)\simeq *$. Then  $CW_{\bz{3^r}}(B\Sigma_3) \simeq CW_{\bz{3^r}}(\bz{3}) \simeq \bz{3}$.
 \end{example}
 

  
 We are now interested in the study of $CW_{BP}(\mathcal{L}p)$ when $S \neq Cl_{\mathcal{F}}(P)$. Our aim is to get tools to identify $\p{P_{BP}(C)}{p}$ in the $p$-completed Chacholsky's fibration describing $CW_{BP}(B\mathcal{F})$.
  
 \begin{prop}\label{p:factor}
  Let $(S,\mathcal{F})$ be a saturated fusion system, $\mathcal{L}$ be the associated centric linking system and let $P$ be a finite $p$-group. Assume that there is a saturated fusion system $(S', \mathcal{F}')$,
  with associated linking system $\mathcal{L}'$, and a factorization (up to homotopy)
  $$
   \xymatrix@C=1.5em@R=1.5em{ |\mathcal L | \ar[r]^\eta \ar[d]_\pi & \mathcal{L}p  \ar@{=}[r] \ar[d]_-{\p{\pi}{p}} & \mathcal{L}p  \ar[d]^-{\p{r}{p}} \\
                              |\mathcal L' | \ar[r]^\eta & \mathcal{L}p' \ar[r]^-{\tilde{r}}      & \p{(P_{\Sigma BP} C)}{p} 
                             }
  $$
  such that
   \begin{enumerate}[(i)]
    \item the map $\tilde{r}$ is injective, i.e., $\ker(\tilde{r}) = \{e\}$,
    \item and or $F$, the homotopy fibre of $\pi \colon |\mathcal{L}| \rightarrow |\mathcal{L}'|$, or $F_p$, the homotopy fibre of $\p{\pi}{p}$, is $\bz{p}$-acyclic. 
   \end{enumerate}
  Then $\tilde{r}$ is a homotopy equivalence. In particular, $CW_{BP}(\mathcal{L}p) \simeq F_p$.
 \end{prop}
  
 \begin{proof}
 We will proceed as follows: first we will construct a map $\p{\tilde{\pi}}{p} \colon \p{(P_{\Sigma BP} C)}{p} \rightarrow \mathcal{L}p'$ under $\mathcal{L}p$, and then we 
  will prove that $\p{\tilde{\pi}}{p}$ is a homotopy inverse of $\p{\tilde{r}}{p}$. This will give us that $\p{CW_{BP}(\mathcal{L}p)}{p} \simeq F_p$. Finally we will show, in this 
  situation, that $CW_{BP}(\mathcal{L}p)$ is $p$-complete and hence $CW_{BP}(\mathcal{L}p) \simeq F_p$.
  
  \begin{enumerate}[(a)]
   \item {\em Construction of $\p{\tilde{\pi}}{p} \colon \p{(P_{\Sigma BP} C)}{p} \rightarrow \mathcal{L}p'$.} In order to apply Zabrodsky lemma \cite[Proposition 3.4]{MR1397728} to the following situation
         $$
          \xymatrix@C=1.5em@R=1.5em{\p{CW_{BP}(\mathcal{L}p)}{p} \ar[d]^-{\p{c}{p}}             &        \\
                                           \mathcal{L}p \ar[r]^-{\p{\pi}{p}} \ar[d]^-{\p{r}{p}} & \mathcal{L}p' \\
                                           \p{(P_{\Sigma BP} C)}{p}                      &
                                   }
         $$
         we need to check that two assumptions are satisifed:
         \begin{itemize}
          \item $\p{\pi}{p} \circ \p{c}{p} \simeq \ast$: or equivalently, $\p{\pi}{p} \circ c \colon CW_{BP}(\mathcal{L}p) \rightarrow \mathcal{L}p'$ is null-homotopic since $\mathcal{L}p'$ is 
                $p$-complete and $CW_{BP}(\mathcal{L}p)$ is $p$-good.
                
                According to Proposition \ref{p:map(CW X, Z) = *} it is sufficient to show that the composite $\p{\pi}{p} \circ c \circ f$ is null-homotopic for all 
                $f \in \mbox{map}p(BP,CW_{BP}(\mathcal{L}p))$. But if we postcompose with $\tilde{r}$,  
                we have $\tilde{r} \circ \p{\pi}{p} \circ c \circ f \simeq \p{r}{p} \circ c \circ f \simeq \ast$. There exists $\rho\in \operatorname{Hom}(P,S')$ such that $\p{\pi}{p} \circ c \circ f\simeq \Theta' \circ B\rho$, then $\rho(P)\leq ker(\tilde{r}) = \{e\}.$
          \item $\mbox{map}_*(\p{CW_{BP}(\mathcal{L}p)}{p},\Omega \mathcal{L}p')\simeq *$: since $\mathcal{L}p'$ and $CW_{BP}(\mathcal{L}p)$ are $p$-good space we have 
                $
                 \mbox{map}p(\p{CW_{BP}(\mathcal{L}p)}{p},\mathcal{L}p') \simeq \mbox{map}p(CW_{BP}(\mathcal{L}p), \mathcal{L}p')
                $ 
                and hence, taking loops, 
                $$
                 \mbox{map}p(\p{CW_{BP}(\mathcal{L}p)}{p}, \Omega \mathcal{L}p') \simeq \mbox{map}p(CW_{BP}(\mathcal{L}p), \Omega \mathcal{L}p').
                $$ 
               The last mapping space is contractible since $\Omega \mathcal{L}p'$ is $BP$-null (p:mapping space) and $CW_{BP}(\mathcal{L}p)$ is $BP$-acyclic (\cite[3.B.1]{MR1392221}).
         \end{itemize}
          
   \item {\em $\tilde{r}$ and $\p{\tilde{\pi}}{p}$ are homotopy inverse.} 
   First note that both $Id_{\p{(P_{\Sigma BP} C)}{p}}$ and $\tilde{r}\circ \tilde{\pi}$ factor $\p{r}{p}$ since $Id_{\p{(P_{\Sigma BP} C)}{p}}\circ \p{r}{p}\simeq \tilde{r}\circ \pi\simeq  \tilde{r}\circ \tilde{\pi}\circ \p{r}{p}$. In order to show  that $\tilde{r} \circ \p{\tilde{\pi}}{p} \simeq Id_{\p{(P_{\Sigma BP} C)}{p}}$, we will apply Zabrodsky lemma to the fibration $\p{CW_{BP}(\mathcal{L}p)}{p} \rightarrow \mathcal{L}p \rightarrow \p{(P_{\Sigma BP} C)}{p}$ and the composite map
          $\p{\tilde{r}}{p} \circ \p{\pi}{p} \colon \mathcal{L}p \rightarrow \p{(P_{\Sigma BP} C)}{p}$. Then uniqueness up to homotopy of the factorization will give the desired equivalence.
          
   We check that $\mbox{map}_*(\p{CW_{BP}(\mathcal{L}p)}{p}, \Omega \p{(P_{\Sigma BP} C)}{p})\simeq *$ and $\p{\tilde{r}}{p} \circ \p{\tilde{\pi}}{p} \circ \p{c}{p} \simeq \ast$. First, by Proposition  \ref{p:pcom-null},  $\p{(P_{\Sigma BP} C)}{p}$ is $\Sigma BP$-null, then $\Omega \p{(P_{\Sigma BP} C)}{p}$ is $BP$-null (\cite[3.A.1]{MR1392221}). Now, since $\p{CW_{BP}(\mathcal{L}p)}{p}$ and $\Omega \p{(P_{\Sigma BP} C)}{p}$ are $p$-good and $CW_{BP}(\mathcal{L}p)$ is $BP$-acyclic, we have 
   $$\mbox{map}_*(\p{CW_{BP}(\mathcal{L}p)}{p}, \Omega \p{(P_{\Sigma BP} C)}{p})\simeq \mbox{map}_*(CW_{BP}(\mathcal{L}p), \Omega \p{(P_{\Sigma BP} C)}{p})\simeq*$$
   Finally, $\p{\tilde{r}}{p} \circ \p{\tilde{\pi}}{p} \circ \p{c}{p} \simeq \p{r}{p}\circ \p{c}{p} \simeq \ast$. 
 
 It remains to prove that $\p{\tilde{\pi}}{p} \circ \p{\tilde{r}}{p} \simeq Id_{\mathcal{L}p'}$. First note that $Id_{\mathcal{L}p'}\circ\pi\simeq \tilde{\pi}\circ \p{r}{p}\simeq \tilde{\pi}\circ \tilde{r}\circ \pi$, then both $Id_{\mathcal{L}p'}$ and $\tilde{\pi}\circ \tilde{r}$ factor $\pi$. In order to show  that $Id_{\mathcal{L}p'}\simeq\tilde{\pi}\circ \tilde{r}$, we will apply again Zabrodsky's lemma to the fibration $F_p\rightarrow B\mathcal{F}\rightarrow B\mathcal{F}'$ (resp. $F\rightarrow |\mathcal L|\rightarrow |\mathcal L'|$, depending on whether $F$ or $F_p$ is $\bz p$-acyclic) and the composite $\tilde{\pi}\circ \p{r}{p}$ (resp. $\tilde{\pi}\circ \p{r}{p}\circ \eta$).  
 
 If $i\colon F_p\rightarrow \mathcal{L}p$, then $\p{r}{p}\circ i\simeq \tilde{r}\circ \pi \circ i \simeq *$. Then since $\Omega\mathcal{L}p'$ is $\bz p$-null and $F_p$ is $\bz p$-acyclic (resp. $F$ is $\bz p$-acyclic), $\mbox{map}_*(F_p,\Omega\mathcal{L}p') \simeq *$ (resp. $\mbox{map}_*(F,\Omega\mathcal{L}p')\simeq *$).

          
   \item {\em $CW_{BP}(\mathcal{L}p)$ is $p$-complete.} By Proposition \ref{p:CW Lin nilp} we obtain the homotopy fibration 
          $$
           CW_{BP}(\mathcal{L}p) \rightarrow F_p \rightarrow (F_p)_{\mathbb{Q}},
          $$ 
          where $(F_p)_{\mathbb{Q}} \simeq \ast$, since $\pi_i (F_p)$ are finite groups for $i\geq 1$.
  \end{enumerate}
 \end{proof}
 
 \begin{remark}
 Proposition \ref{p:factor} also holds without assuming the existence of $\pi$ and only considering the factoritzation on $p$-completed classifying spaces if we know that the homotopy fiber $F_p$ is $\bz p$-acyclic.
 \end{remark}
 
 \begin{coro}
  Let $(S,\mathcal{F})$ be a saturated fusion system and let $P$ be a finite $p$-group. If $\mathcal{O(F)}p \vartriangleleft Cl_{\mathcal{F}}(P)$, then $CW_{BP}(\mathcal{L}p) \simeq \mathcal{L}p'$, where $\mathcal{L}p'$ is the 
  connected cover of $\mathcal{L}p$ with $\pi_1 \mathcal{L}p' \cong Cl_{\mathcal{F}}(P)/\mathcal{O(F)}p$. 

 \end{coro}
 
 \begin{proof}
  The connected cover $\mathcal{L}p'$ is the classifying space of a saturated fusion system over $Cl_{\mathcal{F}}(P)$ by \cite[Theorem A]{MR2302515}. So $\mathcal{L}p'$ is $BP$-cellular by
  Corollary~\ref{c:BS << Lp}. We get the following commutative diagram of homotopy fibrations
  $$
   \xymatrix@C=1.5em@R=1.5em{   B\tilde{\mathcal{F}} \ar[d] \ar@{=}[r] & B\tilde{\mathcal{F}} \ar[d]        &                            \\
                                   \mathcal{L}p'   \ar[d] \ar[r]^-f  &   \mathcal{L}p     \ar[d] \ar[r] & B(S/Cl_{\mathcal{F}}(P)) \ar@{=}[d] \\
                              B(Cl_{\mathcal{F}}(P)/\mathcal{O(F)}p)   \ar[r]     &  B(S/\mathcal{O(F)}p)         \ar[r] & B(S/Cl_{\mathcal{F}}(P))
                             }
  $$
  
  In order to apply Proposition~\ref{p:factor}, we have to find a factorization of $\p{r}{p} \colon \mathcal{L}p \rightarrow \p{(P_{\Sigma BP} C)}{p}$ by a map 
  $\tilde{r} \colon B(S/Cl_{\mathcal{F}}(P)) \rightarrow \p{(P_{\Sigma BP} C)}{p}$ with trivial kernel, where we can assume that $C$ is $1$-connected by 
  step $2$ in the proof of Proposition~\ref{p:pi1finite}. 
  
  By Lemma \ref{l:Sigma null} and Remark \ref{r:BZp null hence BP}, $\Omega \p{(P_{\Sigma BP} C)}{p}$ is $BP$-null and $B\mathcal{F}'$ is  $BP$-cellular space (hence $BP$ acyclic), then $\mbox{map}_*(\mathcal{L}p,\Omega \p{(P_{\Sigma BP} C)}{p})\simeq *$. Moreover, 
  $\p{r}{p} \circ f \simeq \ast$, because $\p{r}{p} \circ f|_{BCl_{\mathcal{F}}(P)} \simeq \ast$ since $\ker(\p{r}{p}) = Cl_{\mathcal{F}}(P)$. We are in the conditions of applying Zabrodsky lemma (see 
  \cite[Proposition 3.4]{MR1397728}) which shows that there is a map 
  $\tilde{r} \colon B(S/Cl_{\mathcal{F}}(P)) \rightarrow \p{(P_{\Sigma BP} C)}{p}$ such that the diagram 
  $$
   \xymatrix@R=1.5em@C=1.5em{
             \mathcal{L}p' \ar[d]^-f                           &                          \\
             \mathcal{L}p  \ar[d]   \ar@{=}[r]                 & \mathcal{L}p \ar[d]^-{\p{r}{p}} \\
             B(S/Cl_{\mathcal{F}}(P)) \ar[r]^-{\tilde{r}} & \p{(P_{\Sigma BP} C)}{p}
                             }
  $$
  commutes up to homotopy. 
  
  Let $[x] \in \ker(\tilde{r}) \leq S/Cl_{\mathcal{F}}(P)$, where $x \in S$. Since the above diagram commutes, 
  $\p{r}{p}|_{B\langle x \rangle} \simeq \tilde{r}|_{B\langle [x] \rangle} \simeq \ast$, then $x \in \ker(\p{r}{p}) = Cl_{\mathcal{F}}(P)$ and hence $[x] = e$. Finally, 
  Proposition~\ref{p:factor} gives us the equivalence $CW_{BP}(\mathcal{L}p) \simeq \mathcal{L}p'$.
 \end{proof}
 
 \begin{coro}\label{c:normal}
  Let $(S,\mathcal{F})$ be a fusion system and let $P$ be a finite $p$-group. If $Cl_{\mathcal{F}}(P) \vartriangleleft \mathcal{F}$, then $CW_{BP} (\mathcal{L}p)$ is homotopy equivalent to the homotopy
  fibre of $\mathcal{L}p \rightarrow B(\mathcal{F}/Cl_{\mathcal{F}}(P))$.
 \end{coro}
 
 \begin{proof}
  Let $K = Cl_{\mathcal{F}}(P)$. Since $K$ is normal in $\mathcal{F}$, there is a saturated fusion system $(S/K, \mathcal{F}/K)$, and a map defined between the nerve of the associated linking systems 
  $\pi \colon |\mathcal{L}| \rightarrow |\mathcal{L}/K|$ whose homotopy fibre is $BK$, and an injective factorization of $\p{r}{p}$ by $\mathcal{L}p/K$ from Proposition \ref{p:K normal}. Then the result follows from Proposition~\ref{p:factor}.
 \end{proof}

 \begin{example}\label{e:NG(S) controls fus}
  Let $G$ be a finite group and let $S$ be a $p$-Sylow subgroup of $G$. Assume that $N_G(S)$ controls fusion in $G$. Then $\p{BN_G(S)}{p} \simeq \p{BG}{p}$ and $S$ is normal in 
  $\mathcal{F}_S(N_G(S))$. On account of Corollary \ref{c:normal}, for all finite $p$-group $P$, $CW_{BP} (\p{BG}{p})$ is equivalent to the homotopy fibre of 
  $\p{BN_G(S)}{p} \rightarrow \p{B(N_G(S)/Cl_{\mathcal{F}_S(N_G(S))}(P))}{p}$.
   
  An example  is given by $G = \z{/p^n} \wr \z{/q} = (\z{/p^n})^q \rtimes \z{/q}$, when $p \neq q$ and $n \geq 1$. The Sylow 
  $p$-subgroup of $G$ is $S = (\z{/p^n})^q$ and $Cl_{\mathcal{F}_S(G)}(\z{/p^r}) = (\z{/p^r})^q$ is abelian and hence normal in $\mathcal{F}_S(G)$ by Corollary \ref{c:normal resistant}. Therefore 
  $\p{BG}{p}$ is $\bz{p^r}$-cellular if and only if $r \geq n$ by Theorem \ref{t:BP << Lp}. Then $CW_{\bz{p^r}}(\p{BG}{p})$ is equivalent to the homotopy fibre of 
  $\p{BG}{p} \rightarrow \p{B(G/(\z{/p^r})^q)}{p}$ by Corollary \ref{c:normal}.
 
  Other explicit examples appear in \cite[Example 5.2]{MR2351607}. The authors proved that the normalizer of the Sylow of the Suzuki group
  $\operatorname{Sz}(2^n)$, with $n$ an odd integer at least $3$, is $N_{\operatorname{Sz}(2^n)}(S) = S \rtimes \z{/(2^n-1)}$ and it controls fusion in $\operatorname{Sz}(2^n)$. In this case, $S$ is $\bz{2^m}$-cellular for
  all $m \geq 2$ and hence $\p{B \! \operatorname{Sz}(2^n)}{2}$ is so. Moreover, $Cl_{\mathcal{F}_S(\operatorname{Sz}(2^n))}(\z{/2}) \cong (\z{/2})^n$ and hence $CW_{\bz{2}}(\p{B \! \operatorname{Sz}(2^n)}{2})$ is equivalent to the 
  homotoy fibre of $\p{B N_{\operatorname{Sz}(2^n)}(S)}{2} \rightarrow \p{B(N_{\operatorname{Sz}(2^n)}(S)/(\z{/2})^n)}{2}$.
 \end{example}
 

\section{The cellularization of the classifying spaces of a family of exotic fusion systems at $p=3$}\label{s:exotic example} 

 
 Let $S=B(3,r;0,\gamma,0)$, for $r \geq 4$ and $\gamma = 0,1,2$, be the family of finite $3$-groups of order $3^r$ (see  \cite[Theorem A.2, Proposition A.9]{MR2272147}) generated by $\{s,s_1, \ldots, s_{r-1}\}$ where
  \begin{itemize}
   \item $s_i = [s_{i-1},s]$ for all $i \in \{2, \ldots, r-1\}$,
   \item $[s_1, s_i] = 1$ for all $i \in \{2, \ldots, r-1\}$,
   \item $s_1^3 s_2^3 s_3 = s_{r-1}^\gamma$,
   \item $s_i^3 s_{i+1}^3 s_{i+2} = 1$ for all $i \in \{2, \ldots, r-1\}$,
   \item $s^3=1$.
  \end{itemize}

 The center of $S$ is $\langle s_{r-1}\rangle$. The normal subgroup $\gamma_1=\langle s_1,\ldots,s_{r-1} \rangle$ of $S$ is of index $3$ and the corresponding group extension is split. There are group isomorphisms
 $$
  B(3,r;0,\gamma,0) = \langle s_1,s_2 \rangle \rtimes \langle s \rangle = \left\{ \begin{array}{ll}
                                                                                   (\z{/3^m} \times \z{/3^m})     \rtimes \z{/3} & \mbox{, if $r = 2m + 1$,} \\
                                                                                   (\z{/3^m} \times \z{/3^{m+1}}) \rtimes \z{/3} & \mbox{, if $r = 2m$.}
                                                                                  \end{array}
                                                                          \right.
 $$

 
In \cite[Theorem 5.10]{MR2272147}, the authors construct families of exotic $3$-local finite groups $\mathcal{F}$ whose Sylow $3$-subgroup is $S=B(3,r;0,\gamma,0)$. 
 
 \begin{prop}\label{c:exotic example}
  Let $\mathcal{F}$ be an exotic fusion system over $B(3,r;0,\gamma,0)$ such that $\mathcal{F}$ has at least one $\mathcal{F}$-Alperin rank two elementary abelian $3$-subgroup given in \cite[Theorem 5.10]{MR2272147}.
  Then 
  \begin{enumerate}[(i)]
   \item If $\gamma = 0$, then $\mathcal{L}p$ is $\bz{3^l}$-cellular for all $l \geq 1$.
   \item Assume $\gamma \neq 0$. Then $\mathcal{L}p$ is $\bz{3^l}$-cellular if and only if $l \geq 2$. If $l=1$, $Cl_{\mathcal{F}}(\z{/3})=\langle s,s_2\rangle$.
   \end{enumerate}
 \end{prop}
 
 \begin{proof}  By Theorem~\ref{t:BP << Lp} we are reduced to the computation of $Cl_{\mathcal{F}}(\z{/3^l})$. Let $N:=\langle s_2,s\rangle$ which is a proper normal subgroup of $S$. This subgroup $N$ contains the center of $S$, $Z(S)=\langle s_{r-1}\rangle\cong \mathbb Z/3$ by  \cite[Lemma A.10]{MR2272147}. Moreover, the description of $\mathcal{F}$ and the computation of automorphisms of $S$ in \cite[Theorem 5.10,Lemma A.14]{MR2272147} show that $N$ is strongly $\mathcal{F}$-closed.

First $s\in Cl_{\mathcal{F}}(\z{/3^l})$ for all $l\geq 1$. If $Cl_{\mathcal{F}}(\z{/3^l})$ is a proper normal subgroup of $S$, again by \cite[Lemma A.10]{MR2272147}, $Cl_{\mathcal{F}}(\z{/3^l})=N$ with $[N : S] = 3$, and 
  $$
   N \cong \left\{ \begin{array}{ll}
                    3^{1+2}_+      & \mbox{, if $r = 4$,} \\
                    B(3,r-1;0,0,0) & \mbox{, if $r > 4$.}
                   \end{array}
           \right.
  $$
  We have then inclusions $N\subset Cl_{\mathcal{F}}(\z{/3^l})\subset S$. And $Cl_{\mathcal{F}}(\z{/3^l})=S$ if and only if there exists $x\in S\setminus N$ such that $x^{3^l}=1$.
  
  Let $x = s^i s_1^j s_2^k \in S$, where $i = 0, 1, 2$. Since the index $[Cl_{\mathcal{F}}(\z{/3^l}) : S]\in \{1,3\}$, $x\in Cl_{\mathcal{F}}(\z{/3^l})$ if and only if $x^2\in Cl_{\mathcal{F}}(\z{/3^l})$. But if $x=s^2 s_1^j s_2^k$, then $x^2=s^4s_1^{j'} s_2^{k'}= s s_1^{j'} s_2^{k'}$, therefore we can assume $i=0,1$. Note that $s_i=[s_{i-1},s]\in \langle s, s_2\rangle \subset N$ for all $i \in \{3, \ldots, r-1\}$ and $s_1^3 = s_{r-1}^\gamma s_{3}^{-1}s_{2}^{-3}\in N$. If $j\equiv 0\mod 3$ then $x\in N$. 
  
 If $i=0$ then $x=s_1^j s_2^k$ and $[s_1,s_2]=1$. Then $o(x)=3$ if and only if $o(s_1^j)=3$ and $o(s_2^k)=3$. In particular, $3j | 3^m$, that is, $3|j$. But then $j\equiv 0\mod 3$ and $x\in N$.
 
  If $i=1$ and $x=ss_1^js_2^k$, one can compute $x^3=(s_{r-1})^{\gamma i}$ using relations $s_1^as=ss_1^as_2^a$, $s_2^b=ss_2^bs_3^b$, $s_3^cs=ss_3^cs_4^c$, $s_1^3s_2^3s_3=s_{r-1}^\gamma$ and $s_2^3s_3^3s_4=1$ with $a,b,c>0$ ( \cite[Proposition A.9]{MR2272147}).  We have that $x^{3^l}=(s_{r-1})^{\gamma i 3^{l-1}}$ and $x^{3^l}=1$ if and only if $\gamma i 3^{l-1} \equiv 0 \mod 3$.  That is, $x^9=1$ always, and $x^3=1$ if $\gamma=0$ or $i\equiv 0 \mod 3$, but in this last case $x\in N$. 
  
 
 Summarizing, if $\gamma =0$ then $x=ss_1\in S\setminus N$ is of order $3$ and then $Cl_{\mathcal{F}}(\z{/3^l})=S$ for all $l\geq 1$. If $\gamma\neq 0$ then $x=ss_1\in S\setminus N$ is of order $9$ and then $Cl_{\mathcal{F}}(\z{/3^l})=S$ for all $l\geq 2$. Finally if $\gamma \neq 0$, $Cl_{\mathcal{F}}(\z{/3})=N$.  
 \end{proof}
 
 We will finish this section by describing $CW_{B\mathbb Z/3}(B\mathcal F)$ when $\mathcal{F}$ is an exotic fusion system over $B(3,r;0,\gamma,0)$ with $\gamma \neq 0$ such that $\mathcal{F}$ has at least one $\mathcal{F}$-Alperin rank two elementary abelian $3$-subgroup given in \cite{MR2272147}. 
 
 \begin{lemma}\label{l:exotic1connected}
 Let $\mathcal{F}$ an exotic fusion system under the hypothesis of Proposition \ref{c:exotic example}, $\mathcal{L}p$ is $1$-connected.
 \end{lemma}
 
 \begin{proof}
 By Proposition \ref{l:hmtpy gps}, the fundamental group of $\mathcal{L}p$ is 
  $\pi_1(\mathcal{L}p) \cong S/\mathcal{O(F)}p$, where 
  $
   \mathcal{O(F)}p := \langle [Q,\mathcal{O}^p(\operatorname{Aut}_{\mathcal{F}}(Q))] \mid Q \leq S \rangle.
  $
The subgroup $\mathcal{O(F)}p$ is a strongly $\mathcal{F}$-closed subgroup of $S$, and the arguments in the proof in \cite[page 1751]{MR2272147} show that it must contain $N=\langle s,s_2\rangle < S$. We will show that $\mathcal{O(F)}p=S$ by proving that $s_1\in \mathcal{O(F)}p$. Checking tables in  \cite[Theorem 5.10,Lemma A.14]{MR2272147}, we see that the automorphisms of order two $\eta$ and/or $\omega$ are group elements in $\operatorname{Aut}_{\mathcal{F}}(S)$. By the description given there $\eta(s_1)=s_1s_2^{f''}$ and $\omega(s_1)=s_1^{-1}s_2^{f''}$. Then $s_1^{-1}\eta(s_1)=s_1^{-2}s_2^{f''}$ or $s_1^{-1}\omega(s_1)=s_1^{-2}s_2^{f''}$ are elements of $\mathcal{O(F)}p$, since $s_2,s_1^3\in N\subset \mathcal{O(F)}p$ then $s_1\in \mathcal{O(F)}p$. 
 \end{proof}
 
  \begin{prop}\label{l:exotic higherlimits}
 Let $\mathcal{F}$ be an exotic fusion system satisfying the hypothesis of Proposition \ref{c:exotic example} with $\gamma\neq 0$. Let $N=\langle s,s_2\rangle< S$, then there exists a unique map (up to homotopy) $f\colon B\mathcal{F}\rightarrow \p{(B\Sigma_3)}{3}$ whose kernel is $N$.
 \end{prop}
 
 \begin{proof}
 
The proof of Proposition \ref{p:exist m and f} shows that the quotient morphism $S\rightarrow S/N\cong \mathbb Z/3$ gives a fusion preserving homomorphism $\rho \colon S\rightarrow \Sigma_3$. We want to show that this morphism extends to a map $f:\mathcal{L}p\rightarrow (B\Sigma_3)^\wedge_3$.
 
 By Proposition \ref{p:L = hocolim}, $\mathcal{L}p \simeq \p{(\operatorname{hocolim}_{\mathcal{O(F)}c} \tilde{B}P)}{3}$, where $\tilde{B}P \simeq BP$ for $P \in \mathcal{F}^c$. The fusion preserving property of $\rho$ shows that $B\rho \in \lim_{\mathcal{O(F)}} [BP,\p{(B\Sigma_3)}{3}]$.  
  
  The obstructions for rigidifying the homotopy commutative diagram in the category of spaces lie in $\lim_{\mathcal{O(F)}c}^{i+1} \pi_{i}(\mbox{map}(BP, \p{(B\Sigma_3)}{3})_{\Theta'\circ B\rho|_P})$, for $i \geq 1$ (see \cite{MR928836}). Note that since the $3$-Sylow subgroup of $\Sigma_3$ is abelian,  we have $\pi_{1}(\mbox{map}(BP, \p{(B\Sigma_3)}{3})_{\Theta'\circ B\rho|_P})$ is abelian being a quocient of $C_{\mathbb Z/3}(\rho(P))$. In fact, it will be trivial or $\mathbb Z/3$ (Proposition \ref{p:mapping space}).
  
  We will show that for any $F\colon \mathcal{O(F)}\rightarrow \mathbb Z_{(p)}-Mod$, $\lim^{i}_{\mathcal{O(F)}}F=0$ for $i>1$.
 From \cite[Proposition 3.2, Corollary 3.3]{MR1992826}, we are reduced to show that derived limits of atomic functors have the same vanishing property. Note that from \cite[Theorem 5.10,Lemma A.14]{MR2272147},  the relevant outomorphism groups $\operatorname{Out}_\mathcal{F}(P)$ are $SL_2(\mathbb F_3)$ or  $GL_2(\mathbb F_3)$. In both cases the $3$-Sylow subgrup is of order $3$, and then \cite[Proposition 6.2(i)]{MR1154593} implies the result.
 
 The obstructions to uniqueness lie in  $\lim_{\mathcal{O(F)}c}^{i} \pi_{i}(\mbox{map}(BP, \p{(B\Sigma_3)}{3})_{\Theta'\circ B\rho|_P})$, for $i \geq 1$ (see \cite{MR928836}). By the previous paragrah we have to look at the first derived functor of atomic functors with value $\mathbb Z/3$. But since $\operatorname{Aut}(\mathbb Z/3)\cong \mathbb Z/2$, by \cite[Proposition 6.1(ii)]{MR1154593} any element of order $3$ will act trivially on $\mathbb Z/3$. Finally note that if a map $g\colon B\mathcal{F}\rightarrow \p{(B\Sigma_3)}{3}$ has kernel $N$, its restriction $g|_{BS}\colon BS\rightarrow \p{(B\Sigma_3)}{3}$ has to be homotopic to $B\rho$.
\end{proof}
 
 \begin{prop}
 Let $\mathcal{F}$ be an exotic fusion system over $B(3,r;0,\gamma,0)$ with $\gamma \neq 0$ such that $\mathcal{F}$ has at least one $\mathcal{F}$-Alperin rank two elementary abelian $3$-subgroup given in \cite[Theorem 5.10]{MR2272147}.  Then there exists a map $f\colon B\mathcal{F} \rightarrow (B\Sigma_3)^\wedge_p$ such that $CW_{B\mathbb Z/3}(B\mathcal F)$ is the homotopy fiber of $f$.
 \end{prop}

\begin{proof}
Let $f$ be the map constructed in Proposition \ref{l:exotic higherlimits} with $\ker(f)=Cl_\mathcal{F}(S)$. Precisely because of this, $f\circ ev\simeq *$ where $ev\colon \bigvee_{[\bz3,\mathcal{L}p]_*} B\mathbb Z/3\rightarrow B\mathcal{F}$. Then $f$ factors through the cofibre $C$ of $ev$ and, since $(B\Sigma_3)^\wedge_3$ is $B\mathbb Z/3$-null, we obtain a factorization of $f$, $f'\colon P_{\Sigma B\mathbb Z/3}(C)\rightarrow (B\Sigma_3)^\wedge_3$ such that the following diagram is homotopy commutative

$$
   \xymatrix@R=1.5em@C=1.5em{
             \mathcal{L}p  \ar[d]   \ar@{=}[r]                 & \mathcal{L}p \ar[d]^-{\p{r}{3}} \\
             P_{\bz3}(C) \ar[r]^-{f'} & \p{(B\Sigma_3)}{3}.
                             }
  $$
  
  The strategy if to construct a homotopy inverse of $f'$, $\bar{\Theta}\colon \p{(B\Sigma_3)}{3}\rightarrow \p{P_{\bz3}(C)}{3}$, up to $3$-completion, which fits in the previous diagram up to homotopy. 
  
  Since $\Sigma_3$ has an abelian normal $3$-Sylow subgroup $\mathbb Z/3$, we have that $(B\mathbb Z/3)_{h\mathbb Z/2}\rightarrow B\Sigma_3$ is an equivalence. Consider the fibre sequence $BN\rightarrow BS\rightarrow B\mathbb Z/3$ and the map $\p{r}{3}|_{BS}\colon BS\rightarrow \p{P_{\bz3}(C)}{3}$, by Zabrodsky's lemma 
  \cite[Proposition 3.4]{MR1397728}, $\p{r}{3}|_{BS}$ factors (uniquely up to homotopy) via $\Theta'\colon B\mathbb Z/3\rightarrow \p{P_{\bz3}(C)}{3}$. In order to get $\bar{\Theta}$, we only need check that $\Theta'$ is $\mathbb Z/2$-equivariant up to homotopy. For any $\mathcal{F}$ in the hypothesis of the proposition, note that there is an element in $\omega'\in \operatorname{Out}_\mathcal{F}(S)$ which project to $\omega\in \operatorname{Out}_{\Sigma_3}(\mathbb Z/3)$ (they are called $\eta$ or $\omega$ in the tables \cite[Theorem 5.10]{MR2272147}). Since $\Theta'$ is unique up to homotopy factoring $\p{r}{3}\circ \Theta$, and $\Theta\circ \omega'\simeq \Theta$, it follows $\omega\circ \Theta'\simeq \Theta'$.
  
 Next we check that $\bar{\Theta}$ is a homotopy inverse to $f$. First consider the following homotopy commutative diagram:
 
 $$
   \xymatrix@C=1.5em@R=1.5em{   CW_{B\mathbb Z/3}(B{\mathcal{F}}) \ar[d]  &         &                            \\
                                   \mathcal{L}p   \ar[d]^r \ar@{=}[r]  &   \mathcal{L}p     \ar[d] \ar@{=}[r] & B\mathcal{F} \ar[d]^{\p{r}{3}} \\
                              P_{\Sigma B\mathbb Z/3}(C)   \ar[r]^{f'}     &  \p{(B\Sigma_3)}{3}         \ar[r]^{\bar{\Theta}} & \p{P_{\Sigma B\mathbb Z/3}(C)}{3}.
                             }
  $$
 Since $3$-completion on the bottom line also gives a homotopy commutative diagram, unicity on Zabrodsky's lemma (see 
  \cite[Proposition 3.4]{MR1397728}) shows that $\bar{\Theta}\circ f'$ is $3$-completion. So $\bar{\Theta}\circ \p{(f')}{3}\simeq id$.
  
  
 Now consider the following homotopy commutative diagram 
 $$
   \xymatrix@C=1em@R=1em{BK \ar[r] \ar[d] & F \ar[d] & \\
                                      BS    \ar[r]^-\Theta  \ar[d] &      \mathcal{L}p \ar@{=}[r]  \ar[d] & \mathcal{L}p \ar[d]^f \\
                                      B\mathbb Z/3   \ar[r]^\iota            &    \p{(B\Sigma_3)}{3}   \ar[r]^{\p{(f')}{3}\circ \bar{\Theta}}        & \p{(B\Sigma_3)}{3}.
                         }
  $$
  The map $\p{(f')}{3}\circ \bar{\Theta}$ is determined by its restriction to the Sylow $3$-subgroup $\mathbb Z/3$ by Proposition \ref{l:exotic higherlimits}. Again $\iota\circ\p{(f')}{3}\circ \bar{\Theta}$ and $\iota$ give homotopy commutative diagrams when placed in the bottom line, then unicity on Zabrodsky's lemma (see 
  \cite[Proposition 3.4]{MR1397728}) shows that they are homotopic.
\end{proof} 
 
 
 

  

  

  

{\small

\begin{minipage}[t]{8 cm}
Nat\`{a}lia Castellana and Alberto Gavira-Romero\\ 
Departament de Matem\`atiques,\\
Universitat Aut\`onoma de Barcelona,\\ 
E-08193 Bellaterra, Spain\\
\textit{E-mail:}\texttt{\,natalia@mat.uab.es}, \\
\phantom{\textit{E-mail:}}\texttt{\,\,gavira@mat.uab.es}
\end{minipage}

}

\end{document}