We define and characterize a class of $p$-complete spaces $X$ which have many of the same properties as the $p$-completions of classifying spaces of finite groups. For example, each such $X$ has a Sylow subgroup $BS\longrightarrow X$, maps $BQ\longrightarrow X$ for a $p$-group$Q$ are described via homomorphisms $Q\longrightarrow S$, and $H^*(X;\mathbb{F}_p)$ is isomorphic to a certain ring of “stable elements” in $H^*(BS;\mathbb{F}_p)$. These spaces arise as the “classifying spaces” of certain algebraic objects which we call “$p$-local finite groups”. Such an object consists of a system of fusion data in $S$, as formalized by L. Puig, extended by some extra information carried in a category which allows rigidification of the fusion data.
The main goal of this paper is to identify and study a certain class of spaces which in many ways behave like $p$-completed classifying spaces of finite groups. These spaces occur as the “classifying spaces” of certain algebraic objects, which we call $p$-local finite groups. A $p$-local finite group consists, roughly speaking, of a finite $p$-group$S$ and fusion data on subgroups of $S$, encoded in a way explained below. Our starting point is our earlier paper ReferenceBLO on $p$-completed classifying spaces of finite groups, together with the axiomatic treatment by Lluís Puig ReferencePu, ReferencePu2 of systems of fusion among subgroups of a given $p$-group.
The $p$-completion of a space $X$ is a space $X{}_{p}^{\wedge }$ which isolates the properties of $X$ at the prime $p$, and more precisely the properties which determine its mod $p$ cohomology. For example, a map of spaces $X\xrightarrow {f}Y$ induces a homotopy equivalence $X{}_{p}^{\wedge }\xrightarrow {\simeq }Y{}_{p}^{\wedge }$ if and only if $f$ induces an isomorphism in mod $p$ cohomology; and $H^*(X{}_{p}^{\wedge };\mathbb{F}_p)\cong {}H^*(X;\mathbb{F}_p)$ in favorable cases (if $X$ is “$p$-good”). When $G$ is a finite group, the $p$-completion$BG{}_{p}^{\wedge }$ of its classifying space encodes many of the properties of $G$ at $p$. For example, not only the mod $p$ cohomology of $BG$, but also the Sylow $p$-subgroup of $G$ together with all fusion among its subgroups, are determined up to isomorphism by the homotopy type of $BG{}_{p}^{\wedge }$.
Our goal here is to give a direct link between $p$-local structures and homotopy types which arise from them. This theory tries to make explicit the essence of what it means to be the $p$-completed classifying space of a finite group, and at the same time yields new spaces which are not of this type, but which still enjoy most of the properties a space of the form $BG{}_{p}^{\wedge }$ would have. We hope that the ideas presented here will have further applications and generalizations in algebraic topology. But this theory also fits well with certain aspects of modular representation theory. In particular, it may give a way of constructing classifying spaces for blocks in the group ring of a finite group over an algebraically closed field of characteristic $p$.
A saturated fusion system$\mathcal {F}$ over a $p$-group$S$ consists of a set $\operatorname {Hom}\nolimits _{\mathcal {F}}(P,Q)$ of monomorphisms, for each pair of subgroups $P,Q\le {}S$, which form a category under composition, include all monomorphisms induced by conjugation in $S$, and satisfy certain other axioms formulated by Puig (Definitions 1.1 and 1.2 below). In particular, these axioms are satisfied by the conjugacy homomorphisms in a finite group. We refer to ReferencePu and ReferencePu2 for more details of Puig’s work on saturated fusion systems (which he calls “full Frobenius systems” in ReferencePu2). The definitions and results given here, in Section 1 and in Appendix A are only a very brief account of those results of Puig used in our paper.
If $\mathcal {F}$ is a saturated fusion system over $S$, then two subgroups $P,P'\le {}S$ are called $\mathcal {F}$-conjugate if $\operatorname {Iso}\nolimits _{\mathcal {F}}(P,P')\ne \emptyset$. A subgroup $P$ is called $\mathcal {F}$-centric if $C_S(P')\le {}P'$ for all $P'$ that are $\mathcal {F}$-conjugate to $P$; this is equivalent to what Puig calls “$\mathcal {F}$-selfcentralizing”. Let $\mathcal {F}^c$ be the full subcategory of $\mathcal {F}$ whose objects are the $\mathcal {F}$-centric subgroups of $S$. A centric linking system associated to $\mathcal {F}$ is a category $\mathcal {L}$ whose objects are the $\mathcal {F}$-centric subgroups of $S$, together with a functor $\mathcal {L}\xrightarrow {\pi }\mathcal {F}^c$ which is the identity on objects and surjective on morphisms, and which satisfies other axioms listed below in Definition 1.7. For example, for each object $P$, the kernel of the induced map $\operatorname {Aut}\nolimits _{\mathcal {L}}(P)\xrightarrow {}\operatorname {Aut}\nolimits _{\mathcal {F}}(P)$ is isomorphic to $Z(P)$, and $\operatorname {Aut}\nolimits _{\mathcal {L}}(P)$ contains a distinguished subgroup isomorphic to $P$.
The motivating examples for these definitions come from finite groups. If $G$ is a finite group and $p$ is a prime, then $\mathcal {F}=\mathcal {F}_S(G)$ is the fusion system over $S\in \operatorname {Syl}_p(G)$ such that for each $P,Q\le {}S$,$\operatorname {Hom}\nolimits _{\mathcal {F}}(P,Q)$ is the set of homomorphisms induced by conjugation in $G$ (and inclusion). The $\mathcal {F}$-centric subgroups of $S$ are the $p$-centric subgroups: those $P\le {}S$ such that $C_G(P)\cong {}Z(P)\times {}C'_G(P)$ for some $C'_G(P)$ of order prime to $p$ (see ReferenceBLO, Lemma A.5). In ReferenceBLO, we defined a category $\mathcal {L}_S^c(G)$ whose objects are the $p$-centric subgroups of $G$ which are contained in $S$, and where $\operatorname {Mor}\nolimits _{\mathcal {L}_S^c(G)}(P,Q)=N_G(P,Q)/C_G'(P)$. Here, $N_G(P,Q)$ is the set of elements of $G$ which conjugate $P$ into $Q$. The category $\mathcal {L}_S^c(G)$, together with its projection to $\mathcal {F}_S(G)$ which sends the morphism corresponding to an element $g\in {}N_G(P,Q)$ to conjugation by $g$, is the example which motivated our definition of an associated centric linking system.
We define a $p$-local finite group to be a triple $(S,\mathcal {F},\mathcal {L})$, where $\mathcal {L}$ is a centric linking system associated to a saturated fusion system $\mathcal {F}$ over a $p$-group$S$. The classifying space of such a triple is the space $|\mathcal {L}|{}_{p}^{\wedge }$, where for any small category $\mathcal {C}$, the space $|\mathcal {C}|$ denotes the geometric realization of the nerve of $\mathcal {C}$. This is partly motivated by the result that $|\mathcal {L}_p^c(G)|{}_{p}^{\wedge }\simeq {}BG{}_{p}^{\wedge }$ for any finite $G$ReferenceBLO, Proposition 1.1. But additional motivation comes from Proposition 2.2 below, which says that if $\mathcal {L}$ is a centric linking system associated to $\mathcal {F}$, then $|\mathcal {L}|\simeq \mathop {\mathop {\mathrm{hocolim}}\limits _{\longrightarrow }}_{\mathcal {O}^c(\mathcal {F})}(\widetilde{B})$, where $\mathcal {O}^c(\mathcal {F})$ is a certain quotient “orbit” category of $\mathcal {F}^c$, and $\widetilde{B}$ is a lifting of the homotopy functor which sends $P$ to $BP$. The classifying space of a $p$-local finite group thus comes equipped with a decomposition as the homotopy colimit of a finite diagram of classifying spaces of $p$-groups.
We now state our main results. Our first result is that a $p$-local finite group is determined up to isomorphism by its classifying space. What is meant by an isomorphism of $p$-local finite groups will be explained later.
A $p$-local finite group $(S,\mathcal {F},\mathcal {L})$ is determined by the homotopy type of $|\mathcal {L}|{}_{p}^{\wedge }$. In particular, if $(S,\mathcal {F},\mathcal {L})$ and $(S',\mathcal {F}',\mathcal {L}')$ are two $p$-local finite groups and $|\mathcal {L}|{}_{p}^{\wedge }\simeq |\mathcal {L}'|{}_{p}^{\wedge }$, then $(S,\mathcal {F},\mathcal {L})$ and $(S',\mathcal {F}',\mathcal {L}')$ are isomorphic.
Next we study the cohomology of $p$-local finite groups. As one might hope, we have the following result, which appears as Theorem 5.8.
Theorem B
For any $p$-local finite group $(S,\mathcal {F},\mathcal {L})$,$H^*(|\mathcal {L}|{}_{p}^{\wedge };\mathbb{F}_p)$ is isomorphic to the ring of “stable elements” in $H^*(BS;\mathbb{F}_p)$; i.e., the inverse limit of the rings $H^*(BP;\mathbb{F}_p)$ as a functor on the category $\mathcal {F}$. Furthermore, this ring is noetherian.
The next theorem gives an explicit description of the mapping space from the classifying space of a finite $p$-group into the classifying space of a $p$-local finite group. It is stated precisely as Corollary 4.5 and Theorem 6.3.
Theorem C
For any $p$-local finite group $(S,\mathcal {F},\mathcal {L})$, and any $p$-group$Q$,
Furthermore, each component of the mapping space has the homotopy type of the classifying space of a $p$-local finite group which can be thought of as the “centralizer” of the image of the corresponding homomorphism $Q\xrightarrow {}S$.
The next result describes the space of self equivalences of the classifying space of a $p$-local finite group. It is a generalization of ReferenceBLO, Theorem C. For a small category $\mathcal {C}$, let $\mathcal {A}ut(\mathcal {C})$ denote the groupoid whose objects are self equivalences of $\mathcal {C}$, and whose morphisms are natural isomorphisms of functors. Let $\mathcal {L}$ be a centric linking system associated to a saturated fusion system $\mathcal {F}$. Self equivalences of $\mathcal {L}$ which are structure preserving, in a sense to be made precise in section 7 below, are said to be isotypical. We let $\mathcal {A}ut_{\text{typ}}(\mathcal {L})$ denote the subgroupoid of $\mathcal {A}ut(\mathcal {L})$ whose objects are the isotypical self equivalences of $\mathcal {L}$. For a space $X$, let $\operatorname {Aut}\nolimits (X)$ denote the topological monoid of all self homotopy equivalences of $X$. The following theorem is restated below as Theorem 8.1.
Theorem D
Fix a $p$-local finite group $(S,\mathcal {F},\mathcal {L})$. Then $\operatorname {Aut}\nolimits (|\mathcal {L}|{}_{p}^{\wedge })$ and $|\mathcal {A}ut_{\text{typ}}(\mathcal {L})|$ are equivalent as topological monoids in the sense that their classifying spaces are homotopy equivalent. In particular, their groups of components are isomorphic, and each component of $\operatorname {Aut}\nolimits (|\mathcal {L}|{}_{p}^{\wedge })$ is aspherical.
The statement of Theorem 8.1 also includes a description of the homotopy groups of $\operatorname {Aut}\nolimits (|\mathcal {L}|{}_{p}^{\wedge })$.
So far, we have not mentioned the question of the existence and uniqueness of centric linking systems associated to a given saturated fusion system. Of course, as pointed out above, any finite group $G$ gives rise to an associated $p$-local finite group. However there are saturated fusion systems which do not occur as the fusion system of any finite group. Thus a tool for deciding existence and uniqueness would be useful. The general obstructions to the existence and uniqueness of associated centric linking systems, which lie in certain higher limits taken over the orbit category $\mathcal {O}^c(\mathcal {F})$ of the fusion system, are described in Proposition 3.1; and a means of computing these groups is provided by Proposition 3.2. The following result is just one special consequence of this, which settles the question for $p$-groups of small rank. Here, for any finite group $G$, we write $\operatorname {rk}\nolimits _p(G)$ for the largest rank of any elementary abelian $p$-subgroup of $G$.
Fix a saturated fusion system $\mathcal {F}$ over a $p$-group$S$. If $\operatorname {rk}\nolimits _p(S)<p^3$, then there exists a centric linking system associated to $\mathcal {F}$, and if $\operatorname {rk}\nolimits _p(S)<p^2$, then the associated centric linking system is unique.
In the last section, we present some direct constructions of saturated fusion systems and associated $p$-local finite groups (see Examples 9.3 and 9.4). The idea is to look at the fusion system over a $p$-group$S$ (for $p$ odd only) generated by groups of automorphisms of $S$ and certain of its subgroups, and show that under certain hypotheses the resulting system is saturated. In all of these cases, the $p$-group$S$ is nonabelian, and has an index $p$ subgroup which is abelian and homocyclic (a product of cyclic groups of the same order). We then give a list of all finite simple groups which have Sylow subgroups of this form, based on the classification theorem, and use that to show that certain of the fusion systems which were constructed are not the fusion systems of any finite groups. In all cases, Theorem E applies to show the existence and uniqueness of centric linking systems, and hence $p$-local finite groups, associated to these fusion systems.
The basic definitions of saturated fusion systems and their associated centric linking systems are given in Section 1. Homotopy decompositions of classifying spaces of $p$-local finite groups are constructed in Section 2. The obstruction theory for the existence and uniqueness of associated centric linking systems, as well as some results about those obstruction groups, are shown in Section 3. Maps from the classifying space of a $p$-group to the classifying space of a $p$-local finite group are studied in Sections 4 and 6, while the cohomology rings of classifying spaces of $p$-local finite groups are dealt with in Section 5. A characterization of classifying spaces of $p$-local finite groups is given in Section 7, and their spaces of self equivalences are described in Section 8. The “exotic” examples of $p$-local finite groups are constructed in Section 9. Finally, some additional results on saturated fusion systems are collected in an appendix.
We would like to thank Dave Benson and Jesper Grodal for their many suggestions throughout the course of this work. In particular, Dave had earlier written and distributed notes which contained some of the ideas of our centric linking systems. We would also like to thank Lluís Puig for giving us a copy of his unpublished notes on saturated fusion systems. Markus Linckelmann, Haynes Miller, Bill Dwyer, and Jon Alperin have all shown interest and made helpful comments and suggestions. Kasper Andersen and Kari Ragnarsson both read earlier versions of this paper in detail, and sent us many suggestions for improvements. Two of the authors would also like to thank Slain’s Castle, a pub in Aberdeen, for their hospitality on New Year’s Day while we worked out the proof that the nerve of a centric linking system is $p$-good.
We would especially like to thank the universities of Aberdeen and Paris-Nord, the CRM and the UAB in Barcelona, and the Max-Planck Institut in Bonn for their hospitality in helping the three authors get together in various combinations; and also the European Homotopy Theory Network for helping to finance these visits.
1. Fusion systems and associated centric linking systems
We begin with the precise definitions of saturated fusion systems and their associated centric linking systems. Additional results about fusion systems due to Puig ReferencePu, ReferencePu2 are in Appendix A.
Given two finite groups $P$,$Q$, let $\operatorname {Hom}\nolimits (P,Q)$ denote the set of group homomorphisms from $P$ to $Q$, and let $\operatorname {Inj}\nolimits (P,Q)$ denote the set of monomorphisms. If $P$ and $Q$ are subgroups of a larger group $G$, then $\operatorname {Hom}\nolimits _G(P,Q)\subseteq \operatorname {Inj}\nolimits (P,Q)$ denotes the subset of homomorphisms induced by conjugation by elements of $G$, and $\operatorname {Aut}\nolimits _G(P)$ the group of automorphisms induced by conjugation in $G$.
Definition 1.1
A fusion system$\mathcal {F}$ over a finite $p$-group$S$ is a category whose objects are the subgroups of $S$, and whose morphism sets $\operatorname {Hom}\nolimits _{\mathcal {F}}(P,Q)$ satisfy the following conditions:
(a)
$\operatorname {Hom}\nolimits _S(P,Q)\subseteq \operatorname {Hom}\nolimits _{\mathcal {F}}(P,Q)\subseteq \operatorname {Inj}\nolimits (P,Q)$ for all $P,Q\le S$.
(b)
Every morphism in $\mathcal {F}$ factors as an isomorphism in $\mathcal {F}$ followed by an inclusion.
Note that what we call a fusion system here is what Puig calls a divisible Frobenius system.
If $\mathcal {F}$ is a fusion system over $S$ and $P,Q\le {}S$, then we write $\operatorname {Hom}\nolimits _{\mathcal {F}}(P,Q)=\operatorname {Mor}\nolimits _{\mathcal {F}}(P,Q)$ to emphasize that morphisms in the category $\mathcal {F}$ are all homomorphisms, and $\operatorname {Iso}\nolimits _{\mathcal {F}}(P,Q)$ for the subset of isomorphisms in $\mathcal {F}$. Thus $\operatorname {Iso}\nolimits _{\mathcal {F}}(P,Q)=\operatorname {Hom}\nolimits _{\mathcal {F}}(P,Q)$ if $|P|=|Q|$, and $\operatorname {Iso}\nolimits _{\mathcal {F}}(P,Q)=\emptyset$ otherwise. Also, $\operatorname {Aut}\nolimits _{\mathcal {F}}(P)=\operatorname {Iso}\nolimits _{\mathcal {F}}(P,P)$ and $\operatorname {Out}\nolimits _{\mathcal {F}}(P)=\operatorname {Aut}\nolimits _{\mathcal {F}}(P)/\operatorname {Inn}\nolimits (P)$. Two subgroups $P,P'\le {}S$ are called $\mathcal {F}$-conjugate if $\operatorname {Iso}\nolimits _{\mathcal {F}}(P,P')\ne \emptyset$.
The fusion systems we consider here will all satisfy the following additional condition. Here, and throughout the rest of the paper, we write $\operatorname {Syl}_p(G)$ for the set of Sylow $p$-subgroups of $G$. Also, for any $P\le {}G$ and any $g\in {}N_G(P)$,$c_g\in \operatorname {Aut}\nolimits (P)$ denotes the automorphism $c_g(x)=gxg^{-1}$.
Definition 1.2
Let $\mathcal {F}$ be a fusion system over a $p$-group$S$.
•
A subgroup $P\le {}S$ is fully centralized in $\mathcal {F}$ if $|C_S(P)|\ge |C_S(P')|$ for all $P'\le {}S$ that are $\mathcal {F}$-conjugate to $P$.
•
A subgroup $P\le {}S$ is fully normalized in $\mathcal {F}$ if $|N_S(P)|\ge |N_S(P')|$ for all $P'\le {}S$ that are $\mathcal {F}$-conjugate to $P$.
•
$\mathcal {F}$ is a saturated fusion system if the following two conditions hold:
(I)
Any $P\le {}S$ which is fully normalized in $\mathcal {F}$ is fully centralized in $\mathcal {F}$, and $\operatorname {Aut}\nolimits _S(P)\in \operatorname {Syl}_p(\operatorname {Aut}\nolimits _{\mathcal {F}}(P))$.
(II)
If $P\le {}S$ and $\varphi \in \operatorname {Hom}\nolimits _{\mathcal {F}}(P,S)$ are such that $\varphi {}P$ is fully centralized, and if we set$$\begin{equation*} N_\varphi = \{ g\in {}N_S(P) \,|\, \varphi c_g\varphi ^{-1} \in \operatorname {Aut}\nolimits _S(\varphi {}P) \}, \end{equation*}$$
then there is $\bar{\varphi }\in \operatorname {Hom}\nolimits _{\mathcal {F}}(N_\varphi ,S)$ such that $\bar{\varphi }|_P=\varphi$.
The above definition is slightly different from the definition of a “full Frobenius system” as formulated by Lluís Puig ReferencePu2, §2.5, but is equivalent to his definition by the remarks after Proposition A.2. Condition (I) can be thought of as a “Sylow condition”. It says that $\operatorname {Out}\nolimits _{\mathcal {F}}(S)$ has order prime to $p$ (just as $N_G(S)/S$ has order prime to $p$ if $S\in \operatorname {Syl}_p(G)$); and more generally it reflects the fact that for any $p$-subgroup$P\le {}G$, there is some $S\in \operatorname {Syl}_p(G)$ such that $N_S(P)\in \operatorname {Syl}_p(N_G(P))$. Another way of interpreting this condition is that if $|N_S(P)|\ge |N_S(P')|$ for $P'$$\mathcal {F}$-conjugate to $P$, then $C_S(P)$ and $\operatorname {Aut}\nolimits _S(P)\cong {}N_S(P)/C_S(P)$ must also be maximal in the same sense. As for condition (II), it is natural to require that some extension property hold for morphisms in $\mathcal {F}$, and $N_\varphi$ is by definition the largest subgroup of $N_S(P)$ to which $\varphi$ could possibly extend.
The motivating example for this definition is the fusion system of a finite group $G$. For any $S\in \operatorname {Syl}_p(G)$, we let $\mathcal {F}_S(G)$ be the fusion system over $S$ defined by setting $\operatorname {Hom}\nolimits _{\mathcal {F}_S(G)}(P,Q)=\operatorname {Hom}\nolimits _G(P,Q)$ for all $P,Q\leq S$.
Proposition 1.3
Let $G$ be a finite group, and let $S$ be a Sylow $p$-subgroup of $G$. Then the fusion system $\mathcal {F}_S(G)$ over $S$ is saturated. Also, a subgroup $P\le {}S$ is fully centralized in $\mathcal {F}_S(G)$ if and only if $C_S(P)\in \operatorname {Syl}_p(C_G(P))$, while $P$ is fully normalized in $\mathcal {F}_S(G)$ if and only if $N_S(P)\in \operatorname {Syl}_p(N_G(P))$.
Proof.
Fix some $P\le {}S$, and choose $g\in {}G$ such that $g^{-1}Sg$ contains a Sylow $p$-subgroup of $N_G(P)$. Then $gPg^{-1}\le {}S$ and $N_{g^{-1}Sg}(P)\in \operatorname {Syl}_p(N_G(P))$, and so $N_S(gPg^{-1})\in \operatorname {Syl}_p(N_G(gPg^{-1}))$. This clearly implies that $|N_S(gPg^{-1})|\ge |N_S(P')|$ for all $P'\le {}S$ that are $G$-conjugate to $P$. Thus $gPg^{-1}$ is fully normalized in $\mathcal {F}_S(G)$, and $P$ is fully normalized in $\mathcal {F}_S(G)$ if and only if $|N_S(P)|=|N_S(gPg^{-1})|$, if and only if $N_S(P)\in \operatorname {Syl}_p(N_G(P))$. A similar argument proves that $P$ is fully centralized in $\mathcal {F}_S(G)$ if and only if $C_S(P)\in \operatorname {Syl}_p(C_G(P))$.
If $P$ is fully normalized in $\mathcal {F}_S(G)$, then since $N_S(P)\in \operatorname {Syl}_p(N_G(P))$, the obvious counting argument shows that
In particular, $P$ is fully centralized in $\mathcal {F}_S(G)$, and this proves condition (I) in Definition 1.2.
To see condition (II), let $P\le {}S$ and $g\in {}G$ be such that $gPg^{-1}\le {}S$ and is fully centralized in $\mathcal {F}_S(G)$, and write $P'=gPg^{-1}$ for short. Set
In particular, $gN_Gg^{-1}=N'_G$ ($N$ and $N'$ are conjugate modulo centralizers), and thus $gNg^{-1}$ and $N'$ are two $p$-subgroups of $N'_G$. Furthermore,
is prime to $p$ (since $P'$ is fully centralized), so $N'\in \operatorname {Syl}_p(N'_G)$. Since $C_G(P')\vartriangleleft {}N'_G$ has $p$-power index, all Sylow $p$-subgroups of $N'_G$ are conjugate by elements of $C_G(P')$, and hence there is $h\in {}C_G(P')$ such that $h(gNg^{-1})h^{-1}\le {}N'$. Thus $c_{hg}\in \operatorname {Hom}\nolimits _{\mathcal {F}_S(G)}(N,S)$ extends $c_g\in \operatorname {Hom}\nolimits _{\mathcal {F}_S(G)}(P,S)$.
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Puig’s original motivation for defining fusion systems came from block theory. Let $k$ be an algebraically closed field of characteristic $p\ne 0$. A block in a group ring $k[G]$ is an indecomposable 2-sided ideal which is a direct summand. Puig showed ReferencePu that the Brauer pairs associated to a block $b$ (the “$b$-subpairs”), together with the inclusion and conjugacy relations defined by Alperin and Broué ReferenceAB, form a saturated fusion system over the defect group of $b$. See, for example, ReferenceAB or ReferenceAlp, Chapter IV, for definitions of defect groups and Brauer pairs of blocks.
In practice, when proving that certain fusion systems are saturated, it will be convenient to replace condition (I) by a modified version of the condition, as described in the following lemma.
Lemma 1.4
Let $\mathcal {F}$ be a fusion system over a $p$-group$S$ which satisfies condition (II) in Definition 1.2, and also satisfies the condition
(I$'$)
Each subgroup $P\le {}S$ is $\mathcal {F}$-conjugate to a fully centralized subgroup $P'\le {}S$ such that $\operatorname {Aut}\nolimits _S(P')\in \operatorname {Syl}_p(\operatorname {Aut}\nolimits _{\mathcal {F}}(P'))$.
Then $\mathcal {F}$ is a saturated fusion system.
Proof.
We must prove condition (I) in Definition 1.2. Assume that $P\le {}S$ is fully normalized in $\mathcal {F}$. By (I$'$), there is $P'\le {}S$ which is $\mathcal {F}$-conjugate to $P$, and such that $P'$ is fully centralized in $\mathcal {F}$ and $\operatorname {Aut}\nolimits _S(P')\in \operatorname {Syl}_p(\operatorname {Aut}\nolimits _{\mathcal {F}}(P'))$. In particular,
and hence the inequalities in (1) are equalities. Thus $P$ is fully centralized and $\operatorname {Aut}\nolimits _S(P)\in \operatorname {Syl}_p(\operatorname {Aut}\nolimits _{\mathcal {F}}(P))$. This proves (I).
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For any pair of fusion systems $\mathcal {F}_1$ over $S_1$ and $\mathcal {F}_2$ over $S_2$, let $\mathcal {F}_1\times \mathcal {F}_2$ be the obvious fusion system over $S_1\times {}S_2$:
for all $P,Q\le {}S_1\times {}S_2$. The following technical result will be needed in Section 5.
Lemma 1.5
If $\mathcal {F}_1$ and $\mathcal {F}_2$ are saturated fusion systems over $S_1$ and $S_2$, respectively, then $\mathcal {F}_1\times \mathcal {F}_2$ is a saturated fusion system over $S_1\times {}S_2$.
Proof.
For any $P\le {}S_1\times {}S_2$, let $P_1\le {}S_1$ and $P_2\le {}S_2$ denote the images of $P$ under projection to the first and second factors. Thus $P\le {}P_1\times {}P_2$, and this is the smallest product subgroup which contains $P$. Similarly, for any $P\le {}S_1\times {}S_2$ and any $\varphi \in \operatorname {Hom}\nolimits _{\mathcal {F}_1{\times }\mathcal {F}_2}(P,S_1\times {}S_2)$,$\varphi _1\in \operatorname {Hom}\nolimits _{\mathcal {F}_1}(P_1,S_1)$ and $\varphi _2\in \operatorname {Hom}\nolimits _{\mathcal {F}_2}(P_2,S_2)$ denote the projections of $\varphi$.
We apply Lemma 1.4, and first check condition (I$'$). Fix $P\le {}S_1\times {}S_2$; we must show that $P$ is $\mathcal {F}_1{\times }\mathcal {F}_2$-conjugate to a subgroup $P'$ which is fully centralized and satisfies $\operatorname {Aut}\nolimits _{S_1\times {}S_2}(P')\in \operatorname {Syl}_p(\operatorname {Aut}\nolimits _{\mathcal {F}_1\times \mathcal {F}_2}(P'))$. We can assume that $P_1$ and $P_2$ are both fully normalized; otherwise replace $P$ by an appropriate subgroup in its $\mathcal {F}_1{\times }\mathcal {F}_2$-conjugacy class. Since $C_{S_1\times {}S_2}(P)=C_{S_1}(P_1)\times {}C_{S_2}(P_2)$, and since (by (I) applied to the saturated fusion systems $\mathcal {F}_i$) the $P_i$ are fully centralized, $P$ is also fully centralized. Also, by (I) again, $\operatorname {Aut}\nolimits _{S_i}(P_i)\in \operatorname {Syl}_p(\operatorname {Aut}\nolimits _{\mathcal {F}_i}(P_i))$, and hence $\operatorname {Aut}\nolimits _{S_1\times {}S_2}(P_1\times {}P_2)\in \operatorname {Syl}_p(\operatorname {Aut}\nolimits _{\mathcal {F}_1{\times }\mathcal {F}_2}(P_1\times {}P_2))$. Thus, if we regard $\operatorname {Aut}\nolimits _{\mathcal {F}_1{\times }\mathcal {F}_2}(P)$ as a subgroup of $\operatorname {Aut}\nolimits _{\mathcal {F}_1{\times }\mathcal {F}_2}(P_1\times {}P_2)$, there is an element $\alpha \in \operatorname {Aut}\nolimits _{\mathcal {F}_1{\times }\mathcal {F}_2}(P_1\times {}P_2)$ such that $\operatorname {Aut}\nolimits _{S_1\times {}S_2}(P_1\times {}P_2)$ contains a Sylow $p$-subgroup of $\alpha \operatorname {Aut}\nolimits _{\mathcal {F}_1{\times }\mathcal {F}_2}(P)\alpha ^{-1}=\operatorname {Aut}\nolimits _{\mathcal {F}_1{\times }\mathcal {F}_2}(\alpha {}P)$. Then
Then $\varphi$ extends to $\varphi _1\times \varphi _2\in \operatorname {Hom}\nolimits _{\mathcal {F}_1{\times }\mathcal {F}_2}(P_1\times {}P_2,S_1\times {}S_2)$ by definition of $\mathcal {F}_1{\times }\mathcal {F}_2$, and hence to $N_{\varphi _1}\times {}N_{\varphi _2}$ by condition (II) applied to the saturated fusion systems $\mathcal {F}_1$ and $\mathcal {F}_2$. So (II) holds for the fusion system $\mathcal {F}_1{\times }\mathcal {F}_2$ ($N_\varphi \le {}N_{\varphi _1}\times {}N_{\varphi _2}$), and this finishes the proof that $\mathcal {F}_1{\times }\mathcal {F}_2$ is saturated.
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In order to help motivate the next constructions, we recall some definitions from ReferenceBLO. If $G$ is a finite group and $p$ is a prime, then a $p$-subgroup$P\le {}G$ is $p$-centric if $C_G(P)=Z(P)\times {}C'_G(P)$, where $C'_G(P)$ has order prime to $p$. For any $P,Q\le {}G$, let $N_G(P,Q)$ denote the transporter: the set of all $g\in {}G$ such that $gPg^{-1}\le {}Q$. For any $S\in \operatorname {Syl}_p(G)$,$\mathcal {L}_S^c(G)$ denotes the category whose objects are the $p$-centric subgroups of $S$, and where $\operatorname {Mor}\nolimits _{\mathcal {L}_S^c(G)}(P,Q)= N_G(P,Q)/C'_G(P)$. By comparison, $\operatorname {Hom}\nolimits _G(P,Q)\cong N_G(P,Q)/C_G(P)$. Hence there is a functor from $\mathcal {L}_S^c(G)$ to $\mathcal {F}_S(G)$ which is the inclusion on objects, and which sends the morphism corresponding to $g\in {}N_G(P,Q)$ to $c_g\in \operatorname {Hom}\nolimits _G(P,Q)$.
Definition 1.6
Let $\mathcal {F}$ be any fusion system over a $p$-group$S$. A subgroup $P\le {}S$ is $\mathcal {F}$-centric if $P$ and all of its $\mathcal {F}$-conjugates contain their $S$-centralizers. Let $\mathcal {F}^c$ denote the full subcategory of $\mathcal {F}$ whose objects are the $\mathcal {F}$-centric subgroups of $S$.
We are now ready to define “centric linking systems” associated to a fusion system.
Definition 1.7
Let $\mathcal {F}$ be a fusion system over the $p$-group$S$. A centric linking system associated to $\mathcal {F}$ is a category $\mathcal {L}$ whose objects are the $\mathcal {F}$-centric subgroups of $S$, together with a functor
and “distinguished” monomorphisms $P\xrightarrow {\delta _P}\operatorname {Aut}\nolimits _{\mathcal {L}}(P)$ for each $\mathcal {F}$-centric subgroup $P\le {}S$, which satisfy the following conditions.
(A)
$\pi$ is the identity on objects and surjective on morphisms. More precisely, for each pair of objects $P,Q\in \mathcal {L}$,$Z(P)$ acts freely on $\operatorname {Mor}\nolimits _{\mathcal {L}}(P,Q)$ by composition (upon identifying $Z(P)$ with $\delta _P(Z(P))\le \operatorname {Aut}\nolimits _{\mathcal {L}}(P)$), and $\pi$ induces a bijection$$\begin{equation*} \operatorname {Mor}\nolimits _{\mathcal {L}}(P,Q)/Z(P) \xrightarrow {\cong } \operatorname {Hom}\nolimits _{\mathcal {F}}(P,Q). \end{equation*}$$
(B)
For each $\mathcal {F}$-centric subgroup $P\le {}S$ and each $g\in {}P$,$\pi$ sends $\delta _P(g)\in \operatorname {Aut}\nolimits _{\mathcal {L}}(P)$ to $c_g\in \operatorname {Aut}\nolimits _{\mathcal {F}}(P)$.
(C)
For each $f\in \operatorname {Mor}\nolimits _{\mathcal {L}}(P,Q)$ and each $g\in {}P$, the following square commutes in $\mathcal {L}$:
One easily checks that for any $G$ and any $S\in \operatorname {Syl}_p(G)$,$\mathcal {L}_S^c(G)$ is a centric linking system associated to the fusion system $\mathcal {F}_S(G)$. Condition (C) is motivated in part because it always holds in $\mathcal {L}_S^c(G)$ for any $G$. Conditions (A) and (B) imply that $P$ acts freely on $\operatorname {Mor}\nolimits _{\mathcal {L}}(P,Q)$. Together with (C), they imply that the $Q$-action on $\operatorname {Mor}\nolimits _\mathcal {L}(P,Q)$ is free, and describe how it determines the action of $P$. Condition (C) was also motivated by the proof of Proposition 2.2 below, where we show that the nerve of any centric linking system is equivalent to the homotopy colimit of a certain functor.
Throughout the rest of this paper, whenever we refer to conditions (A), (B), or (C), it will mean the conditions in the above Definition 1.7.
Definition 1.8
A $p$-local finite group is a triple $(S,\mathcal {F},\mathcal {L})$, where $\mathcal {F}$ is a saturated fusion system over the $p$-group$S$ and $\mathcal {L}$ is a centric linking system associated to $\mathcal {F}$. The classifying space of the $p$-local finite group $(S,\mathcal {F},\mathcal {L})$ is the space $|\mathcal {L}|{}_{p}^{\wedge }$.
Thus, for any finite group $G$ and any $S\in \operatorname {Syl}_p(G)$, the triple $(S,\mathcal {F}_S(G),\mathcal {L}_S^c(G))$ is a $p$-local finite group. Its classifying space is $|\mathcal {L}_S^c(G)|{}_{p}^{\wedge }\simeq |\mathcal {L}_p^c(G)|{}_{p}^{\wedge }$, which by ReferenceBLO, Proposition 1.1 is homotopy equivalent to $BG{}_{p}^{\wedge }$.
The following notation will be used when working with $p$-local finite groups. For any group $G$, let $\mathcal {B}(G)$ denote the category with one object $o_G$, and one morphism denoted $\check{g}$ for each $g\in {}G$.
Notation 1.9
Let $(S,\mathcal {F},\mathcal {L})$ be a $p$-local finite group, where $\pi \colon \mathcal {L}\xrightarrow {}\mathcal {F}^c$ denotes the projection functor. For each $\mathcal {F}$-centric subgroup $P\le {}S$, and each $g\in {}P$, we write
denote the functor which sends the unique object $o_P\in \operatorname {Ob}\nolimits (\mathcal {B}(P))$ to $P$ and which sends a morphism $\check{g}$ (for $g\in {}P$) to $\widehat{g}=\delta _P(g)$. If $f$ is any morphism in $\mathcal {L}$, we let $[f]=\pi (f)$ denote its image in $\mathcal {F}$.
The following lemma lists some easy properties of centric linking systems associated to saturated fusion systems.
Lemma 1.10
Fix a $p$-local finite group $(S,\mathcal {F},\mathcal {L})$, and let $\pi \colon \mathcal {L}\xrightarrow {}\mathcal {F}^c$ be the projection. Fix $\mathcal {F}$-centric subgroups $P,Q,R$ in $S$. Then the following hold.
(a)
Fix any sequence $P\xrightarrow {\varphi }Q\xrightarrow {\psi }R$ of morphisms in $\mathcal {F}^c$, and let $\widetilde{\psi }\in \pi _{Q,R}^{-1}(\psi )$ and $\widetilde{\psi \varphi }\in \pi _{P,R}^{-1}(\psi \varphi )$ be arbitrary liftings. Then there is a unique morphism $\widetilde{\varphi }\in \operatorname {Mor}\nolimits _{\mathcal {L}}(P,Q)$ such that$$\begin{equation*} \widetilde{\psi }\mathchoice {\mathbin {\scriptstyle \circ }}{\mathbin {\scriptstyle \circ }}{\mathbin {\scriptscriptstyle \circ }}{\mathbin {\scriptscriptstyle \circ }}\widetilde{\varphi }= \widetilde{\psi \varphi }, \tag{1} \end{equation*}$$
and furthermore $\pi _{P,Q}(\widetilde{\varphi })=\varphi$.
(b)
If $\widetilde{\varphi },\widetilde{\varphi }'\in \operatorname {Mor}\nolimits _{\mathcal {L}}(P,Q)$ are such that the homomorphisms $\varphi \overset {\text{def}}{=}\pi _{P,Q}(\widetilde{\varphi })$ and $\varphi '\overset {\text{def}}{=}\pi _{P,Q}(\widetilde{\varphi }')$ are conjugate (differ by an element of $\operatorname {Inn}\nolimits (Q)$), then there is a unique element $g\in {}Q$ such that $\widetilde{\varphi }'=\widehat{g}\mathchoice {\mathbin {\scriptstyle \circ }}{\mathbin {\scriptstyle \circ }}{\mathbin {\scriptscriptstyle \circ }}{\mathbin {\scriptscriptstyle \circ }}\widetilde{\varphi }$ in $\operatorname {Mor}\nolimits _{\mathcal {L}}(P,Q)$.
Proof.
(a) Fix any element $\alpha \in \pi _{P,Q}^{-1}(\varphi )$. By (A), there is a unique element $g\in {}Z(P)$ such that $\widetilde{\psi \varphi }= \widetilde{\psi }\mathchoice {\mathbin {\scriptstyle \circ }}{\mathbin {\scriptstyle \circ }}{\mathbin {\scriptscriptstyle \circ }}{\mathbin {\scriptscriptstyle \circ }}\alpha \mathchoice {\mathbin {\scriptstyle \circ }}{\mathbin {\scriptstyle \circ }}{\mathbin {\scriptscriptstyle \circ }}{\mathbin {\scriptscriptstyle \circ }}\widehat{g}$. Hence equation (1) holds if we set $\widetilde{\varphi }=\alpha \mathchoice {\mathbin {\scriptstyle \circ }}{\mathbin {\scriptstyle \circ }}{\mathbin {\scriptscriptstyle \circ }}{\mathbin {\scriptscriptstyle \circ }}\widehat{g}$, and clearly $\pi _{P,Q}(\widetilde{\varphi })=\varphi$. Conversely, for any $\widetilde{\varphi }'\in \operatorname {Mor}\nolimits _\mathcal {L}(P,Q)$ such that $\widetilde{\psi }\mathchoice {\mathbin {\scriptstyle \circ }}{\mathbin {\scriptstyle \circ }}{\mathbin {\scriptscriptstyle \circ }}{\mathbin {\scriptscriptstyle \circ }}\widetilde{\varphi }'= \widetilde{\psi \varphi }$ we have $\pi _{P,Q}(\widetilde{\varphi }')=\varphi$ since they are equal after composing with $\psi$, and so $\widetilde{\varphi }'=\widetilde{\varphi }$ since (by (A) again) the same group $Z(P)$ acts freely and transitively on $\pi _{P,Q}^{-1}(\varphi )$ and on $\pi _{P,R}^{-1}(\psi \varphi )$.
This proves the existence of $g=x{\cdot }\varphi (y)\in {}Q$ such that $\widetilde{\varphi }'=\widehat{g}\mathchoice {\mathbin {\scriptstyle \circ }}{\mathbin {\scriptstyle \circ }}{\mathbin {\scriptscriptstyle \circ }}{\mathbin {\scriptscriptstyle \circ }}\widetilde{\varphi }$. Conversely, if
for $g,h\in {}Q$, then $\widetilde{\varphi }=\widehat{g^{-1}h}\mathchoice {\mathbin {\scriptstyle \circ }}{\mathbin {\scriptstyle \circ }}{\mathbin {\scriptscriptstyle \circ }}{\mathbin {\scriptscriptstyle \circ }}\widetilde{\varphi }$, so $g^{-1}h\in {}C_Q(\varphi (P))$, and $g^{-1}h\in \varphi (P)$ since $P$ (and hence $\varphi (P)$) is $\mathcal {F}$-centric. Write $g^{-1}h=\varphi (y)$ for $y\in {}P$; then $\widetilde{\varphi }=\widetilde{\varphi }\mathchoice {\mathbin {\scriptstyle \circ }}{\mathbin {\scriptstyle \circ }}{\mathbin {\scriptscriptstyle \circ }}{\mathbin {\scriptscriptstyle \circ }}\widehat{y}$ by (C), hence $\widehat{y}=\operatorname {Id}\nolimits$ by (a), and so $y=1$ (and $g=h$) by (A).
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Lemma 1.10(a) implies in particular that all morphisms in $\mathcal {L}$ are monomorphisms in the categorical sense.
The next proposition describes how an associated centric linking system $\mathcal {L}$ over a $p$-group$S$ contains the category with the same objects and whose morphisms are the sets $N_S(P,Q)$.
Proposition 1.11
Let $(S,\mathcal {F},\mathcal {L})$ be a $p$-local finite group, and let $\pi \colon \mathcal {L}\xrightarrow {}\mathcal {F}^c$ be the associated projection. For each $P\le {}S$, fix a choice of “inclusion” morphism $\iota _P\in \operatorname {Mor}\nolimits _{\mathcal {L}}(P,S)$ such that $[\iota _P]=\operatorname {incl}\nolimits \in \operatorname {Hom}\nolimits (P,S)$ (and $\iota _S=\operatorname {Id}\nolimits _S$). Then there are unique injections
defined for all $\mathcal {F}$-centric subgroups $P,Q\le {}S$, which have the following properties.
(a)
For all $\mathcal {F}$-centric$P,Q\le S$ and all $g\in {}N_S(P,Q)$,$[\delta _{P,Q}(g)]=c_g\in \operatorname {Hom}\nolimits _{\mathcal {F}}(P,Q)$.
(b)
For all $\mathcal {F}$-centric$P\le S$ we have $\delta _{P,S}(1)=\iota _P$, and $\delta _{P,P}(g)=\delta _P(g)$ for $g\in {}P$.
(c)
For all $\mathcal {F}$-centric$P,Q,R\le S$ and all $g\in {}N_S(P,Q)$ and $h\in {}N_S(Q,R)$ we have $\delta _{Q,R}(h)\mathchoice {\mathbin {\scriptstyle \circ }}{\mathbin {\scriptstyle \circ }}{\mathbin {\scriptscriptstyle \circ }}{\mathbin {\scriptscriptstyle \circ }}\delta _{P,Q}(g)= \delta _{P,R}(hg)$.
Proof.
For each $\mathcal {F}$-centric$P$ and $Q$ and each $g\in {}N_S(P,Q)$, there is by Lemma 1.10(a) a unique morphism $\delta _{P,Q}(g)\in \operatorname {Mor}\nolimits _{\mathcal {L}}(P,Q)$ such that $[\delta _{P,Q}(g)]=c_g$, and such that the following square commutes: Conditions (b) and (c) above also follow from the uniqueness property in Lemma 1.10(a). The injectivity of $\delta _{P,Q}$ follows from condition (A), since $[\delta _{P,Q}(g)]=[\delta _{P,Q}(h)]$ in $\operatorname {Hom}\nolimits _{\mathcal {F}}(P,Q)$ if and only if $h^{-1}g\in {}C_S(P)=Z(P)$.
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We finish the section with the following proposition, which shows that the classifying space of any $p$-local finite group is $p$-complete, and also provides some control over its fundamental group.
Proposition 1.12
Let $(S,\mathcal {F},\mathcal {L})$ be any $p$-local finite group. Then $|\mathcal {L}|$ is $p$-good. Also, the composite
induced by the inclusion $\mathcal {B}(S)\xrightarrow {\theta _S}\mathcal {L}$, is surjective.
Proof.
For each $\mathcal {F}$-centric subgroup $P\le {}S$, fix a morphism $\iota _P\in \operatorname {Mor}\nolimits _{\mathcal {L}}(P,S)$ which lifts the inclusion (and set $\iota _S=\operatorname {Id}\nolimits _S$). By Lemma 1.10(a), for each $P\le {}Q\le {}S$, there is a unique morphism $\iota _P^Q\in \operatorname {Mor}\nolimits _{\mathcal {L}}(P,Q)$ such that $\iota _Q\mathchoice {\mathbin {\scriptstyle \circ }}{\mathbin {\scriptstyle \circ }}{\mathbin {\scriptscriptstyle \circ }}{\mathbin {\scriptscriptstyle \circ }}{}\iota _P^Q=\iota _P$.
Regard the vertex $S$ as the basepoint of $|\mathcal {L}|$. Define
by sending each $\varphi \in \operatorname {Mor}\nolimits _{\mathcal {L}}(P,Q)$ to the loop formed by the edges $\iota _P$,$\varphi$, and $\iota _Q$ (in that order). Clearly, $\omega (\psi \mathchoice {\mathbin {\scriptstyle \circ }}{\mathbin {\scriptstyle \circ }}{\mathbin {\scriptscriptstyle \circ }}{\mathbin {\scriptscriptstyle \circ }}\varphi )=\omega (\psi ){\cdot }\omega (\varphi )$ whenever $\psi$ and $\varphi$ are composable, and $\omega (\iota _P^Q)=\omega (\iota _P)=1$ for all $P\le {}Q\le {}S$. Also, $\pi _1(|\mathcal {L}|)$ is generated by $\operatorname {Im}\nolimits (\omega )$, since any loop in $|\mathcal {L}|$ can be split up as a composite of loops of the above form.
By Alperin’s fusion theorem for saturated fusion systems (Theorem A.10), each morphism in $\mathcal {F}$, and hence each morphism in $\mathcal {L}$, is (up to inclusions) a composite of automorphisms of fully normalized $\mathcal {F}$-centric subgroups. Thus $\pi _1(|\mathcal {L}|)$ is generated by the subgroups $\omega (\operatorname {Aut}\nolimits _{\mathcal {L}}(P))$ for all fully normalized $\mathcal {F}$-centric$P\le {}S$.
Let $K\vartriangleleft \pi _1(|\mathcal {L}|)$ be the subgroup generated by all elements of finite order prime to $p$. For each fully normalized $\mathcal {F}$-centric$P\le {}S$,$\operatorname {Aut}\nolimits _{\mathcal {L}}(P)$ is generated by its Sylow subgroup $N_S(P)$ together with elements of order prime to $p$. Hence $\pi _1(|\mathcal {L}|)$ is generated by $K$ together with the subgroups $\omega (N_S(P))$; and $\omega (N_S(P))\le \omega (S)$ for each $P$. This shows that $\omega$ sends $S$ surjectively onto $\pi _1(|\mathcal {L}|)/K$, and in particular that this quotient group is a finite $p$-group.
Set $\pi =\pi _1(|\mathcal {L}|)/K$ for short. Since $K$ is generated by elements of order prime to $p$, the same is true of its abelianization, and hence $H_1(K;\mathbb{F}_p)=0$. Thus, $K$ is $p$-perfect. Let $X$ be the cover of $|\mathcal {L}|$ with fundamental group $K$. Then $X$ is $p$-good and $X{}_{p}^{\wedge }$ is simply connected since $\pi _1(X)$ is $p$-perfectReferenceBK, VII.3.2. Also, $H_i(X;\mathbb{F}_p)$ is finite for all $i$ since $|\mathcal {L}|$ and hence $X$ has finite skeleta. Hence $X{}_{p}^{\wedge }\xrightarrow {}|\mathcal {L}|{}_{p}^{\wedge }\xrightarrow {}B\pi$ is a fibration sequence and $|\mathcal {L}|{}_{p}^{\wedge }$ is $p$-complete by ReferenceBK, II.5.2(iv). So $|\mathcal {L}|$ is $p$-good, and $\pi _1(|\mathcal {L}|{}_{p}^{\wedge })\cong \pi$ is a quotient group of $S$. (Alternatively, this follows directly from a “mod $p$ plus construction” on $|\mathcal {L}|$: there are a space $Y$ and a mod $p$ homology equivalence $|\mathcal {L}|\xrightarrow {}Y$ such that $\pi _1(Y)\cong \pi _1(|\mathcal {L}|)/K=\pi$, and $|\mathcal {L}|$ is $p$-good since $Y$ is.)
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2. Homotopy decompositions of classifying spaces
We now consider some homotopy decompositions of the classifying space $|\mathcal {L}|{}_{p}^{\wedge }$ of a $p$-local finite group $(S,\mathcal {F},\mathcal {L})$. The first, and most important, is taken over the orbit category of $\mathcal {F}$.
Definition 2.1
The orbit category of a fusion system $\mathcal {F}$ over a $p$-group$S$ is the category $\mathcal {O}(\mathcal {F})$ whose objects are the subgroups of $S$, and whose morphisms are defined by
We let $\mathcal {O}^c(\mathcal {F})$ denote the full subcategory of $\mathcal {O}(\mathcal {F})$ whose objects are the $\mathcal {F}$-centric subgroups of $S$. If $\mathcal {L}$ is a centric linking system associated to $\mathcal {F}$, then $\widetilde{\pi }$ denotes the composite functor
More generally, if $\mathcal {F}_0\subseteq \mathcal {F}$ is any full subcategory, then $\mathcal {O}(\mathcal {F}_0)$ denotes the full subcategory of $\mathcal {O}(\mathcal {F})$ whose objects are the objects of $\mathcal {F}_0$. Thus, $\mathcal {O}^c(\mathcal {F})=\mathcal {O}(\mathcal {F}^c)$.
Note the difference between the orbit category of a fusion system and the orbit category of a group. If $G$ is a group and $S\in \operatorname {Syl}_p(G)$, then $\mathcal {O}_S(G)$ is the category whose objects are the orbits $G/P$ for all $P\le {}S$, and where $\operatorname {Mor}\nolimits _{\mathcal {O}_S(G)}(G/P,G/Q)$ is the set of all $G$-maps between the orbits. If $\mathcal {F}=\mathcal {F}_S(G)$ is the fusion system of $G$, then morphisms in the orbit categories of $G$ and $\mathcal {F}$ can be expressed in terms of the set $N_G(P,Q)$ of elements which conjugate $P$ into $Q$:
If $P$ is $p$-centric in $G$, then these sets differ only by the action of the group $C'_G(P)$ of order prime to $p$.
We next look at the homotopy type of the nerve of a centric linking system. Here, $$ denotes the category of spaces.
Proposition 2.2
Fix a saturated fusion system $\mathcal {F}$ and an associated centric linking system $\mathcal {L}$, and let $\widetilde{\pi }\colon \mathcal {L}\xrightarrow {}\mathcal {O}^c(\mathcal {F})$ be the projection functor. Let
be the left homotopy Kan extension over $\widetilde{\pi }$ of the constant functor $\mathcal {L}\xrightarrow {*}$. Then $\widetilde{B}$ is a homotopy lifting of the homotopy functor $P\mapsto {}BP$, and
More generally, if $\mathcal {L}_0\subseteq \mathcal {L}$ is any full subcategory, and $\mathcal {F}_0\subseteq \mathcal {F}^c$ is the full subcategory with $\operatorname {Ob}\nolimits (\mathcal {F}_0)=\operatorname {Ob}\nolimits (\mathcal {L}_0)$, then
Recall that we write $\operatorname {Rep}\nolimits _{\mathcal {F}}(P,Q)$ to denote morphisms in $\mathcal {O}^c(\mathcal {F})$. By definition, for each $\mathcal {F}$-centric subgroup $P\le {}S$,$\widetilde{B}(P)$ is the nerve (homotopy colimit of the point functor) of the overcategory $\widetilde{\pi }{\downarrow }P$, whose objects are pairs $(Q,\alpha )$ for $\alpha \in \operatorname {Rep}\nolimits _{\mathcal {F}}(Q,P)$, and where
Since $|\mathcal {L}|\cong \mathop {\mathop {\mathrm{hocolim}}\limits _{\longrightarrow }}_{\mathcal {L}}(*)$, (1) holds by ReferenceHV, Theorem 5.5 (and the basic idea is due to Segal ReferenceSe, Proposition B.1). Similarly, if $\widetilde{B}_0$ denotes the left homotopy Kan extension over $\mathcal {L}_0\xrightarrow {\widetilde{\pi }_0}\mathcal {O}(\mathcal {F}_0)$ of the constant functor $\mathcal {L}_0\xrightarrow {*}$, then
It remains only to show that $\widetilde{B}$ is a lifting of the homotopy functor $P\mapsto {}BP$, and that the inclusion $\widetilde{B}_0(P)\hookrightarrow \widetilde{B}(P)$ is a homotopy equivalence when $P\in \operatorname {Ob}\nolimits (\mathcal {F}_0)$.
Let $\mathcal {B}'(P)\subseteq \widetilde{\pi }{\downarrow }P$ be the subcategory with one object $(P,\operatorname {Id}\nolimits )$ and with morphisms $\{\widehat{g}\,|\,g\in {}P\}$. In particular, $|\mathcal {B}'(P)|\simeq {}BP$. We claim that