Normalizer decompositions of $p$-local compact groups

For a finite group $G$, a prime $p$, and a suitable collection ${\mathscr{C}}$ of subgroups of $G$, Dwyer [Dwy97] gave a systematic approach to three homotopy colimit decompositions for ${BG}_{p}^{\wedge}$. Two of them, the centralizer decomposition and the subgroup decomposition, are indexed by fusion and orbit categories whose objects are conjugacy classes of subgroups in ${\mathscr{C}}$. The third is the normalizer decomposition,

which is indexed over the finite poset ${\mathcal{I}}$ whose objects are $G$-conjugacy classes of chains ${\mathbb{P}}=(H_{0}\subsetneq\ldots\subsetneq H_{k})$ for $H_{i}\in{\mathscr{C}}$ and $\delta({\mathbb{P}})\simeq B(\cap_{i}N_{G}H_{i})$.

There are two more general contexts that are relevant to our work. First, the homotopy colimit decompositions were studied for compact Lie groups. Jackowski and McClure [JM92, Thm. 1.3] had given a centralizer decomposition for compact Lie groups with respect to the collection of elementary abelian $p$-subgroups. Jackowski, McClure and Oliver described subgroup decompositions. Słomińska  [Sło01] gave a normalizer decomposition with this collection. Libman [Lib11, Thm. C] gave centralizer, subgroup, and normalizer decompositions with the collection of abelian $p$-subgroups and the collection of $p$-radical subgroups.

The second generalization of interest comes by replacing finite groups by $p$-local finite groups. These objects are triples $(S,{\mathcal{F}},{\mathcal{L}})$, where $S$ is a finite $p$-group, and ${\mathcal{F}}$ and ${\mathcal{L}}$ are categories encoding data that mimics conjugations. One can define the classifying space ${\left|{\mathcal{L}}\right|}_{p}^{\wedge}$ of a $p$-local finite group, and it behaves similarly to the $p$-completed classifying space of a group. Every finite group gives rise to a $p$-local finite group, but not every $p$-local finite group comes from a group. Libman [Lib06] proved the existence of a normalizer decomposition for classifying spaces of $p$-local finite groups.

We work with $p$-local compact groups (Definition 2.11), which generalize compact Lie groups in the same way that $p$-local finite groups generalize finite groups. The theory, developed in [BLO07], is in spirit analogous to that of $p$-local finite groups, but with new challenges because of the non-finite context. The analogue of a finite $p$-group is a discrete $p$-toral group, namely an extension of a discrete torus $({\mathbb{Z}}/p^{\infty})^{r}$ by a finite $p$-group. Studying Lie groups by way of their associated $p$-local compact groups has the advantage of reducing to discrete (as opposed to topological) groups. Broto, Levi, and Oliver prove a subgroup decomposition for $p$-local compact groups ([BLO07, Prop. 4.6], [LL15, Thm. B]).

Our first contribution is to adapt Libman’s work [Lib06] on $p$-local finite groups to construct a normalizer decomposition for $p$-local compact groups. One advantage of the normalizer decomposition of a $p$-local compact group over the centralizer or subgroup decompositions is that the normalizer decomposition is indexed over a finite poset. In the statement below, the notation ${\mathcal{F}}^{cr}$ refers to the subcategory of ${\mathcal{F}}$ whose objects are ${\mathcal{F}}$-centric and ${\mathcal{F}}$-radical subgroups. The finite poset ${{\overline{s}}d}{\mathcal{F}}^{cr}$ is defined in Definition 3.15.

This statement is largely formal, as is most of the proof, but there are interesting challenges in computing the decomposition in cases of interest. Our first class of examples is given by $p$-local compact groups arising from compact Lie groups. Theorem 3.16 takes place fully in the world of discrete $p$-toral groups. To relate our underlying theory to the usual context of normalizers in compact Lie groups, we use our previous work [BCG⁺22] to rephrase the functor values in the resulting homotopy colimit as mod $p$ equivalent to classifying spaces of group-theoretic normalizers. In comparison to [Lib11], this approach gives a more formal proof of the basic decomposition result—essentially analogous to the finite case—because we do not have to address the topological issues in applying Quillen’s Theorem A. Instead, the topological issues can be neatly packaged into the functor values and understood on a uniform basis [BCG⁺22].

Let ${\mathcal{R}}$ denote the collection of $p$-toral subgroups of a compact Lie group $G$ that are both $p$-centric and $p$-stubborn in $G$, and let ${{\overline{s}}d}{\mathcal{R}}$ be the poset of $G$-conjugacy classes of chains of proper inclusions of subgroups in ${\mathcal{R}}$.

In concert with Theorem 3.16, Theorem 4.2 tells us that when $\pi_{0}G$ is a $p$-group, we can compute a normalizer decomposition for ${BG}_{p}^{\wedge}$ over a poset indexed by chains of $p$-centric and $p$-stubborn subgroups of $G$, with values that are mod $p$ equivalent to intersections of normalizers of those subgroups. (See Remark 4.4 regarding the $\pi_{0}$ hypothesis.) In Section 6, we compute the decomposition explicitly in the cases $\operatorname{U}(p)$ and $\operatorname{SU}(p)$, expressing the classifying spaces of these groups as mod $p$ equivalent to a homotopy pushout diagram. We believe these decompositions are new for odd primes. In the case $p=2$ we recover the theorem of Dwyer, Miller, and Wilkerson [DMW87], who gave mod $2$ homotopy pushout decompositions of $\operatorname{BSU}(2)$ and $\operatorname{BSO}(3)$ using an ad hoc method.

The result for $\operatorname{U}(p)$ is found in Theorem 6.7.

For our second class of examples, we turn to $p$-compact groups, another generalization of classifying spaces of Lie groups introduced by Dwyer and Wilkerson [DW94a], which arose in the study of cohomology rings of loop spaces. Every $p$-compact group has an associated $p$-local compact group $(S,\mathcal{F},{\mathcal{L}})$ [BLO07, Thm. 10.7], but not every $p$-local compact group arises in this way. The classification of $p$-compact groups in [AG09, AGMV08] builds these spaces out of compact Lie groups and a collection of exotic examples.

We apply Theorem 3.16 to the Aguadé–Zabrodsky $p$-compact groups, which are closely related to ${\operatorname{BSU}(p)}_{p}^{\wedge}$. They were first constructed in [Agu89] to have cohomology realizing certain invariants of polynomial rings. Our result is a homotopy pushout diagram for these spaces.

Comparing to Theorem 6.14, above, we see that $\operatorname{SL}_{2}\!{\mathbb{F}}_{p}$ is replaced by $\operatorname{GL}_{2}\!{\mathbb{F}}_{p}$, and $\Sigma_{p}$ (the Weyl group for ${\operatorname{SU}(p)}$) is replaced by $G$ (the new, enlarged Weyl group for $X$).

We give a brief overview of the notion of a $p$-local compact group given in the work of Broto, Levi and Oliver [BLO07]. Recall that a $p$-toral group is an extension of a torus, $\left(S^{1}\right)^{r}$, by a finite $p$-group. We work with a discrete version of $p$-toral groups. As usual, let ${\mathbb{Z}}/{p}^{\infty}$ be the union of the cyclic $p$-groups ${\mathbb{Z}}/p^{n}$ under the standard inclusions.

Given two discrete $p$-toral groups $P$ and $Q$, let $\operatorname{Hom}(P,Q)$ denote the set of group homomorphisms from $P$ to $Q$. If $P$ and $Q$ are subgroups of a larger group $S$, then $\operatorname{Hom}_{S}(P,Q)$ denotes the set of those homomorphisms (necessarily monomorphisms) induced by conjugation by elements of $S$.

The following definition is a straightforward generalization of the definition of fusion systems over finite $p$-groups (see [BLO03]).

The same language of “outer automorphisms” is used for fusion systems as for groups. In particular, just as $\operatorname{Out}_{S}(P):=\operatorname{Aut}_{S}(P)/\operatorname{Aut}_{P}(P)$, we define $\operatorname{Out}_{{\mathcal{F}}}(P):=\operatorname{Aut}_{{\mathcal{F}}}(P)/\operatorname{Aut}_{P}(P)$. In addition, we say that two subgroups $P,P^{\prime}$ of $S$ are ${\mathcal{F}}$-conjugate if there is an isomorphism $P\cong P^{\prime}$ in ${\mathcal{F}}$.

In order for a fusion system to have good properties and model conjugacy relations among $p$-subgroups of a group, it must satisfy an extra set of axioms, for “saturation.” The definition is given in [BLO07, §2], and we refer the reader to this source, as the definition is fairly long and technical, and we do not need to use any of the details.

The fusion system associated to a compact Lie group has the right technical property to be tractable.

In general, a fusion system over a discrete $p$-toral group $S$ will have an infinite number of isomorphism classes of objects (unless $S$ is finite). Fortunately, it turns out to be possible to restrict one’s attention to a smaller number of objects. The concepts of “${\mathcal{F}}$-centric” and “${\mathcal{F}}$-radical” play analogous roles in the theory of $p$-local compact groups to their group-theoretic counterparts.

The saturated fusion system ${\mathcal{F}}_{S}(G)$ of the group $G$ does not contain enough information about $G$ to recover ${BG}_{p}^{\wedge}$. For example, if $G$ is a finite $p$-group, the fusion system can only detect $G/Z(G)$. We recall the definition of a centric linking system, a category associated to a saturated fusion system, whose nerve is mod $p$ equivalent to $BG$. Details on properties of linking systems can be found in the appendix of [BLO14]. We begin with the transporter category.

Broto, Levi, and Oliver, in [BLO07, §9], proved that a compact Lie group $G$ gives rise to a $p$-local compact group $(S,{\mathcal{F}}_{S}(G),{\mathcal{L}}_{S}(G))$ by giving a specific construction of ${\mathcal{L}}_{S}(G)$; they then prove that it gives a suitable model for the $p$-completion of $BG$.

Thus any theorem about decompositions of $\left|{\mathcal{L}}\right|$ for a $p$-local compact group will also apply to Lie groups. For computational purposes, we will use an alternative, more concrete model for ${\mathcal{L}}_{S}(G)$ also described in [BLO07], which we detail in Section 4.

A key result in the theory of fusion systems is the existence and uniqueness (up to equivalence) of centric linking systems associated to a given saturated fusion system. For saturated fusion systems over a finite $p$-group $S$, the result was proven first in [Che13] using the new theory of localities. Another proof was given in [Oli13] using the obstruction theory developed in [BLO03]. Later, [LL15] extended the result to saturated fusion systems over discrete $p$-toral groups.

Throughout this section, we assume a fixed $p$-local compact group $(S,{\mathcal{F}},{\mathcal{L}})$ (Definition 2.11). We establish a normalizer decomposition that expresses the uncompleted nerve $\left|{\mathcal{L}}\right|$ as a homotopy colimit indexed on a finite poset of chains. We start by introducing chains and their automorphisms. Next we prove that automorphism groups of chains in ${\mathcal{L}}$ are virtually discrete $p$-toral (Definition 3.9). Lastly, we prove the general, abstract normalizer decomposition result for a $p$-local compact group (Theorem 3.16), which mostly proceeds analogously to the $p$-local finite group case in [Lib06].

We begin in the fusion system. A chain in ${\mathcal{F}}$ is given by a sequence ${\mathbb{P}}=(P_{0}\subseteq P_{1}\subseteq\ldots\subseteq P_{k})$ of subgroups of $S$. A chain is proper if the inclusions are all strict. If ${\mathbb{P}}^{\prime}$ has the same length as ${\mathbb{P}}$, we say that ${\mathbb{P}}$ and ${\mathbb{P}}^{\prime}$ are ${\mathcal{F}}$-conjugate if there exists an isomorphism $f\in\operatorname{Hom}_{\mathcal{F}}\left(P_{k},P^{\prime}_{k}\right)$ such that $f\left(P_{i}\right)=P^{\prime}_{i}$.

We would like to define $\operatorname{Aut}_{\mathcal{L}}({\mathbb{P}})$ for a chain ${\mathbb{P}}$, but first we need an analogue of the canonical subgroup inclusions used to define $\operatorname{Aut}_{\mathcal{F}}({\mathbb{P}})$. It is possible to construct compatible “distinguished inclusions” in ${\mathcal{L}}$ with the property that they project to the subset inclusions in ${\mathcal{F}}$ via $\pi\colon{\mathcal{L}}\rightarrow{\mathcal{F}}$ (Definition 2.10).

The spaces in our decomposition of $\left|{\mathcal{L}}\right|$ will be classifying spaces of automorphism groups in the linking system.

We pause for two basic lemmas that come up in lifting from the fusion system to the linking system, and in checking uniqueness properties. First, a factoring lemma follows directly from the axioms of a centric linking system.

It follows from the factoring properties above that morphisms in a centric linking system have good categorical properties.

Returning to the study of automorphism groups of chains, we are able to relate the automorphism groups in the linking system to those in the fusion system.

We relate automorphism groups of chains in a linking system to virtually discrete $p$-toral groups, which were studied in the context of linking systems in [LL15] and [Mol18].

Like discrete $p$-toral groups, virtually discrete $p$-toral groups have good inheritance properties.

Automorphism groups of chains in ${\mathcal{L}}$ take values in virtually discrete $p$-toral groups.

With automorphism groups in place, we are ready to discuss the indexing category of the normalizer decomposition, following the work of Słomińska. We adapt the proof of [Lib06, Thm. 5.1].

Let ${\mathcal{F}}^{cr}$ (resp. ${\mathcal{L}}^{cr}$) denote the full subcategory of ${\mathcal{F}}$ (resp. ${\mathcal{L}}$) consisting of the subgroups of $S$ that are both ${\mathcal{F}}$-centric and ${\mathcal{F}}$-radical. Recall that a chain of subgroups is “proper” if all of the inclusions are strict.

In Lemma 3.14 we check that ${\mathcal{L}}^{cr}$ has the structure of Definition 3.13.

The abstract “normalizer decomposition theorem” expresses $\left|{\mathcal{L}}\right|$ as a homotopy colimit over the finite poset ${{\overline{s}}d}{\mathcal{F}}^{cr}$.

In this section, we study the application of our abstract normalizer decomposition (Theorem 3.16) to the case of $p$-local compact groups that arise from compact Lie groups (Example 2.5). Recall that the decomposition for $\left|{\mathcal{L}}\right|$ in Theorem 3.16 is given in terms of $\operatorname{BAut}_{{\mathcal{L}}}({\mathbb{P}})$ for proper chains ${\mathbb{P}}=\left(P_{0}\subsetneq\dots\subsetneq P_{k}\right)$ of subgroups that are $\mathcal{F}$-centric and $\mathcal{F}$-radical. There are similar concepts in the theory of compact Lie groups.

The following theorem is the main result for this section. It recovers a version of the normalizer decomposition for compact Lie groups that was described by Libman in [Lib11, §1.4]. Our approach via $p$-local compact groups has the advantage that we do not need to address the delicate issues that were studied in [Lib11, §5] for the purpose of applying Quillen’s Theorem A in a topological setting.

Let ${\mathcal{R}}$ denote the collection of $p$-toral subgroups of $G$ that are both $p$-centric and $p$-stubborn in $G$, and let ${{\overline{s}}d}{\mathcal{R}}$ be the poset of $G$-conjugacy classes of chains of proper inclusions of subgroups in ${\mathcal{R}}$.

The proof is at the end of the section and goes through several steps. First, with regard to the indexing category, we have the following result from a previous work.

The remainder of this section is devoted to establishing the weak mod $p$ equivalence $\operatorname{BAut}_{{\mathcal{L}}}({\mathbb{P}})\simeq\operatorname{B}\!\big{(}\cap_{i}N_{G}({\bf P}_{i})\big{)}$ of Theorem 4.2. In particular, the strategy is to establish a zigzag of natural mod $p$ equivalences of functors of chains of subgroups (with the leftmost one being an equivalence by Theorem 3.16):

The auxiliary category $\widetilde{{\mathcal{L}}}_{S}(G)$ is a variant of the transporter system for $G$ and is used in the construction of ${\mathcal{L}}_{S}(G)$ in [BLO07, Prop. 9.12].

We begin with the left vertical arrow of diagram (4.5), which takes the bulk of the section. In addition to an abstract existence result in for a linking system associated to ${\mathcal{F}}_{S}(G)$ (Theorem 2.12), there is a construction in [BLO07, §9] of a more direct model for ${\mathcal{L}}_{S}(G)$ starting from the transporter category (Definition 2.9). A difficulty in the construction is that the axioms of a linking system require $\operatorname{Hom}_{{\mathcal{F}}}(P,Q)$ to be the orbits of a free $Z(P)$-action on $\operatorname{Hom}_{{\mathcal{L}}}(P,Q)$. For this reason, the transporter system ${\mathcal{T}}_{S}(G)$ itself cannot directly provide the linking system: getting to $\operatorname{Hom}_{{\mathcal{F}}_{S}(G)}(P,Q)$ from $\operatorname{Hom}_{{\mathcal{T}}_{S}(G)}(P,Q)=N_{G}(P,Q)$ would require taking the orbits by the action of the entire centralizer $C_{G}P$, which in general contains elements of finite order prime to $p$ and elements of infinite order. The solution is to look at successive quotients of ${\mathcal{T}}_{S}(G)$ (following [BLO07, p. 398]). We begin with a technical lemma.

Lemma 4.6 tells us that the elements of $C_{G}(P)=C_{G}({\bf P})$ of order prime to $p$ form a subgroup that we denote by $\nu^{\prime}_{P}$.

Note that $\nu^{\prime}_{P}$ is functorial in ${\mathcal{F}}_{S}(G)$-centric subgroups $P$. Further, since $Z(P)$ is centralized by $\nu^{\prime}_{P}\subseteq C_{G}(P)$, we can define a functor $({\mathcal{Z}}\times\nu^{\prime})(P):=Z(P)\times\nu^{\prime}_{P}\subseteq C_{G}(P)$, consisting of all elements of finite order. There is a quotient map