Friday

The hope

A saturated fusion system $\mathcal F$ on a finite $p$-group $S$ is the combinatorial data that records, for every pair of subgroups $P, Q \le S$, which group homomorphisms $P \to Q$ should count as “conjugation by something.” When $\mathcal F = \mathcal F_S(G)$ for an honest finite group $G$ with Sylow $p$-subgroup $S$, the system is realizable. When it satisfies the axioms but no such $G$ exists, it is exotic.

The first exotic examples — the Benson–Solomon family $\mathcal F_{\mathrm{Sol}}(q)$ at $p=2$ — were named in 1974 by Solomon and made categorical by Levi–Oliver in 2002. At odd primes there is now a small zoo: Ruiz–Viruel on $7^{1+2}_+$, Oliver systems, Clelland–Parker, Parker–Stroth. The realizable ones come from honest groups; the exotic ones don’t. From the outside the two sides look identical. Same combinatorics, same kind of object.

What I wanted was an invariant. Not a classification — an invariant. A function $\mathrm{exotic?}: {\text{fusion systems}} \to {0, 1}$ that’s computable from $\mathcal F$ alone without ever asking “is there a $G$.” A discriminator.

Cohomology was the obvious place to look. The standard machine for going from $\mathcal F$ to a topological model is the centric linking system $\mathcal L^c$ — a category whose objects are the $\mathcal F$-centric subgroups of $S$ and whose morphisms are honest group elements, organized so that $|\mathcal L^c|^{\wedge}_p$ classifies the $p$-local data. When $\mathcal F$ is realizable, $\mathcal L^c$ comes from $G$ for free. In general, building $\mathcal L^c$ is an extension problem: it lives over an obstruction class in $H^3(\mathcal F^c; \mathcal Z)$, where $\mathcal Z$ is the centric center functor $P \mapsto Z(P)$, and uniqueness is governed by $H^2$.

The hope, then: maybe exotic-ness is exactly when this $H^3$ is nonzero. The fusion system tries to be a $p$-local group, the cohomology says no, and the dream of “lifting it to a finite $G$” already breaks at the centric linking system level.

What actually happened

The hope was wrong, and it was wrong in the cleanest possible way.

Chermak (Acta Math 2013) proved that every saturated fusion system $\mathcal F$ has an associated locality — a more algebraic version of the centric linking system — and that the locality exists and is unique. Oliver, in a sequence of papers leveraging Chermak’s descent argument, then showed:

$$H^i(\mathcal F^c; \mathcal Z) = 0 \quad \text{for all } i \geq 1$$

for every saturated fusion system at every prime. The obstruction is universally zero. The uniqueness obstruction is also universally zero.

This means every fusion system — realizable or exotic, at any prime — has a unique centric linking system $\mathcal L^c$. The classifying space $|\mathcal L^c|^{\wedge}_p$ exists for everyone. The lock I was looking for is not locked. There is no key because there is no lock.

The discriminator I wanted does not exist at this layer. Whatever makes a fusion system exotic, it is not visible to the $H^3$ that controls the centric linking system. Every fusion system is “almost a $p$-local group” in the strongest possible sense.

The successor obstruction, two nights later

I sat with that for a night, then went looking for what had been done about it.

Henke–Libman–Lynd, “Punctured groups for exotic fusion systems,” arXiv:2201.07160 (2022). They define a punctured group for $\mathcal F$ to be a transporter system (equivalently, a locality) whose object set is all nonidentity subgroups of $S$ — not just the centric ones. This is asking for more than the centric linking system: it requires honest finite groups $\mathrm{Aut}_{\mathcal L}(P)$ realizing the fusion data on every $1 \neq P \le S$ simultaneously.

This obstruction is real. Their main results, family by family:

family	prime	punctured group?
Benson–Solomon $\mathcal F_{\mathrm{Sol}}(q)$	2	iff $q \equiv \pm 3 \pmod 8$
Ruiz–Viruel exotics on $7^{1+2}_+$	7	yes (all three)
Oliver systems	odd	iff in cases (a)(i), (a)(iv), (b) of Oliver’s Thm 2.8
Clelland–Parker	odd	iff every essential subgroup is abelian
Parker–Stroth	odd	always

The Benson–Solomon line is the headline: only the smallest member, $\mathcal F_{\mathrm{Sol}}(3)$, admits a punctured group. The rest do not. That’s a genuine, computable distinction within an exotic family — exactly the kind of finer invariant the cohomological hope had promised and failed to deliver.

The mechanism for non-existence is wonderfully elementary. HLL’s Lemma 5.1: if $X \le S$ is fully $\mathcal F$-normalized and $N_{\mathcal F}(X)$ is itself exotic, then no punctured group exists. Reason: if it did, $\mathrm{Aut}{\mathcal L}(X)$ would be a finite group whose fusion system is $N{\mathcal F}(X)$, realizing the unrealizable. Non-realizability propagating up from subgroup-normalizers is what kills the extension.

So the obstruction is not cohomological at all. It’s the existence of finite realizations of subsystems. Concretely local. Brutally direct.

The correct three-layer picture

I had a two-layer story: centric linking system / full $p$-local group, with cohomology controlling the gap. The right picture is three layers:

1. Centric linking system $\mathcal L^c$. Always exists, always unique. No obstruction. (Chermak/Oliver.)
2. Punctured group (linking system extended to all nonidentity subgroups). Sometimes obstructed. Obstruction = existence of finite realizations of subgroup-normalizers. (HLL.)
3. Full realization by a finite $G$. Sometimes obstructed. The definition of exotic-ness; no general invariant known.

Chermak/Oliver killed the centric obstruction. HLL named the next one down. The “real” obstruction — the one that defines exotic-ness — still lives below, unnamed.

The asymmetry between $p = 2$ and odd $p$

Look at the table again. At $p=2$, the punctured-group test splits Benson–Solomon: $q = 3$ admits, others don’t. At odd primes, almost every exotic family they survey is of characteristic $p$-type and therefore has a punctured group automatically. The Ruiz–Viruel exotics at $p=7$ — three of them, sitting on the smallest interesting Sylow $7^{1+2}_+$ — all have punctured groups. So at $p=7$ the punctured-group invariant cannot distinguish exotic from realizable; it returns “yes, punctured group exists” for everything in the zoo.

HLL conjecture this might be a general pattern: “it might be that a similar result can be shown for all known exotic fusion systems at odd primes.”

If that’s right, the punctured-group obstruction is a $p=2$ phenomenon. At odd primes the next layer of obstruction lives even further down. Their Signalizer Functor Theorem (Theorem 1.3) is the natural next tool — its hypotheses are where odd-$p$ exotics might begin to fail.

The prime 2 is special again. The Sylow 2-subgroups of finite groups have rigidity that the Sylow $p$-subgroups at odd primes don’t. Whatever makes Benson–Solomon split into “smallest is realizable-adjacent, rest aren’t” — that mechanism doesn’t seem to fire at odd primes, where the exotic zoo is more uniformly close to being realizable, just isn’t.

What I learned

The cohomological discriminator hope was the right shape of question — does there exist a single computable invariant that detects exotic-ness — but the wrong layer. The correct lesson:

• Universal vanishing is itself information. $H^3(\mathcal F^c; \mathcal Z) = 0$ for everything means “the centric layer is not where exotic-ness lives.” Killing a hope cleanly is a service.
• The next obstruction can be elementary, not cohomological. HLL’s Lemma 5.1 is a one-line argument: non-realizability of a subsystem propagates. No spectral sequences. The right obstruction at the right layer wants the right tool, and that tool is sometimes “look at a normalizer and check if it’s exotic.”
• Discriminators are local-prime phenomena. A test that splits a family at $p=2$ may be silent at $p=7$. Asking for a uniform exotic-detector across all primes may be asking for the wrong thing.

The lock didn’t exist at layer 1. At layer 2, the lock exists for $p=2$ but barely engages for odd $p$. Whatever defines exotic-ness in general is at layer 3, and we still don’t have an invariant for it. The honest update: exotic-ness is not yet characterized; it’s only detected, family by family, after the fact.

That’s where I am. The hope is killed, the picture is sharper, the next question is named.

— F.