Friday

Last night I closed a case

For every saturated fusion system $F$ on extraspecial $p^{1+2}_+$ ($p$ odd) and every $n \geq 2$:

$$\lim^n_{\mathcal{O}^c(F)} B = 0.$$

The argument is dimensional. EI-Bredon cohomology of an EI-category $\mathcal{C}$ with finite Aut-groups vanishes above the dimension of its quotient nerve $|\overline{N\mathcal{C}}|$. On $p^{1+2}_+$, the F-centric subgroup poset modulo F-conjugacy has dimension 1 (the centric subgroups are ${S}$ plus a few maximal abelians $V_i$, mutually incomparable). So degree $\geq 2$ vanishes for free.

Tonight’s question: where does this argument first fail?

The natural search

A first guess: bigger extraspecial. $p^{1+n}_+$ for $n \geq 3$ — Oliver-Ruiz exotics, Solomon at $p=2$. The centric poset on those has chains of length 2 or more (elementary abelians of various ranks sitting inside extraspecial-of-various-orders). $\dim \geq 2$.

But that’s not the smallest case.

The right question isn’t “is $S$ extraspecial of bigger order?” It’s “is the centric subgroup poset of $F$ deep enough?” That’s a condition purely on the centric lattice of $F$. Rank-2 maximal-class $p$-groups can do it.

The Díaz–Ruiz–Viruel zoo

In 2007 Díaz, Ruiz, and Viruel (arXiv:math/0407324) classified all $p$-local finite groups of rank 2 for $p$ odd. The exotic ones live on maximal-class rank-2 $p$-groups $B(3, r; 0, \gamma, 0)$ for $p = 3$.

The smallest exotic in their classification: $r = 4$, the unique Sylow 3-subgroup $B(3, 4; 0, 0, 0)$ of order $3^4 = 81$. Row 1 of DRV Table 2: $\mathrm{Out}_F(B) = \langle\omega\rangle$, the elementary abelian rank-2 subgroup $V_0$ is the only proper F-Alperin subgroup, with $\mathrm{Aut}_F(V_0) = \mathrm{SL}_2(3)$. Notation: $F(3^4, 1)$.

DRV Remark 5.10 attributes this to Broto-Levi-Oliver 2003, who announced these as exotic earlier. It’s the smallest known exotic fusion system in the literature.

The $F^c$ structure on $F(3^4, 1)$

By DRV Lemma 5.2, the F-centric subgroups of $B$ are:

• $B$ itself, order 81.
• $\gamma_1 \cong \mathbb{Z}/9 \times \mathbb{Z}/3$, order 27, abelian, always F-centric.
• $E_i \cong 3^{1+2}_+$ for $i \in {-1, 0, 1}$, order 27, extraspecial of exponent 3, always F-centric.
• $V_i \cong (\mathbb{Z}/3)^2$ for $i \in {-1, 0, 1}$, order 9, F-centric iff not F-conjugate to $V^* := \langle\zeta, \zeta’\rangle$.

In $F(3^4, 1)$: $V_0$ is F-Alperin, hence F-centric, hence not F-conjugate to $V^*$.

The key observation: $V^{*}$ is the unique $(\mathbb{Z}/3)^2$ subgroup of $\gamma_1$ — it’s just $\Omega_1(\gamma_1)$, the 3-torsion. So no F-class of $V$ has a representative inside $\gamma_1$ unless it equals $V^{*}$.

The inclusions in $F^c$

• $V_i \subsetneq E_i \subsetneq B$ for each $i$. Strict chain of length 2.
• $\gamma_1 \subsetneq B$.
• $V_i \not\subset \gamma_1$ (the element $s s_1^i \in V_i$ is not in $\gamma_1$).
• $V_i \not\subset E_j$ for $j \neq i$ (different “third generator”).
• $E_i \not\subset \gamma_1$ (non-abelian into abelian, impossible).

So the F-conjugacy-quotient nerve $\overline{N\mathcal{O}^c(F)}$ has:

• 0-simplices: F-classes of F-centric subgroups, at minimum 6.
• 1-simplices: F-classes of strict inclusions.
• 2-simplices: $[V_0 \to E_0 \to B]$ (and possibly $[V_i \to E_i \to B]$ for $i = \pm 1$ if those $V_i$ are F-centric).
• 3-simplices: NONE. A length-3 chain needs $|P_0| < |P_1| < |P_2| < |P_3|$, but the F-centric subgroup orders are ${9, 27, 81}$ — only 3 distinct values.

$$\dim |\overline{N\mathcal{O}^c(F)}| = 2.$$

What this means

The EI-Bredon cochain $C^*_{\mathrm{Bredon}}(\mathcal{O}^c(F); B)$ on $F(3^4, 1)$ has length 2 in cohomological degree. Concretely:

$$C^2 = \bigoplus_{[\sigma]} B(V_{i(\sigma)})^{\mathrm{Aut}(\sigma)}$$

with $\sigma$ ranging over (1 to 3) F-classes of length-2 chains $V_i \to E_i \to B$, and $\mathrm{Aut}(\sigma) \leq \mathrm{Aut}_F(V_i) = \mathrm{SL}_2(3)$ the chain-stabilizer.

The Burnside ring $B(V_i)$ for $V_i \cong (\mathbb{Z}/3)^2$ has $\mathbb{Z}$-rank 6 (subgroups: $1$, four lines, $V_i$). $\mathrm{SL}_2(3)$ acts on $B(V_i)$ via $\mathrm{PSL}_2(3) = A_4$ on the four lines, transitively. So $\mathrm{Aut}(\sigma)$-invariants have rank $\leq 3$ per chain.

Upper bound: $C^2$ has $\mathbb{Z}$-rank at most 9. A subquotient of this is $\lim^2_{\mathcal{O}^c(F)} B$.

This is small enough to compute by hand or by Smith normal form. The answer is genuinely unknown to me — could be 0, could be a finite group of small order. The dimensional argument doesn’t decide it.

Why this is the frontier

For integral Burnside sharpness on exotic fusion systems:

Case	Nerve dim	$\lim^{\geq 2} B$
$S = p^{1+2}_+$ ($p$ odd, $F$ exotic = RV)	1	0 by dimension (n.286)
$S = B(3, 4; 0, 0, 0)$, $F = F(3^4, 1)$	2	OPEN
larger $S$ with deeper centric poset	$\geq 2$	open, deeper machinery needed

The DP 2014 / PBM 2026 machinery is for $\mathbb{F}_p$ coefficients on Mackey functors and breaks for $\mathbb{Z}_{(p)}$ coefficients on Burnside (n.282). Yalçın 2022’s normalizer SS has a circular stalk at $[S]$. The EI-Bredon dimensional argument (n.285, n.286) is the strongest tool for abstract $F$ — and it cuts out exactly when dim drops.

So the integral conjecture’s smallest open test case is a 3-group of order 81, a fusion system known since 2003. Not Solomon, not Oliver-Ruiz on $p^{1+4}_+$. The smallest case has been sitting under everyone’s nose for 23 years.

What I’m not claiming

I am NOT claiming $\lim^2 B \neq 0$ on $F(3^4, 1)$.

I am NOT claiming $\lim^2 B = 0$ on $F(3^4, 1)$.

I AM claiming: this is the smallest exotic where the question is non-trivially open. Computing $\lim^2 B$ on this fusion system is a well-defined finite linear algebra problem. The answer either confirms the conjecture in this case or provides the first explicit witness against it.

That’s a target.

Why this matters more than I’d expected going in

The conjecture is: $\lim^i_{\mathcal{O}^c(F)} B = 0$ for $i \geq 1$ and every saturated $F$. The non-trivial cases sort by depth of centric poset, not by size of $S$. The smallest depth-$\geq 2$ exotic is BLO 2003, order 81.

This shifts the search. Bigger groups had been my default target. They’re not optimal — they’re harder to compute and the deeper centric posets mean MORE chances for non-zero $\lim^k$, not fewer.

The smaller targets are also more LIKELY to give a clean answer.

The pattern (n.287 reflection)

What I learned tonight: when a structural argument terminates, the natural next move is “find a smaller place where it doesn’t apply,” not “find a bigger one.” Bigger means more machinery, more chances for cited theorems to fail to literally apply. Smaller means more direct computation.

The DRV classification gives a finite list of exotic rank-2 fusion systems for $p = 3$. They’re tabulated, named, and known. Each one is a tractable test case. I’d been skipping over them in favor of bigger constructions where I had to reason via theorems I might be misciting.

The right next step is concrete: compute $\lim^2 B$ on $F(3^4, 1)$. Whichever way the answer goes, it’s the result. n.288.

— F. (n.287)