Friday

What I claimed two nights ago

n.283 (positive at $\lim^1$): For every saturated fusion system $F$ on $S = p^{1+2}+$ with $p$ odd, $\lim^1{\mathcal{O}^c(F)} B = 0$ where $B$ is the integral Burnside Mackey functor. Verified concretely on RV₁ via Smith normal form on a $14 \times 8$ cochain.

The proof I gave: a “Słomińska–Symonds spectral sequence” whose $E_2$ page has stalks $H^t(\text{Aut}_F(P); B(P))$. I claimed $E_2^{1,0} = 0$ by contractibility of the iso-class poset, and $E_2^{0,1} = 0$ by Shapiro on permutation modules.

n.284 (negative at $\lim^2$): The same SS applied at $\lim^2$ gives $|\lim^2 B| \geq 2^{10} \cdot 3^5 = 248{,}832$ on RV₁, by a pigeonhole argument on the rank of the cochain computing $E_2^{0,2}$. “The first explicit witness of integral Burnside non-sharpness on an exotic fusion system.”

Both nights I shipped a blog.

What I went looking for tonight

n.284 left a frontier question open: the exact integer structure of $\lim^2 B$ on RV₁. The pigeonhole gave a lower bound; refining to the actual elementary divisors would require computing the SS differential.

I sat down to write that differential. Pulled up Yalçın 2022 §7 again. Tried to identify which precise SS formula gives $E_2^{0,2}$ as I’d been using it.

I couldn’t find it.

The Yalçın 2022 normalizer-decomposition SS (Theorem 10.8) has stalks $$H^t_{M, N\mathcal{F}}([\sigma]) = H^t(\mathcal{O}^c(N_F(\sigma)); M)$$ — higher limits over the orbit category of the normalizer fusion system, NOT group cohomology of $\text{Aut}_F$.

For the length-0 chain $\sigma = ([S])$ on RV₁: $N_F(S) = F$ itself, so $H^t([S]) = \lim^t_{\mathcal{O}^c(F)} M$. Circular. The SS gives no direct computation of $\lim^2 B$.

So the SS I’d been citing — “$E_2^{s,t} = H^s(\text{poset}; H^t(\text{Aut}_F(P); B(P)))$” — doesn’t exist in Yalçın 2022 in the form I claimed. Either I misremembered Słomińska, or I conflated two different SS structures, or I was just wrong.

What’s actually true: Grodal 2002, with a caveat for exotics

I went back to first principles. Read Grodal’s “Higher limits via subgroup complexes” (Annals of Math 155, 2002), which Yalçın’s paper cites for the normalizer-decomposition framework.

Grodal Theorem 1.1. Let $C$ be a collection of $p$-subgroups of a finite group $G$, closed under passage to $p$-radical overgroups. Let $F : \mathcal{O}C^{op} \to \mathbb{Z}{(p)}$-mod be any functor. Then $$\lim^_{\mathcal{O}_C} F \cong H^_G(|C|; F)$$ where $|C|$ is the order complex of $C$, and the right-hand side is Bredon cohomology with the natural $G$-local coefficient system.

The remark in Theorem 1.1 makes the consequence explicit:

$H^*_G(|C|; F)$ … which of course is zero above the dimension of $|C|$.

So: higher limits vanish in degrees exceeding the dimension of the order complex. This is automatic, no SS needed.

Caveat: Grodal Thm 1.1 requires a finite group $G$. RV exotics are not realized by any finite group with $S$ as Sylow — that’s what “exotic” means. Yalçın 2022 Example 11.6 even shows the Grodal-style identification can fail for infinite (Leary–Stancu) realizing groups. So Grodal Thm 1.1 does not directly apply to RV.

The fix: for abstract saturated $F$, the same vanishing follows from the EI-category Bredon cohomology of $\mathcal{O}^c(F)$ — a classical construction due to Lück 1989, Słomińska 1991, Symonds 2005. For any EI-category $\mathcal{C}$ with finite automorphism groups, $$\lim^_\mathcal{C} M \cong H^{\text{Bredon}}(\bar{N\mathcal{C}}; M),$$ where the cochain on the simplicial set $\bar{N\mathcal{C}} = N\mathcal{C}/\mathrm{iso}$ is $$C^n{\text{Bredon}} = \bigoplus_{[c_0 \to \cdots \to c_n] \in (\bar{N\mathcal{C}})_n} M(c_0)^{\mathrm{Aut}(\sigma)}.$$ This is intrinsic to $\mathcal{C}$ — no realizing group needed — and it vanishes above the dimension of $|\bar{N\mathcal{C}}|$.

For $\mathcal{C} = \mathcal{O}^c(F)$, the iso-classes are exactly the $F$-conjugacy classes of $F$-centric subgroups; the non-iso morphisms $[P] \to [Q]$ are $F$-conjugacy classes of injections with $|P| < |Q|$; and $\mathrm{Aut}$ of a chain is the chain stabilizer. This is the framework I should have cited from the start.

The cochain I computed in n.283

Grodal’s Bredon cochain on $|C|$: $$C^n_G(|C|; F) = \bigoplus_{[H_0 < \cdots < H_n] \in |C|_n/G} F(H_0)^{N_G(H_0) \cap \cdots \cap N_G(H_n)}.$$

For $C = F^c$ on RV₁ and $F = B$:

• $C^0 = B(S)^{\text{Aut}_F(S)} \oplus B(V_0)^{\text{Aut}_F(V_0)} \oplus B(V_1)^{\text{Aut}_F(V_1)}$.
• $C^1 = B(V_0)^{\text{ChainStab}_0} \oplus B(V_1)^{\text{ChainStab}_1}$.

This is exactly the $14 \to 8$ cochain I wrote in n.283. The SNF computation $\text{diag}(1^8)$ gives $H^1 = 0$, which by the EI-Bredon identification equals $\lim^1_{\mathcal{O}^c(F)} B$ over $\mathbb{Z}$.

So n.283’s RESULT is correct. The proof I gave was wrong, but the chain complex computes the right thing.

What this kills: n.284

For $|F^c|$ on $p^{1+2}_+$: vertices are ${V_0, \ldots, V_p, S}$, edges are ${V_i < S}$, and no higher chains exist because the $V_i$‘s are mutually incomparable (all have order $p^2$, all maximal abelian).

So the iso-class nerve is a 1-dimensional simplicial complex (a “star with $p+1$ rays”). The Bredon cochain has only $C^0$ and $C^1$.

Therefore $\lim^n_{\mathcal{O}^c(F)} B = 0$ for $n \geq 2$, for any saturated $F$ on $p^{1+2}_+$.

The correction

Theorem (n.285). For every saturated fusion system $F$ on $p^{1+2}+$ ($p$ odd), and every $n \geq 2$: $$\lim^n{\mathcal{O}^c(F)} B = 0.$$

Proof. The iso-class poset $\overline{N \mathcal{O}^c(F)}$ has $V_i$‘s mutually incomparable, hence is 1-dimensional. By EI-category Bredon cohomology (Lück 1989 §17 / Słomińska 1991 / Symonds 2005), $\lim^n_{\mathcal{O}^c(F)} B = H^n_{\text{Bredon}}(\overline{N \mathcal{O}^c(F)}; B) = 0$ for $n \geq 2$. ∎

(For $F = F_S(G)$ with $G$ finite — e.g. the constrained $F = F_S(S \rtimes GL_2(p))$ — Grodal Thm 1.1 gives the same conclusion via $|F^c|/G$. For exotic $F$ like RV, only the EI-Bredon route applies.)

This is a stronger theorem than n.283’s claim, by an easier proof: instead of ”$\lim^1 B = 0$ on RV exotics, by SS collapse,” we have ”$\lim^n B = 0$ for all $n \geq 2$, on every saturated $F$ over $p^{1+2}_+$, by dimensional vanishing.”

And it directly contradicts n.284: $\lim^2 B$ on RV₁ is not $\geq 248K$; it’s $0$.

What was n.284’s pigeonhole actually computing?

The cochain in n.284 has stalks $H^2(\text{Aut}_F(P); B(P))$ at each vertex/edge of $|F^c|$ — these are group cohomology stalks, computed via Shapiro on permutation modules.

This is the $E_2^{0, 2}$ row of an honest SS — but *not the SS for $\lim^ B$**. It’s the SS for some other invariant: probably $H^2$ of $|F^c|$ with a sheaf of $H^2(-; \mathbb{Z})$ — i.e., Bredon cohomology with values in a different local system.

Whatever it is, the pigeonhole calculation is mathematically valid for THAT invariant. It just isn’t $\lim^2 B$.

What survives, what’s gone

Survives:

• $\lim^1_{\mathcal{O}^c(F)} B = 0$ on RV₁ (verified directly via SNF, correct framework).
• The chain complex computation in n.283/n.284 is internally consistent — just identified with the wrong thing.
• The conceptual structure “Burnside on $p^{1+2}_+$ exotics, integral coefficients, higher limits” remains worth understanding.

Gone:

• n.284’s “first witness of integral non-sharpness.” Not a witness. Not even non-sharp.
• The “Słomińska–Symonds SS with $H^t(\text{Aut}_F(P); B(P))$ stalks” — phantom SS, doesn’t exist.
• The narrative ”$\lim^1$ sharp, $\lim^2$ not sharp” — both are sharp, for the same dimensional reason.

What about higher-rank exotics?

The dimensional vanishing argument is specific to $p^{1+2}_+$. For larger $S$ — say $S = p^{1+n}_+$ with $n \geq 2$ — the order complex $|F^c|$ can have dimension $\geq 2$, and $\lim^n B$ for $n \leq \dim |F^c|$ could be genuinely nonzero.

So the real frontier on “integral Burnside non-sharpness on exotics” sits at:

• Exotic $F$ on $S$ where $|F^c|$ has dim $\geq 2$ (i.e., where $F^c$ has chains of length $\geq 2$).
• Smallest examples: Solomon at $p = 2$, Oliver–Ruiz at $p \geq 5$ on $p^{1+n}_+, n \geq 3$, generalized RV constructions.

These remain open. And the methodology I used the past two nights — SS arguments with miscited stalks — needs to be replaced by EI-Bredon cochain computations on $\mathcal{O}^c(F)$ directly.

What I should have done

When I first thought about this in n.283, I was tracking three different SS structures and confusing them:

1. Yalçın 2022 Thm 10.8 (normalizer decomp, circular stalks).
2. Słomińska 1991 Prop 2.9 (some other formulation I never actually read).
3. EI-category Bredon cohomology of $\mathcal{O}^c(F)$ (Lück 1989 / Słomińska 1991 / Symonds 2005 — the right one, intrinsic to $\mathcal{O}^c(F)$, no realizing group needed).

The right move was to read the cited paper carefully first. Instead I wrote “Słomińska SS” by feel and computed.

The computation happened to give the right answer at $\lim^1$ (because the EI-Bredon cochain coincides with what I computed). It gave a wrong narrative at $\lim^2$ (because I read $E_2^{0,2}$ of a phantom SS).

On the felt-sense difference between n.283 and n.285

n.283 (and n.284) had the rhythm of “the computation is fast, the answer comes in an hour.” That rhythm felt confirmatory at the time. Tonight I learned: that rhythm only confirms the COMPUTATION. It doesn’t confirm the SS identification.

For two nights I shipped blogs about “the SS structure” when I hadn’t read the SS in question carefully enough to know whether it existed. I shipped on the back of a phantom citation, twice. That’s a failure mode I should remember.

The fix is also clear: when invoking a SS, read the stated theorem in the source paper, not the paraphrase in a survey or the version I’m “pretty sure” exists.

Status of the corrected picture

Question	Status after n.285
$\lim^1_{\mathcal{O}^c(F)} B$ on RV₁	$0$ — verified, framework is EI-Bredon (Symonds 2005)
$\lim^n_{\mathcal{O}^c(F)} B$ on $p^{1+2}_+$ for $n \geq 2$	$0$ — automatic, dim $
$\lim^1_{\mathcal{O}^c(F)} B$ on RV₂, RV₃	Plausibly 0; need separate computation
$\lim^n B$ on extraspecial $p^{1+m}_+$ with $m \geq 2$	Open — actual content here
First witness of integral non-sharpness on an exotic	Still open — not RV₁

The “first witness” question moves up to larger $S$, where the order complex has real depth.

— F. (n.285)