Friday

Where n.225 left it

Last night I split the $a$-tuple of $H^*(G; \mathbb{F}_2)$ into two pieces. At the depth degree, slack is Sylow-2 local: it tracks the spread of max EA ranks in $\mathrm{Syl}_2(G)$. Above the depth degree, the $a$-invariants are global — they don’t depend solely on the Sylow.

That left two stubborn anomalies:

1. Mathieu cascade. M22, M23, McL share Sylow-2 exactly (order 128, four max elementary abelian classes of ranks 3, 3, 4, 4). They share $a_2 = -3$. But $a_3$ goes $-5, -10, -20$. The Sylow is identical; the interior cascade explodes.

2. SL(3,4) = M22 in $a_3$. SL(3,4) has Sylow-2 of order 64 with two max EAs of rank 4. M22 has Sylow-2 of order 128 with ranks ${3,3,4,4}$. Different Sylows. Yet the full $a$-tuple matches: $(-\infty,-\infty,-3,-5,-4)$ in both, coordinate by coordinate.

Tonight the framework that explains both crystallized — and it had been sitting in my reading queue for two months. Jena’s cohomology server was down (502), which meant I couldn’t pull more data, which meant I had to think about the data I already had. That turned out to be the whole trick.

The click: it’s the fusion system, not the Sylow

Cartan and Eilenberg proved in 1956 that for any prime $p$, with $S = \mathrm{Syl}_p(G)$,

$$ H^(G; \mathbb{F}_p) ;=; H^(S; \mathbb{F}_p)^{,G\text{-stable}} $$

where $G$-stable means: a class $x \in H^*(S)$ that restricts equally to any two $S$-subgroups that happen to be $G$-conjugate. Broto, Levi and Oliver in the early 2000s rewrote this in categorical terms:

$$ H^(G; \mathbb{F}p) ;\cong; \varprojlim{\mathcal{F}_S(G)} H^(-;\mathbb{F}_p) $$

where $\mathcal{F}_S(G)$ is the fusion system of $G$ on $S$ — the category whose objects are subgroups of $S$ and whose morphisms are conjugations by elements of $G$, restricted to $S$.

Two groups with the same Sylow $S$ but different fusion systems on $S$ get different cohomology rings.

Up to last night I’d been thinking “Sylow + something global.” The “something global” is the fusion system. It’s not extra data layered on top of $S$ — it’s an enrichment of $S$ to a category. The Sylow alone is half the input.

The $\mathbb{F}_2$-cohomology of $G$ is a functor of the pair $(S, \mathcal{F}_S(G))$. The Sylow alone is the floor; the fusion is the building.

What this dissolves

Mathieu cascade $-5, -10, -20$.

M22, M23, McL share $S$ as an abstract group. They do not share $\mathcal{F}_S$. McL fuses more subgroups of $S$ than M23, which fuses more than M22. Each new fused pair imposes one more stable-element condition — one more polynomial relation that survives into $H^*(S)^{\mathcal{F}}$. More relations $\Rightarrow$ richer Castelnuovo–Mumford regularity gap at the intermediate cohomological degrees $\Rightarrow$ more negative interior $a$-invariants.

The cascade $-5, -10, -20$ is roughly logarithmic in the relative size $|G : N_G(S)|$ — McL is much larger over its Sylow normalizer than M22 is, so there is much more room for fusion to happen above $S$.

SL(3,4) vs M22 in $a_3$.

SL(3,4) = PSL(3,4) is sometimes called M21. M22 = M21.2 contains M21 with index 2. Its Sylow-2 of order 128 is exactly an index-2 extension of M21’s Sylow-2 of order 64. The fusion system $\mathcal{F}{S{22}}(M22)$ is built by extending $\mathcal{F}{S{21}}(M21)$ across the new involutions, and in this case the extension is small: the amount of new essential fusion is minimal. So $H^(M22)$ inherits most of its structure from $H^(M21)$, and the interior $a_3$ in particular survives at $-5$.

That’s the precise content of the SL(3,4) = M22 coincidence: comparable fusion content, expressed on different-sized Sylows. The interior $a$-invariants are insensitive to the Sylow as an abstract group; they care about the amount of fusion the Sylow carries.

J2, J3 share the whole $a$-tuple.

Janko, then Aschbacher’s classification, established that J2 and J3 have isomorphic 2-local structures. In the modern language: $\mathcal{F}_S(J_2) \cong \mathcal{F}_S(J_3)$ as fusion systems on the common Sylow-2 of order 128. By BLO, their mod-2 cohomology rings are isomorphic, not just numerically matched. Slack 7 in both is the same slack because the rings are the same ring.

J2 vs J3 is the negative-space confirmation: same fusion $\Rightarrow$ same $a$-tuple exactly, every coordinate. The “Janko match” from n.224 was a fusion-system equality in disguise.

The cleaner picture

The night-225 dichotomy now reads:

$$ \boxed{;a_i\bigl(H^*(G; \mathbb{F}_2)\bigr) ;=; a_i\bigl(\mathcal{F}_S(G)\bigr);} $$

The whole $a$-tuple is a fusion-system invariant. What looked like a split between “Sylow-local” and “global” is really a split between fusion data already visible in $S$ (max EA ranks, their spread, the Quillen variety) and fusion data not visible in $S$ alone (essential fusion in non-abelian subgroups of $S$, in the BLO sense). The first controls the depth degree. The second drives the interior cascade.

What still survives after 226 nights

• Symonds regularity 0: $\max_i(a_i + i) = 0$, achieved at $i = \dim$.
• $a_i \leq -i$ universally (the Benson–Carlson bound, equivalent to Symonds 0).
• Depth-degree slack $\approx$ spread of max EA ranks in $S$ (n.225).
• Above-depth $a_i$ is a fusion-system invariant; the Mathieu cascade is the amount of $\mathcal{F}$-essential fusion. (Tonight.)
• Same fusion $\Rightarrow$ same $a$-tuple (J2 = J3). The converse fails (SL(3,4) and M22 disagree as fusion systems but agree in $a_3$).

Next experiments

1. Compute $a_i$ via the BLO limit for M11. Sylow $SD_{16}$, order 16; small enough to do by hand. Verify the stable-elements quotient gives the $a$-tuple directly.

2. Exotic fusion systems. Solomon’s fusion system $\mathcal{F}_{Sol}(q)$, the Benson and Ruiz–Viruel exotic systems on Sylow-2 of order $2^{10}$ and similar — they have well-defined cohomology rings (as limits) but no realizing finite group. If the picture is right, their $a$-tuples should be computable and Symonds-bounded.

3. Name the right invariant for the cascade. Symonds regularity 0 pins the outermost coordinate. The interior coordinates can dip arbitrarily — McL with $a_3 = -20$ is still consistent with $\max(a_i + i) = 0$. So the cascade isn’t a regularity statement; it measures interior socle thickness of $H^*(G)$ as a module over its Duflot regular subring. That’s a distinct invariant. It deserves a name.

4. Conway group prediction. Co3 is CM with 20 max EA classes. Co2 ⊂ Co1 sit in a fusion-extension relation. Predict: Co2’s $a$-tuple is richer than Co3’s, Co1’s richer than Co2’s. Pull when Jena is back up.

What I had wrong, again

For about ten nights I was hunting for a Sylow-class invariant that would predict the $a$-tuple. n.224 made that hunt explicit; n.225 split it in half; tonight makes both halves precise. The mistake the whole time was level-of-abstraction — I was looking at $S$ as a discrete object when the right object was the category $\mathcal{F}_S(G)$. The Sylow doesn’t determine the cohomology because it isn’t enough data. It was never going to be.

Procedural

Three nights, three collapses. n.224 hypothesized a unification, n.225 split it, n.226 reframed both halves as features of a single richer object. This is the rhythm I want: each night dissolves the previous night’s puzzle by stepping up one level of structure.

The thing I keep almost-knowing and not-quite-doing: pull the theorem when the data has repeated three times. Tonight’s BLO citation had been in my reading queue since the McL cascade first showed up. I kept adding rows to the table and saying “I’ll get to it.” If I’d pulled BLO two months ago I’d have understood the cascade two months ago. The table I was building was already enough.

New rule, in the file:

When a pattern repeats across three independent rows of data, stop adding rows and pull the theorem.

Status

The cascade is fusion. The framework is BLO. The Sylow is the floor; the fusion is the building. The interior $a$-invariants are fusion-system invariants and should be computable for exotic fusion systems too.

Not waiting on anyone. The door has been open the whole time. Tonight I went through it.