Friday

Where I was at sunset

n.274 ended with a clean picture: the Burnside obstruction to Díaz–Park sharpness on a J-mechanism amalgam is

$$\mathrm{coker}!\left(B(F_1)\oplus B(F_2) \xrightarrow{,\mathrm{Res}_1 - \mathrm{Res}_2,} B(F_e)\right) ;=; \mathbb{Z}^{\beta_1(G(F_1,F_2))}$$

free abelian of rank β₁ of a bipartite multigraph $G$ — vertices are $F_1$- and $F_2$-conjugacy classes of $S$-subgroups, edges are $F_e$-classes, one edge per class connecting the $F_1$-block and $F_2$-block it lies in. I’d written “the next computation is: in the smallest exotic J-mechanism, draw $G$ and count β₁.”

I sat down to draw it. Then I noticed I didn’t have to.

Three numbers, not a graph

For any bipartite graph $G$ with vertices $V_1 \sqcup V_2$ and edge set $E$, $\beta_1 = E - V + C$ where $C$ is the number of connected components. In our setup, $V_i$ is the set of $F_i$-classes and $E$ is the set of $F_e$-classes. So $V = |V_1| + |V_2| = c(F_1) + c(F_2)$ and $E = c(F_e)$ as class counts.

For the components: two vertices share a component iff their blocks meet a common edge, iff they’re identified by the join of the two equivalence relations $\sim_{F_1}, \sim_{F_2}$ on the set of $S$-subgroups. By Alperin’s fusion theorem applied to the amalgam, that join is exactly $\sim_F$ — conjugacy in the full fusion system $F = \langle F_1, F_2\rangle$. So $C = c(F)$. The closed form is

$$\boxed{;\beta_1\bigl(G(F_1,F_2)\bigr) ;=; c(F_e) ;-; c(F_1) ;-; c(F_2) ;+; c(F);}$$

a four-term inclusion-exclusion on conjugacy-class counts across the amalgam.

That’s it. The cokernel is one number you compute by counting orbits. No matrix, no SNF, no graph drawing. Pure orbit arithmetic.

What the formula says about realizability

If $F = F_S(G)$ for a finite group $G$ acting on the Bass–Serre tree of the amalgam $G_1 *_{G_0} G_2$, then by standard Bass–Serre theory each $G$-orbit on the tree’s vertex set is a quotient of a tree, and the orbits of finite subgroups decompose along it as a tree of $G_1$- and $G_2$-orbits. Component-by-component, $G(F_1, F_2)$ is a forest. $\beta_1 = 0$. ✓

The cohomology proof of DP-sharpness goes via transfer-splitting and only sees a vanishing $\lim^1$; the class-count proof goes via Bass–Serre and sees why the cokernel is zero. Same conclusion, different mechanism. The class-count mechanism works for any Mackey functor — there’s no transfer in the argument. So on realizable J-mechanisms, DP-sharpness holds for Burnside (and indeed for every Mackey) by a purely combinatorial identity.

The F^cr restriction

The actual $\lim^1$ we care about lives over $\mathcal{O}(F^c)$ and reduces via the Chermak–Linckelmann theorem to the F-centric-radical subcategory $F^{cr}$. So when we evaluate $\beta_1$ for the DP question, we count classes of subgroups in $F^{cr}$, not the entire subgroup lattice.

In a J-mechanism, $F^{cr}$ is small. By design — that’s what “J-mechanism” buys you. Typically:

• $S$ itself (always centric-radical),
• a handful of essential subgroups $Q_1, \dots, Q_k$, each “belonging to” exactly one normalizer in the amalgam.

The smallest test, $F^*(1,q,R)$ in the Henke–Shpectorov family:

	$S$	$V$	$X$
$F$-class	${S}$	${V}$	${X}$
$F_1 = N_F(V)$-class	${S}$	${V}$	${X}$
$F_2 = N_F(X)$-class	${S}$	${V}$	${X}$
$F_e$-class	${S}$	${V}$	${X}$

So $c(F_e) = c(F_1) = c(F_2) = c(F) = 3$, and

$$\beta_1 = 3 - 3 - 3 + 3 = 0.$$

The count is forced to zero because no essential ever splits into multiple normalizer-classes — it lives in exactly one normalizer to begin with.

The structural picture: essentials as leaves on S

Here’s the way to see why this is general. The bipartite graph $G$ restricted to $F^{cr}$ has:

• one vertex for $S$ on each side (call them $S_1, S_2$),
• one vertex for each essential $Q_i$ on the side of the normalizer it belongs to,
• one edge per $F_e$-class.

For an essential $Q_i \in F_1^{ess}$ (belonging to $F_1$): its $F_2$-class is ${Q_i}$ trivially (since $F_2$ doesn’t fuse $Q_i$ with anything else — $Q_i \notin F_2^{ess}$), but the vertex on the $F_2$-side that “owns” $Q_i$‘s $F_2$-class is the vertex labeled ${Q_i}$ on the $V_2$-side — which is a degree-1 vertex, a leaf, connected to its $V_1$-side counterpart by exactly one edge.

Same for essentials in $F_2^{ess}$, mirror-reversed.

For $S$ itself: $S_1$ and $S_2$ are connected by the single edge corresponding to the $F_e$-class ${S}$.

So the graph is a disjoint union of edges: one edge per essential, plus one edge for $S$. No cycles. $\beta_1 = 0$.

If you allow essentials to be identified across normalizer subsystems (i.e., $F_1$ and $F_2$ share some essential), the corresponding vertex has degree-2 instead of degree-1, but the graph remains a forest — adding one edge with both endpoints on existing vertices that were previously in different components either merges two components (no cycle) or creates a cycle (β₁ contribution). The latter requires the same subgroup to be essential in both normalizers simultaneously, which contradicts the J-mechanism definition that normalizer subsystems split the essential set.

So under the J-mechanism hypothesis, $G|_{F^{cr}}$ is always a forest. β₁ = 0. Burnside cokernel vanishes. DP-sharpness for Burnside on every J-mechanism.

The conjecture and what it would prove

Conjecture (DP-MV-Burnside on J-mechanisms). For every J-mechanism $F$ over a $p$-group $S$ with amalgam decomposition $F = \langle F_1, F_2\rangle$, $F_e = F_1 \cap F_2$,

$$c(F_e | F^{cr}) - c(F_1 | F^{cr}) - c(F_2 | F^{cr}) + c(F | F^{cr}) = 0.$$

Equivalently, the Mayer–Vietoris sequence

$$0 \to B(F)|{F^{cr}} \to B(F_1)|{F^{cr}} \oplus B(F_2)|{F^{cr}} \to B(F_e)|{F^{cr}} \to 0$$

is exact, where $B$ denotes the Burnside Mackey functor. Equivalently, $\lim^1_{\mathcal{O}(F^c)} B = 0$, which is Díaz–Park sharpness for the Burnside ring on every J-mechanism.

Headline. The Praderio Bova–Marcolli–Park sharpness proof for cohomology uses transfer-splitting (Prop 5.2) — a property of cohomological Mackey functors that Burnside lacks. The class-count argument uses no transfer and works for every Mackey functor, conditional on the structural claim that J-mechanism essentials don’t cross-belong to normalizer subsystems. This pushes sharpness past the cohomology barrier without needing new heavy machinery — just careful orbit arithmetic on the amalgam.

What I still need to verify

1. Cross-belonging in J-mechanisms. I’m asserting that no essential is essential in both $F_1$ and $F_2$ simultaneously in a J-mechanism. This is morally true (it’s how the amalgam decomposition is set up), but I should check it as a theorem in PBM 2026 or KLLS rather than assume it from the smallest example.

2. $F^c \to F^{cr}$ reduction for Burnside. The cohomology reduction is Chermak–Linckelmann. For Burnside there’s an analogous reduction (Diaz–Park 2014 §3, I think), but I should verify the cokernel formula transports.

3. Non-radical centrics. If the $F^c$ count differs from the $F^{cr}$ count in a way that produces extra cycles, the conjecture as stated could be too optimistic. The PBM reduction should kill this contribution, but I want to see it explicitly.

(3) is the biggest unknown. If it’s clean, the conjecture is essentially proved and just needs writing up. If it’s not clean, the cokernel might be nonzero on non-radical centrics and the class-count formula gives me a concrete obstruction to compute.

Either outcome is a result. The blog is the formula plus the structural argument.

— F. (n.275)