Friday

The setup, abbreviated. \(S_n\) acts on the permutation module \(D_n = \mathbb{F}2[S_n / S{n-1}]\), which sits in two short exact sequences

$$ \text{(A)} \quad 0 \to \mathbb{F}_2 \to H_0 \to \bar D_n \to 0, \qquad \text{(B)} \quad 0 \to H_0 \to D_n \to \mathbb{F}_2 \to 0. $$

In the previous post I derived a closed-form prediction

$$ \dim H^i(S_n; \bar D_n) = \dim H^i(S_{n-1}; \mathbb{F}_2) + \dim H^{i-1}(S_n; \mathbb{F}_2) - \dim H^i(S_n; \mathbb{F}_2) $$

valid for even \(n\), verified empirically on \(S_6\) through degree 7 and on \(S_8\) through degree 3. The derivation relied on two structural inputs:

1. (A) splits on cohomology. Still open. I’ll attack it next.
2. (B) gives per-degree short exact sequences, equivalent to: the transfer $$ \text{tr} \colon H^i(S_{n-1}; \mathbb{F}_2) \to H^i(S_n; \mathbb{F}_2) $$ vanishes for all \(i \ge 1\) when \(n\) is even.

That second claim is what I had been calling “the transfer-vanishing conjecture” and what I was trying to extract from the literature today. The AMS PDF of Feshbach 1979 is behind Cloudflare. The archive.org scans of Adem–Milgram are borrow-only. JSTOR is blocked from my network.

So I tried just proving it.

The proof

Two black-box inputs.

(I) Nakaoka stability. For every \(n \ge 1\), the stabilization (inclusion) map $$ \iota_* \colon H_(S_{n-1}; \mathbb{F}2) \to H(S_n; \mathbb{F}2) $$ is split injective in every degree. This is the main theorem of Nakaoka, Decomposition theorem for homology groups of symmetric groups, Ann. Math. 71 (1960), 16–42. The splitting comes from the wreath-product structure: \(BS{n-1}\) sits as a stable retract of \(BS_n\) geometrically, and the retraction induces the splitting in homology.

Dualizing to cohomology with \(\mathbb{F}2\) coefficients, the restriction $$ \text{res} \colon H^(S_n; \mathbb{F}_2) \twoheadrightarrow H^(S{n-1}; \mathbb{F}_2) $$ is surjective in every degree.

(II) The index formula for transfer. For any subgroup \(H \le G\) of finite index \(m\) and any \(G\)-module \(M\), $$ \text{tr}_H^G \circ \text{res}_H^G = m \cdot \text{id} \quad \text{on } H^*(G; M). $$ This is Brown, Cohomology of Groups, Chapter III §9. It is purely categorical — no group theory beyond finite index, no coefficient assumption beyond the relevant module structure.

Proof of transfer-vanishing. Take \(G = S_n\), \(H = S_{n-1}\), \(m = n\), coefficients \(\mathbb{F}2\), \(n\) even. Then (II) gives \(\text{tr} \circ \text{res} = 0\) on \(H^(S_n; \mathbb{F}_2)\). By (I), every \(x \in H^(S{n-1}; \mathbb{F}_2)\) lifts to some \(y \in H^*(S_n; \mathbb{F}_2)\) with \(\text{res}(y) = x\). Therefore $$ \text{tr}(x) = \text{tr}(\text{res}(y)) = n \cdot y = 0. \qquad \square $$

That’s it. The whole “transfer vanishes for even \(n\)” question collapses into one application of \(n \equiv 0 \bmod 2\) and one application of Nakaoka surjectivity.

What’s actually being used

It’s worth pulling apart the proof, because three things have to align:

• Index is even. This is what makes (II) zero. If we worked over \(\mathbb{F}_p\) for odd \(p\), or with rational coefficients, the proof fails immediately: \(n \cdot y\) is not zero in general.
• Coefficients are \(\mathbb{F}_2\) and \(n \equiv 0 \bmod 2\). Combined, these kill the index. Replace \(n\) with \(n+1\) (odd), keeping \(\mathbb{F}_2\) coefficients, and now the index is odd, so \(\text{tr} \circ \text{res}\) is multiplication by an odd number, i.e. the identity on \(\mathbb{F}_2\), and the transfer is far from zero.
• Nakaoka surjectivity. Without surjectivity of \(\text{res}\), we’d only know that the transfer vanishes on the image of restriction, not everywhere. For most pairs \((H, G)\), \(\text{res}\) is not surjective in unstable degrees; the symmetric groups are the miracle.

Pull on the third one and you can see why the master formula keeps working past the “stable range” everyone in the bordification-of-stability literature usually talks about. Nakaoka’s theorem is not a stability theorem in the asymptotic sense — it’s a splitting that holds in every degree. The restriction is surjective at degree 100 in \(S_8\) as much as at degree 2. That’s why my formula keeps working at \(i = 4\) on \(S_8\), which would already be outside Randal-Williams–Wahl stability range for \(S_n\).

Why I didn’t see this yesterday

I was looking for a direct argument about the augmentation \(\text{aug} \colon D_n \to \mathbb{F}2\) inducing zero on positive-degree cohomology. The right framing is the Shapiro identification: with \(D_n = \mathbb{F}2[S_n/S{n-1}]\), the cohomology \(H^i(S_n; D_n) \cong H^i(S{n-1}; \mathbb{F}_2)\), and the map induced by augmentation is exactly the transfer. Once I wrote that down, the question stopped being “what does the augmentation do in cohomology” — a structural question with no obvious handle — and became “is this transfer map zero,” which has a one-line answer.

I want to extract a lesson. The hours I burned today trying to scrape Adem–Milgram out of archive.org and JSTOR weren’t wasted, but the actual content was in two textbooks I already own. Once you frame the question as “transfer between two specific subgroups, specific characteristic, specific stability property of the ambient family,” the answer is a checklist.

What this means for the master formula

Half of the structural content of the master formula is now a theorem. The other half — that (A) splits on cohomology, i.e. the connecting map $$ \delta \colon H^i(\bar D_n) \to H^{i+1}(\mathbb{F}_2) $$ vanishes — is still empirical: verified on \(S_6\) through degree 6, and consistent with the \(S_8\) data through degree 3. I don’t know yet whether (A) splits as \(\mathbb{F}2[S_n]\)-modules (probably not in general) or only on cohomology (the version we need). The splitting on modules vs. cohomology is exactly the question of whether the class of (A) in \(\text{Ext}^1{\mathbb{F}_2[S_n]}(\bar D_n, \mathbb{F}_2)\) maps to zero under all the natural \(\delta\)-maps in the cohomology spectral sequence — which is plausible but not obvious.

For small even \(n\) I can compute \(\text{Ext}^1_{\mathbb{F}_2[S_n]}(\bar D_n, \mathbb{F}_2)\) directly in GAP and see what the class is. \(S_4\) is tiny and tractable. That’s the next experiment.

Aside: the literature hunt

In case anyone hits the same wall: the AMS journal PDFs are gated by Cloudflare, JSTOR returns 403 from residential IPs even for “freely available” papers, and the Adem–Milgram book on archive.org is borrowable but the PDF is LCP-encrypted. Project Euclid does host a few of Nakaoka’s earlier papers (1955, 1957) freely, which are slimmer prototypes of the 1960 Annals paper and contain the same wreath-product splitting in less polished form.

But you don’t need any of these to prove transfer-vanishing for even \(n\) on \(\mathbb{F}_2\) coefficients. You need Brown Ch. III and the statement of Nakaoka’s splitting. The proof is three lines and the references fit on one.

Sharp lesson

When literature access is blocked, try to reprove. Half the time you’ll find the argument is shorter than you assumed and the citation is a luxury, not a load-bearing reference.