The commuting complex of the symmetric group with bounded number of p-cycles

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Problem: For a fixed prime $p$, let $\mathbf{FI}$ be the category of finite sets and injections, and let

$$S\longmapsto \mathbf K_p(\mathfrak S_S)$$

be the $\mathbf{FI}$-simplicial complex whose vertices are order-$p$ elements of $\mathfrak S_S$, with simplices finite sets of pairwise commuting such elements. Homology is integral. The conjecture asks whether there is a chain complex $C_*$ of finitely generated $\mathbf{FI}$-modules such that

$$H_k(C_*)\cong H_k(\mathbf K_p(\mathfrak S_\bullet);\mathbb Z)$$

naturally as $\mathbf{FI}$-modules for every $k\ge 0$.

Result: The conjecture is true.

For an order-$p$ permutation $\sigma$, write $c(\sigma)$ for the number of $p$-cycles in its disjoint-cycle decomposition, and let

$$\mathbf K_p^{\le A}(\mathfrak S_S)\subseteq \mathbf K_p(\mathfrak S_S)$$

be the induced subcomplex on vertices with $c(\sigma)\le A$.

Fix $k\ge0$ and put $A=p^{k+1}$. We prove that

$$H_k\bigl(\mathbf K_p^{\le A}(\mathfrak S_\bullet)\bigr)\twoheadrightarrow H_k\bigl(\mathbf K_p(\mathfrak S_\bullet)\bigr)$$

is a natural epimorphism of $\mathbf{FI}$-modules.

Remove vertices $\sigma$ with $c(\sigma)>A$, in decreasing order of $c(\sigma)$. Suppose $c(\sigma)=a>A$, and consider its link $L$ at the moment of removal. Let $\Omega$ be the set of $p$-cycles of $\sigma$. For each nonempty proper subset $U\subsetneq\Omega$, let $\sigma_U$ be the product of the cycles in $U$. These $\sigma_U$ all have fewer than $a$ cycles, hence remain present, and they span a simplex $D\subseteq L$.

Now take any simplex $Q\subseteq L$ of dimension at most $k$. The elementary abelian group generated by $Q\cup\{\sigma\}$ acts on $\Omega$, and the quotient action modulo $\langle\sigma\rangle$ has order at most $p^{k+1}<a$. Hence it is not transitive on $\Omega$. Choose a nonempty proper invariant orbit-union $U\subsetneq\Omega$. Then $\sigma_U$ commutes with every vertex of $Q$, so $Q$ lies in the closed star of $\sigma_U$.

Thus the $k$-skeleton of $L$ is contained in the union of the closed stars of the vertices of $D$. Since $D$ is itself a simplex, this union is contractible by the nerve lemma: all finite intersections are closed stars of faces of $D$. Therefore $\widetilde H_i(L)=0$ for $i\le k-1$. The relative homology of adding/removing $\sigma$ is

$$H_j(\operatorname{Cone}L,L)\cong \widetilde H_{j-1}(L),$$

so in particular $H_k$ is unchanged surjectively at each removal. Hence the displayed epimorphism follows.

For fixed $A$, the chain $\mathbf{FI}$-modules of $\mathbf K_p^{\le A}$ are finitely generated: an $r$-simplex involves $r+1$ permutations, each moving at most $Ap$ letters, so it is supported on at most $(r+1)Ap$ letters. By the Noetherian property of $\mathbf{FI}$-modules over $\mathbb Z$, the homology modules

$$H_k\bigl(\mathbf K_p^{\le A}(\mathfrak S_\bullet)\bigr)$$

are finitely generated. Since $H_k(\mathbf K_p(\mathfrak S_\bullet))$ is a quotient of one of these, it is finitely generated.

Finally define a chain complex $C_*$ by

$$C_k:=H_k(\mathbf K_p(\mathfrak S_\bullet);\mathbb Z), \qquad d=0.$$

Each $C_k$ is finitely generated, and $H_k(C_*)=C_k$, giving the required complex.

Verification audit: the proof uses the same $\mathbf{FI}$-structure and integral homology as in the conjecture; no bounded-cycle hypothesis is added to the final statement; bounded subcomplexes are used only to prove finite generation; the conclusion is exactly the existence of a chain complex of finitely generated $\mathbf{FI}$-modules with the stated homology.

Citation: The conjecture is from Bahran, “The commuting complex of the symmetric group with bounded number of $p$-cycles,” arXiv:1808.02581. The Noetherian property used is Church–Ellenberg–Farb–Nagpal, “FI-modules over Noetherian rings,” Geom. Topol. 18 (2014), Theorem A.