Cohen--Macaulay Complexes, Duality Groups, and the dualizing module of ${\rm{Out}}(F_N)$

TL;DR

The paper develops a comparative framework between Bieri–Eckmann duality and Cohen–Macaulay duality for discrete groups, and uses this to study the dualizing module of ${\rm Out}(F_N)$ via the spine of Outer space and its local cohomology cosheaf. It shows that CM duality can yield concrete descriptions of dualizing modules even when Lefschetz duality is unavailable, notably by relating $D$ to $H_0(X;h^{n})$ or $\mathrm{Hom}_c(\Gamma(X),\mathbb{Z})$ for appropriate locally CM classifying spaces. The authors construct thickened spines $\mathcal{L}_N$ and $\mathcal{K}_N$ that are homotopy CM and use CM duality to express the dualizing module of ${\rm Out}(F_N)$ in terms of local data on these complexes, establishing a bound on the top-dimensional rational cohomology via stabilizers of roses. They also connect CM properties to notions of visible irreducibility in 2-complexes and present open questions about when CM implies duality and the freeness of dualizing modules. Overall, the work provides a robust, combinatorial toolkit to analyze duality phenomena in groups acting on CM spaces, with explicit outcomes for ${\rm Out}(F_N)$ and broader implications for related groups.

Abstract

We explain how Cohen--Macaulay classifying spaces are ubiquitous among discrete groups that satisfy Bieri--Eckmann duality, and compare Bieri--Eckmann duality to duality results for Cohen--Macaulay complexes. We use this comparison to give a description of the dualizing module of ${\rm{Out}}(F_N)$ in terms of the local cohomology cosheaf of the spine of Outer space.

Figures (3)

• Figure 1: The spine, Jewel space, Outer space, and its simplicial completion are all contractible. ${\rm{Out}}(F_N)$ acts properly and cococompactly on ${\mathcal{X}_N^r}$ and $\mathcal{J}_N$, but does not act cocompactly on $\mathcal{CV}^r_N$ and does not act properly on ${\mathcal{FS}^r_N}$. The boundary of Jewel space is homotopy equivalent to ${\partial\mathcal{FS}_N^r}$, although this homotopy equivalence is also not proper; $\partial\mathcal{J}_N$ is locally finite, whereas ${\partial\mathcal{FS}_N^r}$ is not.
• Figure 2: The group $G=\mathbb{Z}^2 \ast \mathbb{Z}^2$ is not a duality group but is the fundamental group of an aspherical local homotopy CM complex $Y$ which has no free faces. The space $Y$ is given by gluing two tori $T_{1,1}$ each with one boundary to a disc.
• Figure 3: Left: a portion of the free splitting complex $\mathcal{FS}_2$ in rank two. The intersection with the spine $\mathcal{X}_2$ is the (thickened) central tripod. Part of the barycentric subdivision $\mathcal{FS}_2'$ is depicted in the bottom corner, where the thickened spine $\mathcal{K}_N$ is shaded. Right: an enlarged version of the featured part of $\mathcal{K}_N$. Vertices are given by chains of vertices in the free splitting complex, and simplices correspond to increasing chains of chains. The boundary $\partial\mathcal{K}_N$ can be seen in the picture in the chains with single circle as a minimal element.

Theorems & Definitions (47)

• Theorem 1: CM Duality, WW1
• Corollary 2
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6
• Theorem 2.1: Relative CM Duality
• Definition 2.2: Semistability of the local homology sheaf
• Theorem 2.3: WW1, Corollary 3.19
• Theorem 2.4: Theorem \ref{['t:compactly_sup_sections']}