AI 中文总结
该研究通过图论连通性定义的加厚，证明Culler-Vogtmann外空间的单纯边界∂ℱ𝒮到含非3边连通稳定图的子复形C'的包含映射是(2n-3)-连通的，为相关拓扑猜想提供了证据并统一了部分图复形结果。
AI 中文摘要
我们通过图论连通性定义的加厚来研究Culler-Vogtmann外空间的单纯边界∂ℱ𝒮。设C'是自由分裂复形的子复形，由∂ℱ𝒮添加所有非3边连通的稳定图及其面得到。我们证明包含映射∂ℱ𝒮↪C'是(2n-3)-连通的。该证明更精确地表明，添加带割点的图是同伦等价，而2-键加厚中唯一的非可缩纤维出现在θ-图上。这一结果进一步支持∂ℱ𝒮可能是(2n-3)-球面，即Rognes关于公共基复形的连通性猜想在Out(Fₙ)上的类似物，还提供了拓扑通用覆盖视角，统一了现有关于交换图复形的若干结果。
英文摘要
We study the simplicial boundary $\partial\mathcal{FS}$ of Culler-Vogtmann Outer space via thickenings defined by graph-theoretic connectivity. Let $C'$ be the subcomplex of the free splitting complex obtained from $\partial\mathcal{FS}$ by adding all stable graphs that are not $3$-edge connected, together with their faces. We prove that the inclusion $\partial\mathcal{FS}\hookrightarrow C'$ is $(2n-3)$-connected. The proof shows, more precisely, that adding graphs with cut vertices is a homotopy equivalence, while the only non-contractible fibres in the $2$-bond thickening occur over $θ$-graphs. The result gives further evidence that $\partial\mathcal{FS}$ may be $(2n-3)$-spherical, an $\operatorname{Out}(F_n)$-analogue of Rognes's connectivity conjecture for the common basis complex. It also gives a topological, universal-cover perspective that unifies several existing results about the commutative graph complex.
Comments39 pages, 11 figures; v2: added a reference
