An analogue of Rognes' connectivity conjecture for free groups

2025/12/22 by Benjamin Brück, Jeremy Miller, Brück, Benjamin +3
Mathematics · Computer Science · #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Topological and Geometric Data Analysis

paper · doi:10.48550/arxiv.2512.19128

Abstract

We show that the common basis complex of a free group of rank n has the homotopy type of a wedge of spheres of dimension 2n-3. This establishes an Aut(F_n)-analogue of the connectivity conjecture that Rognes originally stated for GL_n(R). To prove this, we provide several homotopy-equivalent models of the common basis complex, both in terms of free factors in free groups and in terms of sphere systems in 3-manifolds.

Citations

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Related

• Connectivity of partial basis complexes of freely decomposable groups
• Relative free splitting and free factor complexes I: Hyperbolicity
• On rank filtrations of algebraic K-theory and Steinberg modules
• Configuration spaces on a wedge of spheres and Hochschild-Pirashvili homology
• The common basis complex and the partial decomposition poset