An analogue of Rognes' connectivity conjecture for free groups

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 $\mathrm{A}\mathrm{u}\mathrm{t}(F_n)$-analogue of the connectivity conjecture that Rognes originally stated for ${\mathrm{G}\mathrm{L}}_n(R)$. 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.