An analogue of Rognes' connectivity conjecture for free groups

TL;DR

The paper proves that CB(F_n) is homotopy equivalent to a wedge of (2n−3)-spheres, providing an Aut(F_n)-analogue of Rognes' connectivity conjecture for free groups. It develops parallel algebraic and geometric models (CB ≃ PD ≃ FCD ≃ FCD^{g}) and constructs a key map α between a geometric and an algebraic model, then shows α is a homotopy equivalence by Quillen fiber arguments. A degree theorem for sphere systems, adapted from Hatcher–Vogtmann and extended by Aygun–Miller, yields the necessary high connectivity of the geometric model, which translates into the desired spherical wedge decomposition. The work also situates CB(F_n) in the broader context of Outer space, graph homology, and Grothendieck–Teichmüller theory, highlighting potential pathways to related conjectures for Z^n and connections to stable homology phenomena.

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 $\mathrm{Aut}(F_n)$-analogue of the connectivity conjecture that Rognes originally stated for $\mathrm{GL}_n(R)$. 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.

Figures (5)

• Figure 1: Left: Manifold $M_{2,0}$ with embedded sphere system $\sigma = \left\{ [S_1], [S_2]\right\}$ in blue. Right: The manifold $M_{2,0}-\sigma$ has two connected components, diffeomorphic to $M_{0,3}$ and $M_{1,1}$.
• Figure 2: Top left and right: A sphere system $\tau\in \mathrm{FCD}^{\mathrm{g}}$ (in blue) that is cut and the graph $\Gamma(\tau)$ with $C_1$, $C_2$, $C_3$ the connected components of $|\Gamma(\tau)|\setminus \{M_0(\tau)\}$. Below: The open cover $\left\{ M_{C_1}, M_{C_2}, M_{C_3}\right\}$ from the proof of \ref{['lm:tau_to_decomp']}.
• Figure 3: The sphere system $\tau$ (in blue) from \ref{['fig:MCcovering']}, and additional spheres (green, orange and pink) in the link of $\tau$. Adding the green or the orange sphere to $\tau$ gives rise to the same decomposition, while adding the pink sphere gives rise to a coarser decomposition.
• Figure 4: On the left: The manifold $M\cong M_{4,1}$, a sphere system $\tau\in F_{\mathrm{cut}}$ and the system $\tau_d$ for a decomposition $d$ of the form $\left\{ D_1\cong F_3, D_2\cong F_1 \right\}$. On the right: The corresponding decomposition of $M_0(\tau)$ from the proof of \ref{['lem:F_1_comp_with_tau']}.
• Figure 5: The manifold $M$ from the proof of \ref{['lem:tau_d_A_empty']}. The four orange circles depict $\tau_d$ and the part below the orange circles is $M_0(\tau_d)$. The blue circle depicts a sphere $S$ such that $[S]\in \mathcal{S}(M_0(\tau_d))\subset \mathcal{S}$.

Theorems & Definitions (50)

• Theorem 1.1
• Theorem 1.2
• Theorem 2.1: Quillen
• Example 2.2
• Theorem 2.3
• Lemma 3.1
• proof
• Theorem 3.2
• proof
• Remark 3.3