Shareshian's EL-labeling conjecture for solvable subgroup lattices

Let $G$ be a finite solvable group, and let $L(G)$ denote its subgroup lattice. Choose a chief series

$$1=N_{0}\subset N_{1}\subset\dots\subset N_{k}=G.$$

A complement to a subgroup $N$ is a subgroup $H$ with $HN=G$ and $H\cap N=1$; a chain of complements to this chief series is a chain

$$1=H_{k}\subset H_{k-1}\subset\dots\subset H_{0}=G$$

where $H_i$ is a complement to $N_i$ for each $i$. Shareshian's EL-labeling conjecture. The lattice $L(G)$ admits an $EL$-labeling whose descending chains are precisely the chains of complements to a chief series. This conjecture seeks an explicit shelling-type labeling whose descending chains realize the complement-chain basis underlying the topology of solvable subgroup lattices; the paper presents an EL-labeling, resolving the conjecture.