Chains of modular elements and shellability

TL;DR

This paper proves shellability of certain skeleta of lattices with left-modular chains, extends EL-labeling theory, and applies these results to characterize finite solvable groups through subgroup lattice topology.

Contribution

It introduces a relaxed EL-labeling condition, proves shellability of skeleta under specific modularity conditions, and characterizes finite solvable groups via subgroup lattice topology.

Findings

The (r-2)-skeleton of certain lattices is vertex-decomposable and shellable.

Extended EL-labeling allows multiple ascending chains, broadening applicability.

Shellability results lead to new topological characterizations of finite solvable groups.

Abstract

Let L be a lattice admitting a left-modular chain of length r, not necessarily maximal. We show that if either L is graded or the chain is modular, then the (r-2)-skeleton of L is vertex-decomposable (hence shellable). This proves a conjecture of Hersh. Under certain circumstances, we can find shellings of higher skeleta. For instance, if the left-modular chain consists of every other element of some maximum length chain, then L itself is shellable. We apply these results to give a new characterization of finite solvable groups in terms of the topology of subgroup lattices. Our main tool relaxes the conditions for an EL-labeling, allowing multiple ascending chains as long as they are lexicographically before non-ascending chains. We extend results from the theory of EL-shellable posets to such labelings. The shellability of certain skeleta is one such result. Another is that a poset with such a labeling is homotopy equivalent (by discrete Morse theory) to a cell complex with cells in correspondence to weakly descending chains.