  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.
