Poset topology of s weak order via SB-labelings

Ceballos and Pons generalized weak order on permutations to a partial order on certain labeled trees, thereby introducing a new class of lattices called s-weak order. They also generalized the Tamari lattice by defining a particular sublattice of s-weak order called the s-Tamari lattice. We prove that the homotopy type of each open interval in s-weak order and in the s-Tamari lattice is either a ball or sphere. We do this by giving s-weak order and the s-Tamari lattice a type of edge labeling known as an SB-labeling. We characterize which intervals are homotopy equivalent to spheres and which are homotopy equivalent to balls; we also determine the dimension of the spheres for the intervals yielding spheres.