A Class of Lattices Whose Intervals are Spherical or Contractible

We study a class of lattices called weak* complemented lattices which are shown to have the property that the order complex of any interval of the lattice is either contractible or homotopy equivalent to a sphere. The two main examples are lattices generated by intervals in a total order and the lattices of partitions of integers under dominance order. The proofs are done mainly using homotopy complementation formulas for lattices and with a method called B-labeling. We also show that a class of lattices called Greene lattices are either contractible or spherical. Lattices generated by multisets are also discussed.