ABSTRACT
The topology of a semisimple orbit is quite simple: semisimple orbits are closed simply connected submanifolds of g. This chapter explores the topology of nilpotent orbits, and confines to two topics: fundamental groups of nilpotent orbits, and the partial order on orbits by containment of their closures. There is a remarkable order-reversing map d from the set of nilpotent orbits to itself that restricts to an involution on its range. Orbits in the range of d are called special; there is a unique smallest special orbit lying above any given orbit in the partial order. The chapter defines the map d and the set of special orbits for every classical algebra g. The map d has also been defined and completely computed in the exceptional cases. It is an easy matter to compute the dimension of any nilpotent orbit in a classical Lie algebra. The chapter describes partial order for classical g in terms of partitions.
