Nonnegative Grassman Chambers Are Balls

Classically, the notion of total positivity referred to matrices all of whose minors had positive determinants. Lusztig generalized this notion substantially ([L1],[L2],[L3]) introducing the nonnegative part of an arbitrary reductive group, as well as the nonnegative part of a flag variety. Lusztig proved that the latter is always contractible and it has been conjectured to always be homeomorphic to a closed ball. Some work in this direction may be found in [W1],[W2]. However, even the case of Grassmannians remained open. In this paper, we present an elementary proof that the nonnegative part of a Grassmannian is homeomorphic to a ball. We would like to thank Patricia Hersh, Chuck Livingston, and James Davis for helpful discussions. We would like to especially thank Lauren Williams for correcting some errors in an early version of the paper.