Euler characteristics for links of Schubert cells in the space of complete flags

Let Fn be the space of complete flags in k (where k is R or C). With an arbitrary complete flag f ∈ Fn we associate the standard Schubert cell decomposition Schf of the space Fn whose cells are enumerated by elements from Sn while the dimension over k of such a cell equals the number of inversions in the corresponding permutation (see for example [FF] §5.4). Definition. The train Tnf of the flag f ∈ Fn is the union of all cells of Schf of positive codimension. Let cσ be the cell of the decomposition Schf corresponding to the permutation σ and B a sufficiently small n(n− 1)/2-dimensional (over k) ball with the origin at some point of cσ. Definition. The manifold Aσ = B \ Tnf is called the link of the cell cσ. By χσ we denote the Euler characteristic of Aσ :