ELEMENTARY ABELIAN 2-GROUP ACTIONS ON FLAG MANIFOLDS AND APPLICATIONS

Let N- denote the unoriented cobordism ring. Let G = (Z/2)(n) and let Z()(G) denote the equivariant cobordism ring of smooth manifolds wit h smooth G-actions having finite stationary points. In this paper we s how that the unoriented cobordism class of the (real) flag manifold M = O(m)/(O(m(1)) X...X O(m(s))) is in the subalgebra generated by +(i<2 n) Ni, where m = Sigma m(j), and 2(n-1) < m less than or equal to 2(n) . We obtain sufficient conditions for indecomposability of an element in Z()(G). We also obtain a sufficient condition for algebraic indepe ndence of any set of elements in Z()(G). Using our criteria, we const ruct many indecomposable elements in the kernel of the forgetful map Z (d)(G) --> N-d in dimensions 2 less than or equal to d < n, for n > 2, and show that they generate a polynomial subalgebra of Z()(G).