Elementary abelian 2-group actions on flag manifolds and applications

Abstract

Let N_∗ denote the unoriented cobordism ring. Let G=(Z=2)ⁿ and let Z_∗(G) denote the equivariant cobordism ring of smooth manifolds with smooth G-actions having finite stationary points. In this paper we show that the unoriented cobordism class of the (real) flag manifold M=O(m)=(O(m₁)× . . . ×O(m_s)) is in the subalgebra generated by ⊕_i<2ⁿ N_i, where m=Σ m_j, and 2ⁿ⁻¹ < m≤ 2ⁿ. We obtain sufficient conditions for in-decomposability of an element in Z_∗(G). We also obtain a sufficient condition for algebraic independence of any set of elements in Z_∗(G). Using our criteria, we construct many indecomposable elements in the kernel of the forgetful map Z_d(G)→N_din dimensions 2 ≤ d < n, for n > 2, and show that they generate a polynomial subalgebra of Z_*(G).