Elementary abelian 2-group actions on flag manifolds and applications

Let $\mathcal N_\ast$ denote the unoriented cobordism ring. Let $G=(\mathbb Z/2)^n$ and let $Z_\ast(G)$ 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 $M=O(m)/(O(m_1)\times\dots\times O(m_s))$ is in the subalgebra generated by $\bigoplus _{i<2^n}\mathcal N_i$, where $m= \sum m_j$, and $2^{n-1}<m\le 2^n$. We obtain sufficient conditions for indecomposability of an element in $Z_\ast(G)$. We also obtain a sufficient condition for algebraic independence of any set of elements in $Z_\ast(G)$. Using our criteria, we construct many indecomposable elements in the kernel of the forgetful map $Z_d(G)\to\mathcal N_d$ in dimensions $2\le d<n$, for $n>2$, and show that they generate a polynomial subalgebra of $Z_\ast(G)$.