Elementary abelian 2-group actions on flag manifolds and applications
Proc. Amer. Math. Soc. 126 (1998), 595-606.
 
Goutam Mukherjee 
Stat-Math Division, Indian Statistical Institute, 203 B. T. Road, 
Calcutta-700 035, India. 

and Parameswaran Sankaran
SPIC Mathematical Institute, 92 G. N. Chetty Road, Madras-600 017, India.


Abstract. Let $\mathcal N_*$ denote the unoriented cobordism ring. Let 
$G = (\mathbb Z/2)^n$ 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 $M = O(m)/(O(m_1) \times \dots \times O(M_s))$ is in 
the subalgebra generated by $\bigoplus_{i < 2^n} \mathcal N_i$, where 
$m = \sum m_j$, and $2^{n-1} < m < 2^n$. We obtain sufficient conditions 
for indecomposability 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 $Z_d(G) \rightarrow \mathcal N_d$ in dimensions 
$2 \le d < n$, for $n > 2$, and show that they generate a polynomial 
subalgebra of $Z_*(G)$.