(Z2)k-actions and the minimal data of the normal bundle

Let J n , k r denote the group of the n-dimensional unoriented cobordism classes containing a representative that admits a ( Z 2 ) k -action with the fixed point set of constant dimension n − r and J ⁎ , k r = ∑ n ⩾ r J n , k r . In this paper, we introduce the minimal data ‖ J ⁎ , k r ‖ of the normal bundle which reflects the complexity of the ( Z 2 ) k -action. Such minimal data or their upper bounds are determined for some r and k.