(Z₂)^k-actions With Fixed Point Set Of Constant Codimension 2^k+7

Letφ: (Z₂)^k×Mⁿ→Mⁿ denote a smooth action of the group (Z₂)^k = {T₁,..., T_k|T_i² = 1, T_iT_j = T_jT_i} on a closed manifold Mⁿ. The fixed point set F of the action is the disjoint union of closed submanifolds of Mⁿ, which are finite in number. If each component of F is of constant dimension n - r, we say that F is of constant codimension r. Let MO_n denote the unoriented cobordism group of dimension n and JJ_n,k^r the set of unoriented cobordism classes of Mⁿ that admits a (Z₂)^k-action with fixed point set of constant codimension r. J_n,k^r is a group of MO_n and J_*,k^r =∑_n≥rJ_n,k^r is an ideal of the unoriented cobordism ring MO_* =∑_n≥0MO_n. In this paper, we determine J_*,k^2k+7 by constructing indecomposable manifolds M and defining appropriate (Z₂)^k -action on M.