(Z₂)～k-actions With Fixed Point Set Of Constant Codimension 2～k+34

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 n-dimensional manifold Mⁿ. Here (Z₂)^k is considered as the group generated by k commuting involutions. The fixed point set F of the action of (Z₂)^k on Mⁿ is a 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 J_n,k^r denote the set of n-dimensional cobordism classα_n containing a representative Mⁿ admitting a (Z₂)^k-action with fixed point set of constant codimension r.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+34 by constructing ingeniously indecomposable manifolds M, which can be generators in MO_*, and defining appropriate (Z₂)^k-action on M.