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

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+2vby constructing ingeniously indecomposable manifolds M,which can be generators in MO_*,and defining appropriate(Z₂)^k-action on M.