(Z₂)～k-actions With Fixed Point Set Of Constant Codimension 2～k+6

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^r,k denote the set of n-dimensional cobordism class α_n containing a representative Mⁿ admitting a (Z₂)^k-action with fixed point set of constant codimension r. J_*^r,k = Σ_n≥r J_n^r,k is an ideal of the unoriented cobordism ring MO_* = ∑_n≥0 MO_n.In this paper,we determine J_*,k^2k+6 by constructing ingeniously indecomposable manifolds M, which can be generators in MO_*,and defining appropriate (Z₂)k-action on M. Mean-while,for k = 2,we determine J_n,k^2k+4(n≥13) by the same method.