Involutions Fixing P(6,2n+1 )and The Structureof J_*,k^2k+15

Let (M ,T) be a smooth closed manifold with a smooth involution T whose fixed pointset is F = {x T(x) = x ,x∈M} , then F is the disjoint union of smooth closed submani?odof M. For F = P(6 , 2n + 1)(n odd) , we prove (M ,T) is bounded.Let Mn be a closed n-dimensional manifold , (?)→Mn denote a smoothaction of the group (Z2)k = {T₁,T₂,···,T_k|T_j² = 1,T_jT_i = T_iT_j} on Mn . The fixed pointset F is the disjoint union of closed submanifolds of Mn , which are finite in number. We saythat F is of constant codimension r , if each component of F is of constant n ? r . Let MOndenote the unoriented cobordism group of dimension n and Jnr,k the set of n dimensional unori-ented cobordism classαn containing a representative Mn adminting a (Z2)k-action with fixedpoint set of constant codimension r . The unoriented cobordism ring M(?)is an ideal of MO? . In this paper , we determine (?).