The Classification Of A Special Kind Of Closed Manifolds With （Z₂）^k-actions Up To Cobordism

LetΦ:(Z2)κ×Mn→Mn denote a smooth action of the group(Z2)κ={T1,…,Tκ|Ti2=1,TiTj=TjTi} on a closed n-dimensional manifold Mn.Here(Z2)κ is considered as the group generated by k commuting involutions.The fixed point set F of the action of(Z2)κ 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一r,we say that F is of constant codimension r.Let Jn,kr denote the set of n-dimensional cobordism class αn containing a representative Mn admitting a(Z2)κ-action with fixed point set of constant codimension r.J*,κr=∑n≥r Jn,κ is an ideal of the unoriented cobordism ring MO*=∑n≥0MOn.In this paper,we determine J*,κ2κ|2l4,J*,κ2κ|2l6,J*,κ2κ|36by constructing ingeniously indecomposable manifolds M,and defining appropriate(Z2)κ-action on M.