The Determination Of A Special Kind Of Ideals In Cobordism Ring

Letφ:(Z2)k×Mnâ†'Mn be a smooth action of the group(Z2)k={T1,…,Tk|Ti2=1,TiTj=TjTi)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 Jn,kt be the set of n-dimensional unoriented cobordism class αn containing a representative Mn admitting a(Z2)k-action with fixed point set of constant codimension r. J*,kt=∑n≥r Jn,kt is an ideal of the unoriented cobordism ring MO*=∑n≥0MOn.In this paper,we determine J*,k2k-2l-12, J*,k2k2l16å'ŒJ*,k2k2l18by constructing ingeniously indecomposable manifolds M,and defining appropriate(Z2)k-action on M.