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

The article consists of two parts.In the first chapter,we discuss the problem of (Z₂)^k-actions with fixed point set of constant codimension. Let (?) : (Z₂)^k × Mⁿ → Mⁿ denote the smooth action of the group (Z₂)^k = on a closed manifold Mⁿ.The fixed point set F of the action is the disjoint union of closed submanifolds of Mⁿ,which are finite in number. If each component of F is of constant dimension n — r,we may say that F is of constant codimension r. If a n-dimensional unoriented cobordism class α_n containing a representative that admits a (Z₂)^k-action with fixed point set of constant codimension r,we may say that α ∈is an ideal of the unoricnted cobordism ring MO_* = . In this chapter, we determine and by choosing appropriate representatives of indecomposibles in MO_* and defining appropriate (Z₂)^k-action on it.In the second chapter,we determine1 the possible form of the total Stiefel-Whitney classes of vector bundles on RP(j) × CP(k) by using Steenrod square and Wu formula.This would be a base to the study of involutions fixing RP(j)x CP(k).