Characteristic Classes Of Vector Bundles Over Productive Space And Bordism Classification Of Manifolds With Group Action

This dissertation consists of three chapters.In the first chapter, we discuss the involutions fixing the product of the real pro-jective space RP(2m+1) with the complex projective space CP(k). Let (M, T) bea smooth closed manifold with a smooth involution T whose fixed point set is F. ForF=RP(2m + 1)×CP(k), we prove that every involution bounds.In the second chapter, we determine the total Stiefel-Whitney classes of vectorbundles over the product of the real projective space RP(h) with the quaternionicprojective space HP(k). As an application we discuss the involutions fixing RP(2m +1)×HP(k) and prove that every involution fixing RP(2m + 1)×HP(k) bounds.In the third chapter, we discuss the problem of (Z₂)^k-actions with fixed point setof constant codimension. Let(?):(Z₂)^k×Mⁿ→Mⁿ be a smooth action of the group(?) on a closed n-dimensional manifold Mⁿ.Here (Z₂)^k is considered as the group generated by k commuting involutions. Thefixed point set F of the action is a disjoint union of closed sub-manifolds of Mⁿ. Ifeach component of F is (n-r)-dimensional, then F has constant codimension r. LetJ_n,k^r denote the set of n-dimensional cobordism classesα_n containing a representativeMⁿ admitting a (Z₂)^k-action with fixed point set of constant codimension r. J_*,k^r =Σ_n≥rJ_n,k^r is an ideal of the unoriented cobordism ring MO_* =Σ_n≥0MO_n. In thischapter, we determine the ideal J_*,k^2k+8 by constructing ingeniously indecomposablemanifolds M, which can be generators in MO_*, and defining appropriate (Z₂)^k-actionon M.