The union of a connected manifold and a point as fixed set of commuting involutions

Let A be the family of all equivariant bordism classes of involutions containing a representative (M, T) with M connected and with the fixed point set of T being the disjoint union of a fixed connected n-dimensional manifold Vn and a point. In this paper we analyse the influence of A in the determination of the equivariant bordism classes of (Z2)κ-actions which contain a representative fixing Vn ∪ point. The results obtained are used to determine, up to bordism, all possible (Z2)κ-actions fixing RP(n) ∪ point with n odd.