Some Immersion Theorems for Manifolds

Abstract

Abstract. In this paper we obtain several results on immersing manifolds into Euclidean spaces. For example, a spin manifold Mn immerses in if2"-3 for dimension n = 0mod4 and n not a power of 2. A spin manifold Mn immerses in Ä2""4 for n=l mod 8 and «>7. Let M" be a 2-connected manifold for n = 6 mod 8 and n>6 such that H3(M;Z) has no 2-torsion. Then M immerses in /?2"~5 and embeds in /?2""4. The method of proof consists of expressing ¿-invariants in Postnikov resolutions for the stable normal bundle of a manifold by means of higher order cohomology operations. Properties of the normal bundle are used to evaluate the operations.1. Preliminaries. By a manifold M" we mean that M is a closed connected smooth manifold of dimension n. We write M^RS and M^R1 to denote the existence of a differentiable immersion of M into Euclidean s-space and a smooth embedding of M into Euclidean /-space respectively. A manifold M is called a spin manifold iff w^M) = w2iM) = 0. The geometric dimension of a stable vector bundle