Embeddings and Immersions of Manifolds in Euclidean Space

The problem of computing the number of embeddings or immersions of a manifold in Euclidean space is treated from a different point of view than is usually taken. Also, a theorem dealing with the existence of an embedding of Mm in ä ¡s given. Introduction. This paper studies the existence and classification of differential immersions and embeddings of a differentiable manifold into Euclidean space from a different point of view than is frequently taken. Most authors use Hirsch's results [11] and study a Postnikov factorization of the map BO(k) —► BO and try to lift the stable normal map of the manifold Mmv: M—+BO back to BO(k) for various k. See Gitler's article [4] for references of this method. There is another technique for obtaining embeddings and immersions which is due to Haefliger and Hirsch and is described in §2. It involves studying a Postnikov factorization of the map F" —► P°° and (for embeddings) trying to lift a map g:MxMAfZ2 =M*^P°° back to P". See [3], [6], [23], and [24] for authors who have used this approach. One difficulty with this method is in calculating the cohomology of M* in terms of something known, namely the cohomology of M. Now, Haefliger proved in [7] that the inclusion map /: M* —► S°° xz M x M induces an epimorphism for Z2-cohomology. In §7 I prove that if m is odd /* induces isomorphisms of the Zi^-terms of the Bockstein spectral sequences of the two spaces for r > 2. §6 contains a calculation of the Bockstein spectral sequence of 5°° xz X x X for a finite C. W. complex X. Thus a good hold on H*(M*, Z) is obtained (when m is odd). Similar results are obtained for the twisted integer cohomology of M* when m is even. §3 describes a modified Postnikov factorization of the map p: P" —*■ P°° obtained by killing the first four nonzero homotopy groups of the fiber of p. It also describes the set of isotopy classes of differentiable embeddings ofMm in Received by the editors September 25, 1974. AMS (MOS) subject classifications (1970). Primary 57D40; Secondary 55G45.