Handle attaching on generic maps

Abstract:

Using the handle attaching technique along the singular value set of generic maps in the stable range together with the handle subtraction of Haefliger, smooth immersions and embeddings are studied. We generalize Whitney’s immersion theorem, and Haefliger and Hirsh’s result on embedding and classification of embeddings of $k$-connected ($(k + 1)$-connected for the classification) smooth $n$-manifolds into ${{\mathbf {R}}^{2n - k}}$. For example, we obtain the following as a generalization of Whitney’s immersion theorem. If $f: {V^n} \to {M^m}, {3n} < {2m}$, is a generic map such that each component of its double point set is either a closed manifold or diffeomorphic to the $(2n - m)$-disk, then $f$ is homotopic to an immersion.