Immersions of Manifoldsc)

Let M and N be differentiable manifolds of dimensions k and n respectively, kN is called an immersion if / is of class C1 and the Jacobian matrix of/ has rank k at each point of M. Such a map is also called regular. Until recently, very little was known about the existence and classification of immersions of one manifold in another. The present work addresses itself to this problem and reduces it to the problem of constructing and classifying cross-sections of fibre bundles. In 1944, Whitney [15] proved that every ^-dimensional manifold can be immersed in Euclidean space of 2k — 1 dimensions, P2*-1. The WhitneyGraustein theorem [13] classifies immersions of the circle S1 inthe plane E2 up to regular homotopy, which is a homotopy /< with the property that for each t,ft is an immersion, and the induced homotopy/(* of the tangent bundle of M into the tangent bundle of N is continuous. In his thesis [8], Smale generalizes the Whitney-Graustein theorem to the case of immersions of S1 in an arbitrary manifold. In [9] Smale classifies immersions of S* in E" for arbitrary kEn, k/') and (g, g') are "regularly homotopic" (in a sense to be defined later). Given two immersions/, g: £>*—»£" that agree on S*_1 and have the same first derivatives at points of S*_1, Q(f, g) is an element of a certain homotopy group, and has the following properties: (1) Q(f, g) =0 if and only if/and g are regularly homotopic "rel S*-1," i.e., the homotopy agrees with / and g on Si_1 at each stage, up to the first derivative; (2) fl(/, g) enjoys the usual algebraic properties of a difference cochain. At this point we should like to be able to make the following statement: "If / is an immersion of the