A classification of immersions of the two-sphere

Abstract

An immersion of one C' differentiable manifold in another is a regular map (a C' map whose Jacobian is of maximum rank) of the first into the second. A homotopy of an immersion is called regular if at each stage it is regular and if the induced homotopy of the tangent bundle is continuous. Little is known about the general problem of classification of immersions under regular homotopy. Whitney [5] has shown that two immersions of a k-dimensional manifold in an n-dimensional manifold, n>2k+2, are regularly homotopic if and only if they are homotopic. The Whitney-Graustein Theorem [4] classifies immersions of the circle S in the plane E2. In my thesis [3] this theorem is extended to the case where E2 is replaced byany C2 manifold Mn, n> 1. As far as I know, these are the only known results. In this paper we give a classification of immersions of the 2-sphere S2 in Euclidean n-space E , n>2, with respect to regular homotopy. Let V.,2 be the Stiefel manifold of all 2-frames in En. If f and g are two immersions of S2 in En, an invariant Q(f, g) Cw2(Vn,2) is defined.