1959/02/01 by Stephen T. Smale · 3 citations
Mathematics · #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Algebraic Geometry and Number Theory #Mathematics #Immersion (mathematics) #Regular homotopy #Tangent bundle #Homotopy #Pure mathematics #Combinatorics #Homotopy group #Tangent space
paper · pdf · doi:10.2307/1993205
openalex publication_date 1959/02/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/02
An immersion of one C1 differentiable manifold in another is a regular map (a C1 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 -dimensional manifold in an ra-dimensional manifold, n2k+2, are regularly homotopic if and only if they are homotopic.The Whitney-Graustein Theorem[4] classifies immersions of the circle S1 in the plane E2.In my thesis [3 ] this theorem is extended to the case where E2 is replaced by any C2 manifold Af, ra>l.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 ra-space En, ra>2, with respect to regular homotopy.Let Vn.i he the Stiefel manifold of all 2-frames in En.If / and g are two immersions of S2 in En, an invariant 2(/, g)Ciri(Vn,i) is defined.Theorem A. If f and g are C2 immersions of S2 in En, they are regularly homotopic if and only if Q,(f, g) =0.Furthermore let fioG7r2(Fn,2) and let a C2 immersion f: S2->En be given.Then there exists an immersion g: S2->7n such that fl(/, g) =fio-Thus there is a 1-1 correspondence between elements of ir2(Vni2) and regular homotopy classes of immersions of S2 in En.Since 2(3,2) =0, Theorem A implies: Theorem B. 4ray two C2 immersions of S2 in E3 are regularly homotopic.That this should be so, is not obvious.For example, it is not trivial to see that a reflection of the unit sphere in E3 is regularly homotopic to the identity on the unit sphere.Since 7^(4,2) =Z, there are an infinite number of regular homotopy classes of S2 in E*.In fact we are able to obtain using results of [l], TheoremC. Given yC.H2(S2), 7 even, then there is an immersion of S2 in E4 such that the characteristic class of the normal bundle is 7. Furthermore, any two such C2 immersions are regularly homotopic.There is no immersion of S2 in E4 with odd normal class.In [2] it is proved for say S2 in 4 that the normal class of the immersion