Geometric Formulas for Smale Invariants of Codimension Two Immersions
نویسنده
چکیده
We give three formulas expressing the Smale invariant of an immersion f of a (4k− 1)-sphere into (4k + 1)-space. The terms of the formulas are geometric characteristics of any generic smooth map g of any oriented 4k-dimensional manifold, where g restricted to the boundary is an immersion regularly homotopic to f in (6k − 1)-space. The formulas imply that if f and g are two non-regularly homotopic immersions of a (4k − 1)-sphere into (4k + 1)-space then they are also non-regularly homotopic as immersions into (6k−1)-space. Moreover, any generic homotopy in (6k−1)-space connecting f to g must have at least ak(2k−1)! cusps, where ak = 2 if k is odd and ak = 1 if k is even.
منابع مشابه
Regular homotopy and total curvature I: circle immersions into surfaces
An immersion of manifolds is a map with everywhere injective differential. Two immersions are regularly homotopic if there exists a continuous 1–parameter family of immersions connecting one to the other. The Smale–Hirsch h–principle [8, 4] says that the space of immersions M → N , dim(M) < dim(N) is homotopy equivalent to the space of injective bundle maps TM → TN . In contrast to differential...
متن کاملDifferential 3-knots in 5-space with and without Self Intersections
Regular homotopy classes of immersions S → R constitute an infinite cyclic group. The classes containing embeddings form a subgroup of index 24. The obstruction for a generic immersion to be regularly homotopic to an embedding is described in terms of geometric invariants of its self intersection. Geometric properties of self intersections are used to construct two invariants J and St of generi...
متن کاملNew Viewpoints in the Geometry of Submanifolds of R
0. Introduction. The geometry of submanifolds of euclidean space is the oldest branch of differential geometry. The subject was the original source of most of the classical and modern ideas in the field, and still is the setting in which seemingly complicated general phenomena are most easily understood. In fact as Allendoerfer [1] once said " . . . an excellent way of discovering [theorems] is...
متن کاملCobordism of Fold Maps, Stably Framed Manifolds and Immersions
We give complete geometric invariants of cobordisms of fold maps with oriented singular set and cobordisms of even codimensional fold maps. These invariants are given in terms of cobordisms of stably framed manifolds and cobordisms of immersions with prescribed normal bundles.
متن کاملFold Maps and Immersions from the Viewpoint of Cobordism
We obtain complete geometric invariants of cobordism classes of oriented simple fold maps of (n+ 1)-dimensional manifolds into an n-dimensional manifold N in terms of immersions with prescribed normal bundles. We compute that for N = R the cobordism group of simple fold maps is isomorphic to the direct sum of the (n−1)th stable homotopy group of spheres and the (n − 1)th stable homotopy group o...
متن کامل