Basic Notions Seminar talk on November 1, 2002

Speaker:   Yuri Burman (University of Oregon and Independent Moscow University)

Title:   Classification of immersions

Abstract:      An immersion is a smooth mapping   f: M -> N   between two manifolds whose image has no local singularities. Formally, it is defined by the condition that the rank of its derivative at any point of M is equal to the dimension of M. Thus an immersion of a circle into R^2 is just a closed smooth plane curve (self-intersections are allowed).

    Two immersions f_0 and f_1 of M into N are called equivalent if they can be joined by a continuous one-parameter family f_t whose elements are all immersions. The principal classification of immersions with respect to this equivalence was obtained by M.Hirsch and S.Smale in the 1950s. For example, the plane smooth curves f(t) are classified by a single integer, the winding number, which is a number of full turns made by the nonzero plane vector f'(t) around the origin, as t goes along the entire circle (this is in fact an earlier theorem by H.Whitney).
    Probably the most publicised case of the general Smale's result is the following theorem:

    Any two immersions of the 2-sphere into   R^3   are equivalent.

In particular, this implies a possibility to turn the sphere inside out in a 3-space allowing self-intersections during the process, but not local singularities!

The talk will contain an exact formulation of the general Smale's result, and a discussion of some applications.