Immersion (mathematics)
Hassler Whitney initiated the systematic study of immersions and regular homotopies in the 1940s, proving that for 2m < n + 1 every map f : M m → N n of an m-dimensional manifold to an n-dimensional manifold is homotopic to an immersion, and in fact to an embedding for 2m < n; these are the Whitney immersion theorem and Whitney embedding theorem.
Stephen Smale expressed the regular homotopy classes of immersions
Morris Hirsch generalized Smale's expression to a homotopy theory description of the regular homotopy classes of immersions of any m-dimensional manifold M m in any n-dimensional manifold N n. The Hirsch-Smale classification of immersions was generalized by Mikhail Gromov.
Since characteristic classes multiply under direct sum of vector bundles, this obstruction can be stated intrinsically in terms of the space M and its tangent bundle and cohomology algebra.
For example, the Möbius strip has non-trivial tangent bundle, so it cannot immerse in codimension 0 (in
William S. Massey (1960) showed that these characteristic classes (the Stiefel–Whitney classes of the stable normal bundle) vanish above degree n − α(n), where α(n) is the number of "1" digits when n is written in binary; this bound is sharp, as realized by real projective space.
Further, codimension 0 immersions do not behave like other immersions, which are largely determined by the stable normal bundle: in codimension 0 one has issues of fundamental class and cover spaces.
despite the circle being parallelizable, which can be proven because the line has no fundamental class, so one does not get the required map on top cohomology.
Note, however, that the converse is false: there are injective immersions that are not embeddings.
The nature of the multiple points classifies immersions; for example, immersions of a circle in the plane are classified up to regular homotopy by the number of double points.
At a key point in surgery theory it is necessary to decide if an immersion
of an m-sphere in a 2m-dimensional manifold is regular homotopic to an embedding, in which case it can be killed by surgery.
Wall associated to f an invariant μ(f ) in a quotient of the fundamental group ring
which counts the double points of f in the universal cover of N. For m > 2, f is regular homotopic to an embedding if and only if μ(f ) = 0 by the Whitney trick.
This was first done by André Haefliger, and this approach is fruitful in codimension 3 or more – from the point of view of surgery theory, this is "high (co)dimension", unlike codimension 2 which is the knotting dimension, as in knot theory.
It is studied categorically via the "calculus of functors" by Thomas Goodwillie Archived 2009-11-28 at the Wayback Machine, John Klein, and Michael S. Weiss.
Immersed plane curves have a well-defined turning number, which can be defined as the total curvature divided by 2π.
This is invariant under regular homotopy, by the Whitney–Graustein theorem – topologically, it is the degree of the Gauss map, or equivalently the winding number of the unit tangent (which does not vanish) about the origin.
Further, this is a complete set of invariants – any two plane curves with the same turning number are regular homotopic.
While immersed plane curves, up to regular homotopy, are determined by their turning number, knots have a very rich and complex structure.
[5] In some cases the obstruction is 2-torsion, such as in Koschorke's example,[6] which is an immersed surface (formed from 3 Möbius bands, with a triple point) that does not lift to a knotted surface, but it has a double cover that does lift.
A far-reaching generalization of immersion theory is the homotopy principle: one may consider the immersion condition (the rank of the derivative is always k) as a partial differential relation (PDR), as it can be stated in terms of the partial derivatives of the function.
