Immersion (mathematics) - Multiple Points

Multiple Points

A k-tuple point (double, triple, etc.) of an immersion f : M → N is an unordered set {x₁, ..., x_k} of distinct points x_i ∈ M with the same image f(x_i) ∈ N. If M is an m-dimensional manifold and N is an n-dimensional manifold then for an immersion f : M → N in general position the set of k-tuple points is an n−k(n−m)-dimensional manifold. An embedding is an immersion without multiple points (where k > 1). 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 f : Sm → N2m 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 Z 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.

One can study embeddings as "immersions without multiple points", since immersions are easier to classify. Thus, one can start from immersions and try to eliminate multiple points, seeing if one can do this without introducing other singularities – studying "multiple disjunctions". 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, John Klein, and Michael S. Weiss.