Degree of a continuous mapping

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This article is about the term "degree" as used in algebraic topology. For alternate meanings, see degree (mathematics) or degree.

In topology, the term degree is applied to continuous maps between manifolds of the same dimension. The degree of a map can be defined in terms of homology groups or, for smooth maps, in terms of preimages of regular values. It is a generalization of winding number. For example, consider the map zn on the complex plane. Viewed as a map from S2 to itself, it has degree n. It wraps the sphere n times around itself.

In physics, the degree of a continuous map is usually called a topological quantum number.

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[edit] From a circle to itself

The simplest and most important case is the degree of a continuous map from the circle to itself (this is called the winding number):

f\colon S^1\to S^1. \,

There is a projection

\mathbb R \to S^1= \mathbb R/ \mathbb Z \,, x\mapsto [x],

where [x] is the equivalence class of x modulo1 (i.e. x\sim y if and only if x − y is an integer).

If

f : S^1 \to S^1 \,

is continuous then there exists a continuous

F : \mathbb R \to \mathbb R,

called a lift of f to \mathbb R, such that f([z]) = [F(z)]. Such a lift is unique up to an additive integer constant and

deg(f)= F(x + 1)-F(x). \,

Note that

F(x + 1)-F(x) \,

is an integer and it is also continuous with respect to x; locally constant functions on the real line must be constant. Therefore the definition does not depend on choice of x.

[edit] Between manifolds

[edit] Algebraic topology

Let X and Y be closed connected oriented m-dimensional manifolds. Orientability of a manifold implies that its top homology group is isomorphic to Z. Choosing an orientation means choosing a generator of the top homology group.

A continuous map f : X→Y induces a homomorphism f* from Hm(X) to Hm(Y). Let [X] be the chosen generator of Hm(X), or the fundamental class of X. Then the degree of f is defined to be f*([X]). In other words,

f_m([X])=\deg(f)[Y] \, .

If y in Y and f-1(y) is a finite set, the degree of f can be computed by considering the m-th local homology groups of X at each point in f-1(y).

[edit] Differential topology

In the language of differential topology, the degree of a smooth map can be defined as follows: If f is a smooth map whose domain is a compact manifold and p is a regular value of f, consider the finite set

f^{-1}(p)=\{x_1,x_2,..,x_n\} \,.

By p being a regular value, in a neighborhood of each xi the map f is a local diffeomorphism (it is a covering map). Diffeomorphisms can be either orientation preserving or orientation reversing. Let r be the number of points xi at which f is orientation preserving and s be the number at which f is orientation reversing. When the domain of f is connected, the number r - s is independent of the choice of p and one defines the degree of f to be r - s. This definition coincides with the algebraic topological definition above.

The same definition works for compact manifolds with boundary but then f should send the boundary of X to the boundary of Y.

One can also define degree modulo 2 (deg2(f)) the same way as before but taking the fundamental class in Z2 homology. In this case deg2(f) is element of Z2, the manifolds need not be orientable and if n is the number of preimages of p as before then deg2(f) is n modulo 2.

Integration of differential forms gives a pairing between (C∞-)singular homology and de Rham cohomology: <[c], [ω]> = ∫cω, where [c] is a homology class represented by a cycle c and ω a closed form representing a de Rham cohomology class. For a smooth map f : X→Y between orientable m-manifolds, one has

\langle f_* [c], [\omega] \rangle = \langle [c], f^*[\omega] \rangle,

where f* and f* are induced maps on chains and forms respectively. Since f*[X] = deg f · [Y], we have

\deg f \int_Y \omega = \int_X f^*\omega \,

for any m-form ω on Y.

[edit] Properties

A Degree two map of a sphere onto itself.
A Degree two map of a sphere onto itself.

The degree of map is a homotopy invariant; moreover for continuous maps from the sphere to itself it is a complete homotopy invariant, i.e. two maps f,g:S^n\to S^n \, are homotopic if and only if deg(f) = deg(g).

In other words, degree is an isomorphism [S^n,S^n]=\pi_n S^n \to \mathbf{Z}.

[edit] References

  • Flanders, H. (1989). Differential forms with applications to the physical sciences. Dover. 
  • Hirsch, M. (1976). Differential topology. Springer-Verlag. ISBN 0-387-90148-5.