Degree of a continuous mapping

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

A degree two map of a sphere onto itself.

In topology, the degree is a numerical invariant that describes a continuous mapping between two compact oriented manifolds of the same dimension. Intuitively, the degree represents the number of times that the domain manifold wraps around the range manifold under the mapping. The degree is always an integer, but may be positive or negative depending on the orientations.

The degree of a map was first defined by Brouwer, who showed that the degree is a homotopy invariant, and used it to prove the Brouwer fixed point theorem. In modern mathematics, the degree of a map plays an important role in topology and geometry. In physics, the degree of a continuous map (for instance a map from space to some order parameter set) is one example of a topological quantum number.

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 modulo 1 (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 $x$.

Between manifolds

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 H_m(X) to H_m(Y). Let [X] be the chosen generator of H_m(X), or the fundamental class of X. Then the degree of f is defined to be f_*([X]). In other words,

$f_{*}([X])=\deg(f)[Y]\,.$

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).

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 x_i 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 x_i 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.

Properties

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 $[S^{n},S^{n}]=\pi _{n}S^{n}\to \mathbf {Z}$.

See also

• Topological degree theory

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.