The simplest and most important case is the degree of a continuous map from the -sphere to itself : Let be a continuous map. Then induces a homomorphism, where is the th homology group. Considering the fact that, we see that must be of the form for some fixed. This is then called the degree of.
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 Hm to Hm. Let , resp. be the chosen generator of Hm, resp. Hm. Then the degree of f is defined to be f*. In other words, If y in Y and f−1 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.
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 By p being a regular value, in a neighborhood of each xi the map f is a local diffeomorphism. 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 the same way as before but taking the fundamental class in Z2 homology. In this case deg2 is an element of Z2, the manifolds need not be orientable and if n is the number of preimages of p as before then deg2 is n modulo 2. Integration of differential forms gives a pairing between singular homology and de Rham cohomology:, where is a homology class represented by a cycle and a closed form representing a de Rham cohomology class. For a smooth map f : X→Y between orientable m-manifolds, one has where f* and f* are induced maps on chains and forms respectively. Since f* = deg f · , we have for any m-form ω on Y.
If is a bounded region, smooth, a regular value of and , then the degree is defined by the formula where is the Jacobi matrix of in. This definition of the degree may be naturally extended for non-regular values such that where is a point close to. The degree satisfies the following properties:
These properties characterise the degree uniquely and the degree may be defined by them in an axiomatic way. In a similar way, we could define the degree of a map between compact oriented manifolds with boundary.
Properties
The degree of a map is a homotopy invariant; moreover for continuous maps from the sphere to itself it is a complete homotopy invariant, i.e. two maps are homotopic if and only if. In other words, degree is an isomorphism between and. Moreover, the Hopf theorem states that for any -dimensional closed oriented manifoldM, two maps are homotopic if and only if A self-map of the n-sphere is extendable to a map from the n-ball to the n-sphere if and only if.
Calculating the degree
There is an algorithm for calculating the topological degree deg of a continuous functionf from an n-dimensional box B to, where f is given in the form of arithmetical expressions. An implementation of the algorithm is available in - a software tool for computing the degree.
