Degree of a continuous mapping
In topology, the degree of a continuous mapping between two compact oriented manifolds of the same dimension is a number that 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 between general manifolds was first defined by Brouwer,[1] who showed that the degree is homotopy invariant and used it to prove the Brouwer fixed point theorem.
Less general forms of the concept existed before Brouwer, such as the winding number and the Kronecker characteristic (or Kronecker integral).
[2] 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.
The simplest and most important case is the degree of a continuous map from the
Let X and Y be closed connected oriented m-dimensional manifolds.
Poincare duality implies that the manifold's top homology group is isomorphic to Z.
Choosing an orientation means choosing a generator of the top homology group.
[Y] be the chosen generator of Hm(X), resp.
In other words, 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).
, then 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.
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 codomain of f is connected, the number r − s is independent of the choice of p (though n is not!)
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 an element of Z2 (the field with two elements), 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:
is a homology class represented by a cycle
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.
This definition of the degree may be naturally extended for non-regular values
The topological degree can also be calculated using a surface integral over the boundary of
is a connected n-polytope, then the degree can be expressed as a sum of determinants over a certain subdivision of its facets.
In a similar way, we could define the degree of a map between compact oriented manifolds with boundary.
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
-dimensional closed oriented manifold M, two maps
(Here the function F extends f in the sense that f is the restriction of F to
There is an algorithm for calculating the topological degree deg(f, B, 0) of a continuous function f from an n-dimensional box B (a product of n intervals) to
[6] An implementation of the algorithm is available in TopDeg - a software tool for computing the degree (LGPL-3).
