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 was first defined by Brouwer, who showed that the degree is homotopy invariant (invariant among homotopies), 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.
Definitions of the degree.
From "S""n" to "S""n".
The simplest and most important case is the degree of a continuous map from the formula_1-sphere formula_2 to itself (in the case formula_3, this is called the winding number):
Let formula_4 be a continuous map. Then formula_5 induces a homomorphism formula_6, where formula_7 is the formula_1th homology group. Considering the fact that formula_9, we see that formula_10 must be of the form formula_11 for some fixed formula_12.
This formula_13 is then called the degree of formula_5.
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"("X") to "Hm"("Y"). Let ["X"], resp. ["Y"] be the chosen generator of "Hm"("X"), resp. "Hm"("Y") (or the fundamental class of "X", "Y"). Then the degree of "f" is defined to be "f"*(["X"]). 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").
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 "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" (though "n" is not!) 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 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: formula_17, where formula_18 is a homology class represented by a cycle formula_18 and formula_20 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"*["X"] = deg "f" · ["Y"], we have
for any "m"-form "ω" on "Y".
Maps from closed region.
If formula_23is a bounded region, formula_24 smooth, formula_25 a regular value of formula_5 and
formula_27, then the degree formula_28 is defined
by the formula
where formula_30 is the Jacobi matrix of formula_5 in formula_32. 
This definition of the degree may be naturally extended for non-regular values formula_25 such that formula_34 where formula_35 is a point close to formula_25.
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 formula_54 are homotopic if and only if formula_55.
In other words, degree is an isomorphism between formula_56 and formula_57.
Moreover, the Hopf theorem states that for any formula_1-dimensional closed oriented manifold "M", two maps formula_59 are homotopic if and only if formula_60
A self-map formula_61 of the "n"-sphere is extendable to a map formula_62 from the "n"-ball to the "n"-sphere if and only if formula_63. (Here the function "F" extends "f" in the sense that "f" is the restriction of "F" to formula_2.)
Calculating the degree.
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 formula_65, where "f" is given in the form of arithmetical expressions. An implementation of the algorithm is available in TopDeg - a software tool for computing the degree (LGPL-3).