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A degree two map of a sphere onto itself.

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,^[1] 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 Sⁿ to Sⁿ

The simplest and most important case is the degree of a continuous map from the ndisplaystyle n-sphere Sndisplaystyle S^n to itself (in the case n=1displaystyle n=1, this is called the winding number):

Let f:Sn→Sndisplaystyle fcolon S^nto S^n be a continuous map. Then fdisplaystyle f induces a homomorphism f∗:Hn(Sn)→Hn(Sn)displaystyle f_*colon H_nleft(S^nright)to H_nleft(S^nright), where Hn(⋅)displaystyle H_nleft(cdot right) is the ndisplaystyle nth homology group. Considering the fact that Hn(Sn)≅Zdisplaystyle H_nleft(S^nright)cong mathbb Z , we see that f∗displaystyle f_* must be of the form f∗:x↦αxdisplaystyle f_*colon xmapsto alpha x for some fixed α∈Zdisplaystyle alpha in mathbb Z .
This αdisplaystyle alpha is then called the degree of fdisplaystyle f.

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

f∗([X])=deg⁡(f)[Y].displaystyle 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)=x1,x2,…,xn.displaystyle f^-1(p)=x_1,x_2,ldots ,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 (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: ⟨c,ω⟩=∫cωdisplaystyle langle c,omega rangle =int _comega , where cdisplaystyle c is a homology class represented by a cycle cdisplaystyle c and ωdisplaystyle omega a closed form representing a de Rham cohomology class. For a smooth map f : X→Y between orientable m-manifolds, one has

⟨f∗[c],[ω]⟩=⟨[c],f∗[ω]⟩,displaystyle 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∫Yω=∫Xf∗ωdisplaystyle deg fint _Yomega =int _Xf^*omega ,

for any m-form ω on Y.

Maps from closed region

If Ω⊂Rndisplaystyle Omega subset mathbb R ^nis a bounded region, f:Ω¯→Rndisplaystyle f:bar Omega to mathbb R ^n smooth, pdisplaystyle p a regular value of fdisplaystyle f and
p∉f(∂Ω)displaystyle pnotin f(partial Omega ), then the degree deg⁡(f,Ω,p)displaystyle deg(f,Omega ,p) is defined
by the formula

deg⁡(f,Ω,p):=∑y∈f−1(p)sgn⁡detDf(y)displaystyle deg(f,Omega ,p):=sum _yin f^-1(p)operatorname sgn det Df(y)

where Df(y)displaystyle Df(y) is the Jacobi matrix of fdisplaystyle f in ydisplaystyle y.
This definition of the degree may be naturally extended for non-regular values pdisplaystyle p such that deg⁡(f,Ω,p)=deg⁡(f,Ω,p′)displaystyle deg(f,Omega ,p)=deg(f,Omega ,p') where p′displaystyle p' is a point close to pdisplaystyle p.

The degree satisfies the following properties:^[2]

• If deg⁡(f,Ω¯,p)≠0displaystyle deg(f,bar Omega ,p)neq 0, then there exists x∈Ωdisplaystyle xin Omega such that f(x)=pdisplaystyle f(x)=p.


• 
  deg⁡(id,Ω,y)=1displaystyle deg(operatorname id ,Omega ,y)=1 for all y∈Ωdisplaystyle yin Omega .


• Decomposition property:

deg⁡(f,Ω,y)=deg⁡(f,Ω1,y)+deg⁡(f,Ω2,y)displaystyle deg(f,Omega ,y)=deg(f,Omega _1,y)+deg(f,Omega _2,y), if Ω1,Ω2displaystyle Omega _1,Omega _2 are disjoint parts of Ω=Ω1∪Ω2displaystyle Omega =Omega _1cup Omega _2 and y∉f(Ω¯∖(Ω1∪Ω2))displaystyle ynot in f(overline Omega setminus (Omega _1cup Omega _2)).

• 
  Homotopy invariance: If fdisplaystyle f and gdisplaystyle g are homotopy equivalent via a homotopy F(t)displaystyle F(t) such that F(0)=f,F(1)=gdisplaystyle F(0)=f,,F(1)=g and p∉F(t)(∂Ω)displaystyle pnotin F(t)(partial Omega ), then deg⁡(f,Ω,p)=deg⁡(g,Ω,p)displaystyle deg(f,Omega ,p)=deg(g,Omega ,p)
  


• The function p↦deg⁡(f,Ω,p)displaystyle pmapsto deg(f,Omega ,p) is locally constant on Rn−f(∂Ω)displaystyle mathbb R ^n-f(partial Omega )
  

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 f,g:Sn→Sndisplaystyle f,g:S^nto S^n, are homotopic if and only if deg⁡(f)=deg⁡(g)displaystyle deg(f)=deg(g).

In other words, degree is an isomorphism between [Sn,Sn]=πnSndisplaystyle [S^n,S^n]=pi _nS^n and Zdisplaystyle mathbf Z .

Moreover, the Hopf theorem states that for any ndisplaystyle n-dimensional closed oriented manifold M, two maps f,g:M→Sndisplaystyle f,g:Mto S^n are homotopic if and only if deg⁡(f)=deg⁡(g).displaystyle deg(f)=deg(g).

A self-map f:Sn→Sndisplaystyle f:S^nto S^n of the n-sphere is extendable to a map F:Bn→Sndisplaystyle F:B_nto S^n from the n-ball to the n-sphere if and only if deg⁡(f)=0displaystyle deg(f)=0. (Here the function F extends f in the sense that f is the restriction of F to Sndisplaystyle S^n.)

See also

• 
  Covering number, a similarly named term. Note that it does not generalize the winding number but describes covers of a set by balls


• 
  Density (polytope), a polyhedral analog


• Topological degree theory

Notes

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.
  


• 
  Milnor, J.W. (1997). Topology from the Differentiable Viewpoint. Princeton University Press. ISBN 978-0-691-04833-8.
  


• 
  Outerelo, E.; Ruiz, J.M. (2009). Mapping Degree Theory. American Mathematical Society. ISBN 978-0-8218-4915-6.
  

External links

• 
  Hazewinkel, Michiel, ed. (2001) [1994], "Brouwer degree", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
  


• 
  TopDeg: Software tool for computing the topological degree of a continuous function (LGPL-3)