From Sⁿ to Sⁿ edit

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

Let ${\displaystyle f\colon S^{n}\to S^{n}}$  be a continuous map. Then ${\displaystyle f}$  induces a homomorphism ${\displaystyle f_{*}\colon H_{n}\left(S^{n}\right)\to H_{n}\left(S^{n}\right)}$ , where ${\displaystyle H_{n}\left(\cdot \right)}$  is the ${\displaystyle n}$ th homology group. Considering the fact that ${\displaystyle H_{n}\left(S^{n}\right)\cong \mathbb {Z} }$ , we see that ${\displaystyle f_{*}}$  must be of the form ${\displaystyle f_{*}\colon x\mapsto \alpha x}$  for some fixed ${\displaystyle \alpha \in \mathbb {Z} }$ . This ${\displaystyle \alpha }$  is then called the degree of ${\displaystyle f}$ .

Between manifolds edit

Algebraic topology edit

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.

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,

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

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). Namely, if ${\displaystyle f^{-1}(y)=\{x_{1},\dots ,x_{m}\}}$ , then

${\displaystyle \deg(f)=\sum _{i=1}^{m}\deg(f|_{x_{i}})\,.}$ 

Differential topology edit

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

${\displaystyle f^{-1}(p)=\{x_{1},x_{2},\ldots ,x_{n}\}\,.}$ 

By p being a regular value, in a neighborhood of each x_i the map f is a local diffeomorphism. 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 codomain 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: ${\textstyle \langle c,\omega \rangle =\int _{c}\omega }$ , where ${\displaystyle c}$  is a homology class represented by a cycle ${\displaystyle c}$  and ${\displaystyle \omega }$  a closed form representing a de Rham cohomology class. For a smooth map f : X →Y between orientable m-manifolds, one has

${\displaystyle \left\langle f_{*}[c],[\omega ]\right\rangle =\left\langle [c],f^{*}[\omega ]\right\rangle ,}$ 

where f_∗ and f^∗ are induced maps on chains and forms respectively. Since f_∗[X] = deg f · [Y], we have

${\displaystyle \deg f\int _{Y}\omega =\int _{X}f^{*}\omega \,}$ 

for any m-form ω on Y.

Maps from closed region edit

If ${\displaystyle \Omega \subset \mathbb {R} ^{n}}$ is a bounded region, ${\displaystyle f:{\bar {\Omega }}\to \mathbb {R} ^{n}}$  smooth, ${\displaystyle p}$  a regular value of ${\displaystyle f}$  and ${\displaystyle p\notin f(\partial \Omega )}$ , then the degree ${\displaystyle \deg(f,\Omega ,p)}$  is defined by the formula

${\displaystyle \deg(f,\Omega ,p):=\sum _{y\in f^{-1}(p)}\operatorname {sgn} \det(Df(y))}$ 

where ${\displaystyle Df(y)}$  is the Jacobian matrix of ${\displaystyle f}$  in ${\displaystyle y}$ .

This definition of the degree may be naturally extended for non-regular values ${\displaystyle p}$  such that ${\displaystyle \deg(f,\Omega ,p)=\deg \left(f,\Omega ,p'\right)}$  where ${\displaystyle p'}$  is a point close to ${\displaystyle p}$ .

The degree satisfies the following properties:^[2]

• If ${\displaystyle \deg \left(f,{\bar {\Omega }},p\right)\neq 0}$ , then there exists ${\displaystyle x\in \Omega }$  such that ${\displaystyle f(x)=p}$ .
• ${\displaystyle \deg(\operatorname {id} ,\Omega ,y)=1}$  for all ${\displaystyle y\in \Omega }$ .
• Decomposition property:
  $${\displaystyle \deg(f,\Omega ,y)=\deg(f,\Omega _{1},y)+\deg(f,\Omega _{2},y),}$$

   
  if ${\displaystyle \Omega _{1},\Omega _{2}}$  are disjoint parts of ${\displaystyle \Omega =\Omega _{1}\cup \Omega _{2}}$  and ${\displaystyle y\not \in f{\left({\overline {\Omega }}\setminus \left(\Omega _{1}\cup \Omega _{2}\right)\right)}}$ .
• Homotopy invariance: If ${\displaystyle f}$  and ${\displaystyle g}$  are homotopy equivalent via a homotopy ${\displaystyle F(t)}$  such that ${\displaystyle F(0)=f,\,F(1)=g}$  and ${\displaystyle p\notin F(t)(\partial \Omega )}$ , then ${\displaystyle \deg(f,\Omega ,p)=\deg(g,\Omega ,p)}$ 
• The function ${\displaystyle p\mapsto \deg(f,\Omega ,p)}$  is locally constant on ${\displaystyle \mathbb {R} ^{n}-f(\partial \Omega )}$ 

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.

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 ${\displaystyle f,g:S^{n}\to S^{n}\,}$  are homotopic if and only if ${\displaystyle \deg(f)=\deg(g)}$ .

In other words, degree is an isomorphism between ${\displaystyle \left[S^{n},S^{n}\right]=\pi _{n}S^{n}}$  and ${\displaystyle \mathbf {Z} }$ .

Moreover, the Hopf theorem states that for any ${\displaystyle n}$ -dimensional closed oriented manifold M, two maps ${\displaystyle f,g:M\to S^{n}}$  are homotopic if and only if ${\displaystyle \deg(f)=\deg(g).}$ 

A self-map ${\displaystyle f:S^{n}\to S^{n}}$  of the n-sphere is extendable to a map ${\displaystyle F:B_{n+1}\to S^{n}}$  from the n+1-ball to the n-sphere if and only if ${\displaystyle \deg(f)=0}$ . (Here the function F extends f in the sense that f is the restriction of F to ${\displaystyle S^{n}}$ .)

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 ${\displaystyle \mathbb {R} ^{n}}$ , where f is given in the form of arithmetical expressions.^[3] An implementation of the algorithm is available in TopDeg - a software tool for computing the degree (LGPL-3).

• 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

• 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.

• "Brouwer degree", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Let's get acquainted with the mapping degree, by Rade T. Zivaljevic.