Degree of a continuous mapping

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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.

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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 ${\displaystyle n}$-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

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 : XY 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,

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

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

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

By p being a regular value, in a neighborhood of each xi 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 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 (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: ${\displaystyle \langle c,\omega \rangle =\int _{c}\omega }$, 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 : XY between orientable m-manifolds, one has

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

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

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

for any m-form ω on Y.

Maps from closed region

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

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

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

The degree satisfies the following properties: [2]

• If ${\displaystyle \deg(f,{\bar {\Omega }},p)\neq 0}$, 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 ${\displaystyle y\not \in f({\overline {\Omega }}\setminus (\Omega _{1}\cup \Omega _{2}))}$.
• 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.

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

In other words, degree is an isomorphism between ${\displaystyle [S^{n},S^{n}]=\pi _{n}S^{n}}$ 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 ${\displaystyle F:B_{n}\to S^{n}}$ from the n-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 ${\displaystyle S^{n}}$.)

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 ${\displaystyle \mathbb {R} ^{n}}$, 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).

Notes

1. Brouwer, L. E. J. (1911). "Über Abbildung von Mannigfaltigkeiten". Mathematische Annalen. 71 (1): 97–115. doi:10.1007/bf01456931.
2. Dancer, E. N. (2000). Calculus of Variations and Partial Differential Equations. Springer-Verlag. pp. 185–225. ISBN   3-540-64803-8.
3. Franek, Peter; Ratschan, Stefan (2015). "Effective topological degree computation based on interval arithmetic". Mathematics of Computation. 84 (293): 1265–1290. doi:10.1090/S0025-5718-2014-02877-9. ISSN   0025-5718.

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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.