Degree of a continuous mapping

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

Let $f\backslash colon\; S^n\backslash to\; S^n$ be a continuous map. Then $f$ induces a pushforward homomorphism $f\_*\backslash colon\; H\_n\backslash left(S^n\backslash right)\; \backslash to\; H\_n\backslash left(S^n\backslash right)$, where $H\_n\backslash left(\backslash cdot\backslash right)$ is the $n$th homology group. Considering the fact that $H\_n\backslash left(S^n\backslash right)\backslash cong\backslash mathbb\{Z\}$, we see that $f\_*$ must be of the form $f\_*\backslash colon\; x\backslash mapsto\backslash alpha\; x$ for some fixed $\backslash alpha\backslash in\backslash mathbb\{Z\}$.

This $\backslash alpha$ is then called the degree of $f$.

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,

:$f\_*([X])\; =\; \backslash deg(f)[Y]\; \backslash ,\; .$

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 $f^\{-1\}(y)=\backslash \{x\_1,\backslash dots,x\_m\backslash \}$, then

:$\backslash deg(f)\; =\; \backslash sum\_\{i=1\}^\{m\}\backslash deg(f|\_\{x\_i\})\; \backslash ,\; .$

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

:$f^\{-1\}(p)\; =\; \backslash \{x\_1,\; x\_2,\; \backslash ldots,\; x\_n\backslash \}\; \backslash ,.$

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: $\backslash langle\; c,\; \backslash omega\backslash rangle\; =\; \backslash int\_c\; \backslash omega$, where $c$ is a homology class represented by a cycle $c$ and $\backslash omega$ a closed form representing a de Rham cohomology class. For a smooth map f: X →Y between orientable m-manifolds, one has

:$\backslash left\backslash langle\; f\_*\; [c],\; [\backslash omega]\; \backslash right\backslash rangle\; =\; \backslash left\backslash langle\; [c],\; f^*[\backslash omega]\; \backslash right\backslash rangle,$

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

:$\backslash deg\; f\; \backslash int\_Y\; \backslash omega\; =\; \backslash int\_X\; f^*\backslash omega\; \backslash ,$

for any m-form ω on Y.

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

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

where $Df(y)$ is the Jacobian matrix of $f$ in $y$.

This definition of the degree may be naturally extended for non-regular values $p$ such that $\backslash deg(f,\; \backslash Omega,\; p)\; =\; \backslash deg\backslash left(f,\; \backslash Omega,\; p\text{'}\backslash right)$ where $p\text{'}$ is a point close to $p$. The topological degree can also be calculated using a surface integral over the boundary of $\backslash Omega$,{{cite journal |last1=Polymilis |first1=C. |last2=Servizi |first2=G. |last3=Turchetti |first3=G. |last4=Skokos |first4=Ch. |last5=Vrahatis |first5=M. N. |journal=Libration Point Orbits and Applications |title=Locating Periodic Orbits by Topological Degree Theory |date=May 2003 |pages=665–676 |doi=10.1142/9789812704849_0031 |arxiv=nlin/0211044 |isbn=978-981-238-363-1 }} and if $\backslash Omega$ is a connected n-polytope, then the degree can be expressed as a sum of determinants over a certain subdivision of its facets.{{cite journal |last1=Stynes |first1=Martin |title=A simplification of Stenger's topological degree formula |journal=Numerische Mathematik |date=June 1979 |volume=33 |issue=2 |pages=147–155 |doi=10.1007/BF01399550 |url=https://cs.nyu.edu/~exact/pap/mesh/collection/stynes_simplificationTopDeg79.pdf |access-date=21 September 2024}}

The degree satisfies the following properties:{{cite book| last=Dancer|first=E. N.|title=Calculus of Variations and Partial Differential Equations| year=2000|publisher=Springer-Verlag| isbn=3-540-64803-8|pages=185–225}}

• If $\backslash deg\backslash left(f,\; \backslash bar\backslash Omega,\; p\backslash right)\; \backslash neq\; 0$, then there exists $x\; \backslash in\; \backslash Omega$ such that $f(x)\; =\; p$.
• $\backslash deg(\backslash operatorname\{id\},\; \backslash Omega,\; y)\; =\; 1$ for all $y\; \backslash in\; \backslash Omega$.
• Decomposition property: $$\backslash deg(f,\; \backslash Omega,\; y)\; =\; \backslash deg(f,\; \backslash Omega\_1,\; y)\; +\; \backslash deg(f,\; \backslash Omega\_2,\; y),$$ if $\backslash Omega\_1,\; \backslash Omega\_2$ are disjoint parts of $\backslash Omega\; =\; \backslash Omega\_1\; \backslash cup\; \backslash Omega\_2$ and $y\; \backslash not\backslash in\; f\{\backslash left(\backslash overline\{\backslash Omega\}\backslash setminus\backslash left(\backslash Omega\_1\; \backslash cup\; \backslash Omega\_2\backslash right)\backslash right)\}$.
• Homotopy invariance: If $f$ and $g$ are homotopy equivalent via a homotopy $F(t)$ such that $F(0)\; =\; f,\backslash ,\; F(1)\; =\; g$ and $p\; \backslash notin\; F(t)(\backslash partial\backslash Omega)$, then $\backslash deg(f,\; \backslash Omega,\; p)\; =\; \backslash deg(g,\; \backslash Omega,\; p)$.
• The function $p\; \backslash mapsto\; \backslash deg(f,\; \backslash Omega,\; p)$ is locally constant on $\backslash R^n\; -\; f(\backslash partial\backslash Omega)$.

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

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

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

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

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 $\backslash R^n$, where f is given in the form of arithmetical expressions.{{Cite journal|last1=Franek|first1=Peter|last2=Ratschan|first2=Stefan|date=2015|title=Effective topological degree computation based on interval arithmetic|journal=Mathematics of Computation|language=en|volume=84|issue=293|pages=1265–1290|doi=10.1090/S0025-5718-2014-02877-9|s2cid=17291092|issn=0025-5718|arxiv=1207.6331}} An implementation of the algorithm is available in [http://sourceforge.net/projects/topdeg/ 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

{{reflist}}

• {{cite book|author=Flanders, H.|title=Differential forms with applications to the physical sciences|publisher=Dover|year=1989}}
• {{cite book|author=Hirsch, M.|title=Differential topology|publisher=Springer-Verlag|year=1976|isbn=0-387-90148-5}}
• {{cite book|author=Milnor, J.W.|title=Topology from the Differentiable Viewpoint|publisher=Princeton University Press|year=1997|isbn=978-0-691-04833-8}}
• {{cite book|author1=Outerelo, E. |author2=Ruiz, J.M. |title=Mapping Degree Theory|publisher=American Mathematical Society|year=2009|isbn=978-0-8218-4915-6}}

• {{springer|title=Brouwer degree|id=p/b130260}}
• [http://elib.mi.sanu.ac.rs/files/journals/tm/27/tm1426.pdf Let's get acquainted with the mapping degree], by Rade T. Zivaljevic.

Category:Algebraic topology

Category:Differential topology

Category:Theory of continuous functions