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| {{About|the term "degree" as used in algebraic topology||Degree (disambiguation){{!}}Degree}}
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| [[File:Sphere wrapped round itself.png|200px|thumb|right|A degree two map of a [[sphere]] onto itself.]]
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| In [[topology]], the '''degree''' of a [[continuous function (topology)|continuous mapping]] between two [[Compact space|compact]] [[Orientability|oriented]] [[manifold]]s of the same [[dimension]] is a number that represents the number of times that the [[Domain of a function|domain]] manifold wraps around the [[Range (mathematics)|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 [[Luitzen Egbertus Jan Brouwer|Brouwer]],<ref>{{cite journal | last = Brouwer | first = L. E. J. | authorlink = Luitzen Egbertus Jan Brouwer | title = Über Abbildung von Mannigfaltigkeiten | journal = Mathematische Annalen | volume = 71 | issue = 1 | pages = 97–115 | year = 1911 | url = http://www.springerlink.com/content/h15uqp1w28862q47}}</ref> who showed that the degree is a [[homotopy]] invariant ([[invariant (mathematics)|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]].
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| ==Definitions of the degree==
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| ===From ''S''<sup>''n''</sup> to ''S''<sup>''n''</sup>===
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| The simplest and most important case is the degree of a [[continuous map]] from <math>S^n</math> to itself (in the case <math>n=1</math>, this is called the [[winding number]]):
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| Let <math>f\colon S^n\to S^n</math> be a continuous map. Then <math>f</math> induces a homomorphism <math>f_*\colon H_n\left(S^n\right)\to H_n\left(S^n\right)</math>. Considering the fact that <math>H_n\left(S^n\right)\cong\mathbb{Z}</math>, we see that <math>f_*</math> must be of the form <math>f_*\colon x\mapsto\alpha x</math> for some fixed <math>\alpha\in\mathbb{Z}</math>.
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| This <math>\alpha</math> is then called the degree of <math>f</math>.
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| ===Between manifolds===
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| ==== Algebraic topology ====
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| Let ''X'' and ''Y'' be closed [[connected space|connected]] [[orientation (mathematics)|oriented]] ''m''-dimensional [[manifold]]s. 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.
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| A continuous map ''f'' : ''X''→''Y'' induces a homomorphism ''f''<sub>*</sub> from ''H<sub>m</sub>''(''X'') to ''H<sub>m</sub>''(''Y''). Let [''X''], resp. [''Y''] be the chosen generator of ''H<sub>m</sub>''(''X''), resp. ''H<sub>m</sub>''(''Y'') (or the [[fundamental class]] of ''X'', ''Y''). Then the '''degree''' of ''f'' is defined to be ''f''<sub>*</sub>([''X'']). In other words,
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| :<math>f_*([X])=\deg(f)[Y] \, .</math>
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| If ''y'' in ''Y'' and ''f'' <sup>−1</sup>(''y'') is a finite set, the degree of ''f'' can be computed by considering the ''m''-th [[Relative homology|local homology groups]] of ''X'' at each point in ''f'' <sup>−1</sup>(''y'').
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| ==== Differential topology ====
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| 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
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| :<math>f^{-1}(p)=\{x_1,x_2,\ldots,x_n\} \,.</math>
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| By ''p'' being a regular value, in a neighborhood of each ''x''<sub>''i''</sub> 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''<sub>''i''</sub> 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.
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| The same definition works for compact manifolds with [[Boundary (topology)|boundary]] but then ''f'' should send the boundary of ''X'' to the boundary of ''Y''.
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| One can also define '''degree modulo 2''' (deg<sub>2</sub>(''f'')) the same way as before but taking the ''fundamental class'' in '''Z'''<sub>2</sub> homology. In this case deg<sub>2</sub>(''f'') is an element of '''Z'''<sub>2</sub> (the [[GF(2)|field with two elements]]), the manifolds need not be orientable and if ''n'' is the number of preimages of ''p'' as before then deg<sub>2</sub>(''f'') is ''n'' modulo 2.
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| Integration of [[differential form]]s gives a pairing between (C<sup>∞</sup>-)[[singular homology]] and [[de Rham cohomology]]: <[''c''], [''ω'']> = ∫<sub>''c''</sub>''ω'', where [''c''] is a homology class represented by a cycle ''c'' and ''ω'' a closed form representing a de Rham cohomology class. For a smooth map ''f'' : ''X''→''Y'' between orientable ''m''-manifolds, one has
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| :<math>\langle f_* [c], [\omega] \rangle = \langle [c], f^*[\omega] \rangle,</math>
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| where ''f''<sub>*</sub> and ''f''* are induced maps on chains and forms respectively. Since ''f''<sub>*</sub>[''X''] = deg ''f'' · [''Y''], we have | |
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| :<math>\deg f \int_Y \omega = \int_X f^*\omega \,</math>
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| for any ''m''-form ''ω'' on ''Y''.
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| ===Maps from closed region===
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| If <math>\Omega\subset\R^n</math>is a bounded [[Region (mathematical analysis)|region]], <math>f:\bar\Omega\to\R^n</math> smooth, <math>p</math> a [[regular value]] of <math>f</math> and
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| <math>p\notin f(\partial\Omega)</math>, then the degree <math>\deg(f,\Omega,p)</math> is defined
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| by the formula
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| :<math>\deg(f,\Omega,p):=\sum_{y\in f^{-1}(p)} \sgn \det Df(y)</math>
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| where <math>Df(y)</math> is the [[Jacobi matrix]] of <math>f</math> in <math>y</math>.
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| This definition of the degree may be naturally extended for non-regular values <math>p</math> such that <math>\deg(f,\Omega,p)=\deg(f,\Omega,p')</math> where <math>p'</math> is a point close to <math>p</math>.
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| The degree satisfies the following properties:<ref name=dancer>{{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}}</ref>
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| * If <math>\deg(f,\bar\Omega,p)\neq 0</math>, then there exists <math>x\in\Omega</math> such that <math>f(x)=p</math>.
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| * <math>\deg(\operatorname{id}, \Omega, y) = 1</math> for all <math>y \in \Omega</math>.
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| *Decomposition property:
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| :<math>\deg(f, \Omega, y) = \deg(f, \Omega_1, y) + \deg(f, \Omega_2, y)</math>, if <math>\Omega_1, \Omega_2</math> are disjoint parts of <math>\Omega=\Omega_1\cup\Omega_2</math> and <math>y \not\in f(\overline{\Omega}\setminus(\Omega_1\cup\Omega_2))</math>.
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| * ''Homotopy invariance'': If <math>f</math> and <math>g</math> are homotopy equivalent via a homotopy <math>F(t)</math> such that <math>F(0)=f,\,F(1)=g</math> and <math>p\notin F(t)(\partial\Omega)</math>, then <math>\deg(f,\Omega,p)=\deg(g,\Omega,p)</math>
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| * The function <math>p\mapsto \deg(f,\Omega,p)</math> is locally constant on <math>\R^n-f(\partial\Omega)</math>
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| These properties characterise the degree uniquely and the degree may be defined by them in an axiomatic way.
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| In a similar way, we could define the degree of a map between compact oriented [[Manifold#Manifold with boundary|manifolds with boundary]].
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| ==Properties==
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| The degree of a map is a [[homotopy]] invariant; moreover for continuous maps from the [[n-sphere|sphere]] to itself it is a ''complete'' homotopy invariant, i.e. two maps <math>f,g:S^n\to S^n \,</math> are homotopic if and only if <math>\deg(f) = \deg(g)</math>.
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| In other words, degree is an isomorphism <math>[S^n,S^n]=\pi_n S^n \to \mathbf{Z}</math>.
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| Moreover, the [[Hopf theorem]] states that for any <math>n</math>-[[manifold]] ''M'', two maps <math>f,g: M\to S^n</math> are homotopic if and only if <math>\deg(f)=\deg(g).</math>
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| A map <math>f:S^n\to S^n</math> is extendable to a map <math>F:B_n\to S^n</math> if and only if <math>\deg(f)=0</math>.
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| ==See also==
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| *[[Covering number]], a similarly named term
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| *[[density (polytope)]], a polyhedral analog
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| *[[Topological degree theory]]
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| ==Notes==
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| {{reflist}}
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| ==References==
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| * {{cite book|author=Flanders, H.|title=Differential forms with applications to the physical sciences|publisher=Dover|year=1989}}
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| * {{cite book|author=Hirsch, M.|title=Differential topology|publisher=Springer-Verlag|year=1976|isbn=0-387-90148-5}}
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| * {{cite book|author=Milnor, J.W.|title=Topology from the Differentiable Viewpoint|publisher=Princeton University Press|year=1997|isbn=978-0-691-04833-8}}
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| == External links ==
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| * {{springer|title=Brouwer degree|id=p/b130260}}
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| * [http://sourceforge.net/projects/topdeg/ TopDeg]: Software tool for computing the topological degree of a continuous function (LGPL-3)
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| [[Category:Algebraic topology]]
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| [[Category:Differential topology]]
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| [[Category:Continuous mappings]]
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