Degree of a continuous mapping

<h2 id="definitions-of-the-degree">Definitions of the degree</h2>
<h3>From Sn to Sn</h3>
The simplest and most important case is the degree of a <a href="/facts/Continuous_map/sKbl02pB">continuous map</a> from the <a href="/facts/N-sphere/bFu2S7E2">
 
 
 
 n
 
 
 {\displaystyle n}
 
-sphere</a> 
 
 
 
 
 S
 
 n
 
 
 
 
 {\displaystyle S^{n}}
 
 to itself (in the case 
 
 
 
 n
 =
 1
 
 
 {\displaystyle n=1}
 
, this is called the winding number):
Let 
 
 
 
 f
 :
 
 S
 
 n
 
 
 →
 
 S
 
 n
 
 
 
 
 {\displaystyle f\colon S^{n}\to S^{n}}
 
 be a continuous map. Then 
 
 
 
 f
 
 
 {\displaystyle f}
 
 induces a <a href="/facts/Pushforward_(homology)/7oWGC08l">pushforward</a> homomorphism 
 
 
 
 
 f
 
 ∗
 
 
 :
 
 H
 
 n
 
 
 
 (
 
 S
 
 n
 
 
 )
 
 →
 
 H
 
 n
 
 
 
 (
 
 S
 
 n
 
 
 )
 
 
 
 {\displaystyle f_{*}\colon H_{n}\left(S^{n}\right)\to H_{n}\left(S^{n}\right)}
 
, where 
 
 
 
 
 H
 
 n
 
 
 
 (
 ⋅
 )
 
 
 
 {\displaystyle H_{n}\left(\cdot \right)}
 
 is the 
 
 
 
 n
 
 
 {\displaystyle n}
 
th <a href="/facts/Homology_group/VFJmryEW">homology group</a>. Considering the fact that 
 
 
 
 
 H
 
 n
 
 
 
 (
 
 S
 
 n
 
 
 )
 
 ≅
 
 Z
 
 
 
 {\displaystyle H_{n}\left(S^{n}\right)\cong \mathbb {Z} }
 
, we see that 
 
 
 
 
 f
 
 ∗
 
 
 
 
 {\displaystyle f_{*}}
 
 must be of the form 
 
 
 
 
 f
 
 ∗
 
 
 :
 x
 ↦
 α
 x
 
 
 {\displaystyle f_{*}\colon x\mapsto \alpha x}
 
 for some fixed 
 
 
 
 α
 ∈
 
 Z
 
 
 
 {\displaystyle \alpha \in \mathbb {Z} }
 
.
This 
 
 
 
 α
 
 
 {\displaystyle \alpha }
 
 is then called the degree of 
 
 
 
 f
 
 
 {\displaystyle f}
 
.

<h3>Between manifolds</h3>
<h4>Algebraic topology</h4>
Let X and Y be closed <a href="/facts/Connected_space/5CUTxfoA">connected</a> <a href="/facts/Orientation_(mathematics)/lGXs42Nw">oriented</a> m-dimensional <a href="/facts/Manifold/rXPERJtu">manifolds</a>. <a href="/facts/Poincare_duality/lwNjUCIa">Poincare duality</a> implies that the manifold's top <a href="/facts/Homology_group/VFJmryEW">homology group</a> 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 Hm(X) to Hm(Y). Let [X], resp. [Y] be the chosen generator of Hm(X), resp. Hm(Y) (or the <a href="/facts/Fundamental_class/bqmdWEEc">fundamental class</a> 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 <a href="/facts/Relative_homology/K3HAWAF8">local homology groups</a> of X at each point in f −1(y). Namely, if 
 
 
 
 
 f
 
 −
 1
 
 
 (
 y
 )
 =
 {
 
 x
 
 1
 
 
 ,
 …
 ,
 
 x
 
 m
 
 
 }
 
 
 {\displaystyle f^{-1}(y)=\{x_{1},\dots ,x_{m}\}}
 
, then

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

<h4>Differential topology</h4>
In the language of <a href="/facts/Differential_topology/Y21eGQe0">differential topology</a>, 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 <a href="/facts/Regular_value/ARWB4sfr">regular value</a> of f, consider the finite set

f
          
            −
            1
          
        
        (
        p
        )
        =
        {
        
          x
          
            1
          
        
        ,
        
          x
          
            2
          
        
        ,
        …
        ,
        
          x
          
            n
          
        
        }
        
        .
      
    
    {\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 <a href="/facts/Diffeomorphism/855N5YYj">diffeomorphism</a>. 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 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 <a href="/facts/Boundary_(topology)/WgdH18kp">boundary</a> 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 <a href="/facts/GF(2)/KIKKpYua">field with two elements</a>), 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 <a href="/facts/Differential_form/T9gyHtMv">differential forms</a> gives a pairing between (C∞-)<a href="/facts/Singular_homology/6c2AHr85">singular homology</a> and <a href="/facts/De_Rham_cohomology/4qGKeJq0">de Rham cohomology</a>: 
 
 
 
 ⟨
 c
 ,
 ω
 ⟩
 =
 
 ∫
 
 c
 
 
 ω
 
 
 {\textstyle \langle c,\omega \rangle =\int _{c}\omega }
 
, where 
 
 
 
 c
 
 
 {\displaystyle c}
 
 is a homology class represented by a cycle 
 
 
 
 c
 
 
 {\displaystyle c}
 
 and 
 
 
 
 ω
 
 
 {\displaystyle \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 \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

deg
        ⁡
        f
        
          ∫
          
            Y
          
        
        ω
        =
        
          ∫
          
            X
          
        
        
          f
          
            ∗
          
        
        ω
        
      
    
    {\displaystyle \deg f\int _{Y}\omega =\int _{X}f^{*}\omega \,}

for any m-form ω on Y.

<h3>Maps from closed region</h3>
If 
 
 
 
 Ω
 ⊂
 
 
 R
 
 
 n
 
 
 
 
 {\displaystyle \Omega \subset \mathbb {R} ^{n}}
 
 is a bounded <a href="/facts/Region_(mathematical_analysis)/sC90ftKL">region</a>, 
 
 
 
 f
 :
 
 
 
 Ω
 ¯
 
 
 
 →
 
 
 R
 
 
 n
 
 
 
 
 {\displaystyle f:{\bar {\Omega }}\to \mathbb {R} ^{n}}
 
 smooth, 
 
 
 
 p
 
 
 {\displaystyle p}
 
 a <a href="/facts/Regular_value/ARWB4sfr">regular value</a> of 
 
 
 
 f
 
 
 {\displaystyle f}
 
 and 
 
 
 
 p
 ∉
 f
 (
 ∂
 Ω
 )
 
 
 {\displaystyle p\notin 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
        ⁡
        det
        (
        D
        f
        (
        y
        )
        )
      
    
    {\displaystyle \deg(f,\Omega ,p):=\sum _{y\in f^{-1}(p)}\operatorname {sgn} \det(Df(y))}

where 
 
 
 
 D
 f
 (
 y
 )
 
 
 {\displaystyle Df(y)}
 
 is the <a href="/facts/Jacobian_matrix_and_determinant/0tNv6o8j">Jacobian matrix</a> of 
 
 
 
 f
 
 
 {\displaystyle f}
 
 in 
 
 
 
 y
 
 
 {\displaystyle y}
 
.
This definition of the degree may be naturally extended for non-regular values 
 
 
 
 p
 
 
 {\displaystyle p}
 
 such that 
 
 
 
 deg
 ⁡
 (
 f
 ,
 Ω
 ,
 p
 )
 =
 deg
 ⁡
 
 (
 
 f
 ,
 Ω
 ,
 
 p
 ′
 
 
 )
 
 
 
 {\displaystyle \deg(f,\Omega ,p)=\deg \left(f,\Omega ,p'\right)}
 
 where 
 
 
 
 
 p
 ′
 
 
 
 {\displaystyle p'}
 
 is a point close to 
 
 
 
 p
 
 
 {\displaystyle p}
 
. The topological degree can also be calculated using a <a href="/facts/Surface_integral/oPk5IRI1">surface integral</a> over the boundary of 
 
 
 
 Ω
 
 
 {\displaystyle \Omega }
 
,<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a> and if 
 
 
 
 Ω
 
 
 {\displaystyle \Omega }
 
 is a connected n-<a href="/facts/Polytope/a76nXrL0">polytope</a>, then the degree can be expressed as a sum of determinants over a certain subdivision of its <a href="/facts/Facet_(geometry)/IreRHGJ7">facets</a>.<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a>
The degree satisfies the following properties:<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a> 

<ul><li>If 
 
 
 
 deg
 ⁡
 
 (
 
 f
 ,
 
 
 
 Ω
 ¯
 
 
 
 ,
 p
 
 )
 
 ≠
 0
 
 
 {\displaystyle \deg \left(f,{\bar {\Omega }},p\right)\neq 0}
 
, then there exists 
 
 
 
 x
 ∈
 Ω
 
 
 {\displaystyle x\in \Omega }
 
 such that 
 
 
 
 f
 (
 x
 )
 =
 p
 
 
 {\displaystyle f(x)=p}
 
.</li>
<li>
 
 
 
 deg
 ⁡
 (
 id
 ,
 Ω
 ,
 y
 )
 =
 1
 
 
 {\displaystyle \deg(\operatorname {id} ,\Omega ,y)=1}
 
 for all 
 
 
 
 y
 ∈
 Ω
 
 
 {\displaystyle y\in \Omega }
 
.</li>
<li>Decomposition property: 
 
 
 
 deg
 ⁡
 (
 f
 ,
 Ω
 ,
 y
 )
 =
 deg
 ⁡
 (
 f
 ,
 
 Ω
 
 1
 
 
 ,
 y
 )
 +
 deg
 ⁡
 (
 f
 ,
 
 Ω
 
 2
 
 
 ,
 y
 )
 ,
 
 
 {\displaystyle \deg(f,\Omega ,y)=\deg(f,\Omega _{1},y)+\deg(f,\Omega _{2},y),}
 
 if 
 
 
 
 
 Ω
 
 1
 
 
 ,
 
 Ω
 
 2
 
 
 
 
 {\displaystyle \Omega _{1},\Omega _{2}}
 
 are disjoint parts of 
 
 
 
 Ω
 =
 
 Ω
 
 1
 
 
 ∪
 
 Ω
 
 2
 
 
 
 
 {\displaystyle \Omega =\Omega _{1}\cup \Omega _{2}}
 
 and 
 
 
 
 y
 ∉
 f
 
 
 (
 
 
 
 Ω
 ¯
 
 
 ∖
 
 (
 
 
 Ω
 
 1
 
 
 ∪
 
 Ω
 
 2
 
 
 
 )
 
 
 )
 
 
 
 
 {\displaystyle y\not \in f{\left({\overline {\Omega }}\setminus \left(\Omega _{1}\cup \Omega _{2}\right)\right)}}
 
.</li>
<li>Homotopy invariance: If 
 
 
 
 f
 
 
 {\displaystyle f}
 
 and 
 
 
 
 g
 
 
 {\displaystyle g}
 
 are <a href="/facts/Homotopy_equivalent/1nWrPxdj">homotopy equivalent</a> via a homotopy 
 
 
 
 F
 (
 t
 )
 
 
 {\displaystyle F(t)}
 
 such that 
 
 
 
 F
 (
 0
 )
 =
 f
 ,
 
 F
 (
 1
 )
 =
 g
 
 
 {\displaystyle F(0)=f,\,F(1)=g}
 
 and 
 
 
 
 p
 ∉
 F
 (
 t
 )
 (
 ∂
 Ω
 )
 
 
 {\displaystyle p\notin F(t)(\partial \Omega )}
 
, then 
 
 
 
 deg
 ⁡
 (
 f
 ,
 Ω
 ,
 p
 )
 =
 deg
 ⁡
 (
 g
 ,
 Ω
 ,
 p
 )
 
 
 {\displaystyle \deg(f,\Omega ,p)=\deg(g,\Omega ,p)}
 
.</li>
<li>The function 
 
 
 
 p
 ↦
 deg
 ⁡
 (
 f
 ,
 Ω
 ,
 p
 )
 
 
 {\displaystyle p\mapsto \deg(f,\Omega ,p)}
 
 is locally constant on 
 
 
 
 
 
 R
 
 
 n
 
 
 −
 f
 (
 ∂
 Ω
 )
 
 
 {\displaystyle \mathbb {R} ^{n}-f(\partial \Omega )}
 
.</li></ul>
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 <a href="/facts/Manifold/rXPERJtu">manifolds with boundary</a>.

<h2 id="properties">Properties</h2>
The degree of a map is a <a href="/facts/Homotopy/1nWrPxdj">homotopy</a> invariant; moreover for continuous maps from the <a href="/facts/N-sphere/bFu2S7E2">sphere</a> to itself it is a complete homotopy invariant, i.e. two maps 
 
 
 
 f
 ,
 g
 :
 
 S
 
 n
 
 
 →
 
 S
 
 n
 
 
 
 
 
 {\displaystyle f,g:S^{n}\to 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 
 
 
 
 
 [
 
 
 S
 
 n
 
 
 ,
 
 S
 
 n
 
 
 
 ]
 
 =
 
 π
 
 n
 
 
 
 S
 
 n
 
 
 
 
 {\displaystyle \left[S^{n},S^{n}\right]=\pi _{n}S^{n}}
 
 and 
 
 
 
 
 Z
 
 
 
 {\displaystyle \mathbf {Z} }
 
.
Moreover, the <a href="/facts/Hopf_theorem/ajIhL09K">Hopf theorem</a> states that for any 
 
 
 
 n
 
 
 {\displaystyle n}
 
-dimensional closed oriented <a href="/facts/Manifold/rXPERJtu">manifold</a> M, two maps 
 
 
 
 f
 ,
 g
 :
 M
 →
 
 S
 
 n
 
 
 
 
 {\displaystyle f,g:M\to S^{n}}
 
 are homotopic if and only if 
 
 
 
 deg
 ⁡
 (
 f
 )
 =
 deg
 ⁡
 (
 g
 )
 .
 
 
 {\displaystyle \deg(f)=\deg(g).}

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

<h2 id="calculating-the-degree">Calculating the degree</h2>
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 
 
 
 
 
 
 R
 
 
 n
 
 
 
 
 {\displaystyle \mathbb {R} ^{n}}
 
, where f is given in the form of arithmetical expressions.<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a> An implementation of the algorithm is available in <a href="http://sourceforge.net/projects/topdeg/">TopDeg</a> - a software tool for computing the degree (LGPL-3).

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Covering_number/pocU9b3r">Covering number</a>, a similarly named term. Note that it does not generalize the winding number but describes covers of a set by balls</li>
<li><a href="/facts/Density_(polytope)/LLQmbxty">Density (polytope)</a>, a polyhedral analog</li>
<li><a href="/facts/Topological_degree_theory/YnxAP32b">Topological degree theory</a></li></ul>
<h2 id="notes">Notes</h2>

<ul><li>Flanders, H. (1989). Differential forms with applications to the physical sciences. Dover.</li>
<li>Hirsch, M. (1976). Differential topology. Springer-Verlag. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-387-90148-5.</li>
<li>Milnor, J.W. (1997). Topology from the Differentiable Viewpoint. Princeton University Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-691-04833-8.</li>
<li>Outerelo, E.; Ruiz, J.M. (2009). Mapping Degree Theory. American Mathematical Society. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-8218-4915-6.</li></ul>
<h2 id="external-links">External links</h2>
<ul><li><a href="https://www.encyclopediaofmath.org/index.php?title=Brouwer_degree">"Brouwer degree"</a>, <a href="/facts/Encyclopedia_of_Mathematics/WC6mGtPm">Encyclopedia of Mathematics</a>, <a href="/facts/European_Mathematical_Society/B3h7b672">EMS Press</a>, 2001 [1994]</li>
<li><a href="http://elib.mi.sanu.ac.rs/files/journals/tm/27/tm1426.pdf">Let's get acquainted with the mapping degree</a>, by Rade T. Zivaljevic.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Brouwer, L. E. J. (1911). "Über Abbildung von Mannigfaltigkeiten". Mathematische Annalen. 71 (1): 97–115. doi:10.1007/bf01456931. S2CID 177796823. <a href="/wiki/Luitzen_Egbertus_Jan_Brouwer" target="_blank">/wiki/Luitzen_Egbertus_Jan_Brouwer</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Siegberg, Hans Willi (1981). "Some Historical Remarks Concerning Degree Theory". The American Mathematical Monthly. 88 (2): 125. doi:10.2307/2321135. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Polymilis, C.; Servizi, G.; Turchetti, G.; Skokos, Ch.; Vrahatis, M. N. (May 2003). "Locating Periodic Orbits by Topological Degree Theory". Libration Point Orbits and Applications: 665–676. arXiv:nlin/0211044. doi:10.1142/9789812704849_0031. ISBN 978-981-238-363-1. <a href="978-981-238-363-1" target="_blank">978-981-238-363-1</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Stynes, Martin (June 1979). "A simplification of Stenger's topological degree formula" (PDF). Numerische Mathematik. 33 (2): 147–155. doi:10.1007/BF01399550. Retrieved 21 September 2024. <a href="https://cs.nyu.edu/~exact/pap/mesh/collection/stynes_simplificationTopDeg79.pdf" target="_blank">https://cs.nyu.edu/~exact/pap/mesh/collection/stynes_simplificationTopDeg79.pdf</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Dancer, E. N. (2000). Calculus of Variations and Partial Differential Equations. Springer-Verlag. pp. 185–225. ISBN 3-540-64803-8. <a href="3-540-64803-8" target="_blank">3-540-64803-8</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Franek, Peter; Ratschan, Stefan (2015). "Effective topological degree computation based on interval arithmetic". Mathematics of Computation. 84 (293): 1265–1290. arXiv:1207.6331. doi:10.1090/S0025-5718-2014-02877-9. ISSN 0025-5718. S2CID 17291092. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
</ol>

Degree of a continuous mapping open-in-new

Degree of a continuous mapping