View source for Brouwer degree

''topological degree''

A fundamental concept in [[Algebraic topology|algebraic topology]], [[Differential topology|differential topology]] and [[Mathematical analysis|mathematical analysis]]. It is rooted in the fundamental work of L. Kronecker [[#References|[a5]]] for systems of smooth real-valued functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b1302601.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b1302602.png" /> real variables such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b1302603.png" /> is a regular value for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b1302604.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b1302605.png" /> is bounded and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b1302606.png" /> do not vanish simultaneously on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b1302607.png" />. Letting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b1302608.png" />, Kronecker showed in 1869 that the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b1302609.png" /> defined (in modern notation) by the integral

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b13026010.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b13026011.png" />, is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b13026012.png" />, when this sum makes sense, i.e. when the Jacobian of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b13026013.png" /> does not vanish on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b13026014.png" /> (cf. also [[Jacobian|Jacobian]]). The special case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b13026015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b13026016.png" /> is a closed simple curve was already considered by A. Cauchy in 1837 (the [[Winding number|winding number]]). After several interesting applications to differential equations and function theory by H. Poincaré in 1882–1886 and P.G. Bohl in 1904, in 1910–1912, L.E.J. Brouwer [[#References|[a2]]] and J. Hadamard [[#References|[a3]]] made this Kronecker integral a topological tool by extending it to continuous mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b13026017.png" /> and more general sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b13026018.png" />. Hadamard refined Kronecker's analytical approach, but Brouwer created and used new simplicial techniques to define a (global) degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b13026019.png" /> for continuous mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b13026020.png" /> between two oriented compact boundaryless connected manifolds of the same finite dimension. He used it to prove the theorems on invariance of dimension and invariance of domain (cf. also [[Brouwer theorem|Brouwer theorem]]). Kronecker's integral can be seen as a special case of the Brouwer degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b13026021.png" />, or of the (local) Brouwer degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b13026022.png" />, defined as follows (cf. also [[Degree of a mapping|Degree of a mapping]]).

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b13026023.png" /> is open and bounded, the Brouwer degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b13026024.png" /> of a continuous mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b13026025.png" /> can be defined for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b13026026.png" /> using an approximation scheme introduced by M. Nagumo [[#References|[a8]]] in 1950. The idea consists in defining it first for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b13026027.png" /> smooth and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b13026028.png" /> a regular value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b13026029.png" />, through the formula

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b13026030.png" /></td> </tr></table>

and then to approximate the continuous function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b13026031.png" /> and the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b13026032.png" /> above by a sequence of such functions and points for which this definition holds. This is possible by the Weierstrass approximation theorem (cf. [[Weierstrass theorem|Weierstrass theorem]]) and the [[Sard theorem|Sard theorem]]. The degrees of the approximations stabilize to a common value, denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b13026033.png" /> and being an algebraic count of the number of counter-images of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b13026034.png" /> under <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b13026035.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b13026036.png" />, which is stable for small perturbations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b13026037.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b13026038.png" />. A similar approach can be used to define <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b13026039.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b13026040.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b13026041.png" /> are oriented boundaryless differentiable manifolds.

==Properties and axiomatic characterization.==
The first basic property of the Brouwer degree is its additivity-excision: if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b13026042.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b13026043.png" /> are disjoint open subsets such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b13026044.png" />, then one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b13026045.png" />.

The second property is its homotopy invariance: let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b13026046.png" /> be a bounded open set, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b13026047.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b13026048.png" /> be continuous, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b13026049.png" />; then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b13026050.png" /> is independent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b13026051.png" />.

It has been shown in the 1970s (see [[#References|[a9]]] for references) that the Brouwer degree can be uniquely characterized as the integer-valued function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b13026052.png" /> on the set

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b13026053.png" /></td> </tr></table>

by the additivity-excision and the homotopy invariance properties, together with the following direct consequence of the definition (the normalization property): if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b13026054.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b13026055.png" />.

The additivity-excision property implies the existence property: if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b13026056.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b13026057.png" />.

Easy consequences of the homotopy invariance are the equalities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b13026058.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b13026059.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b13026060.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b13026061.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b13026062.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b13026063.png" /> belong to the same component <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b13026064.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b13026065.png" /> (with the common value written <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b13026066.png" />).

The existence and homotopy properties have many important applications in studying the existence and bifurcation of solutions of various types of equations.

==Product theorem.==
In 1934, J. Leray [[#References|[a6]]] proved a useful product theorem for the Brouwer degree: let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b13026067.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b13026068.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b13026069.png" />, be continuous functions such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b13026070.png" />. Denoting by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b13026071.png" /> the bounded components of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b13026072.png" />, one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b13026073.png" />, where only finitely many terms are different from zero.

This result has deep applications in topology, for example the Jordan separation theorem: for homeomorphic compact subsets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b13026074.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b13026075.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b13026076.png" />, the sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b13026077.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b13026078.png" /> have the same number of connected components.

==Fixed-point theorems.==
An easy consequence of the Brouwer degree is the following Knaster–Kuratowski–Mazurkiewicz fixed-point theorem, first stated and proved in 1929 [[#References|[a4]]]: let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b13026079.png" /> be the closed ball of centre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b13026080.png" /> and radius <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b13026081.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b13026082.png" /> be a continuous function such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b13026083.png" />. Then there is at least one <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b13026084.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b13026085.png" />.

The special case where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b13026086.png" /> is the Brouwer fixed-point theorem [[#References|[a2]]], which has many different and useful equivalent forms.

==Degree of symmetric mappings.==
Useful computational results hold under symmetry assumptions. The oldest one, which corresponds to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b13026087.png" />-symmetry, was conjectured by S.M. Ulam and proved by K. Borsuk in 1933 [[#References|[a1]]]: let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b13026088.png" /> be a bounded open symmetric neighbourhood of the origin in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b13026089.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b13026090.png" /> be a continuous odd function such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b13026091.png" />; then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b13026092.png" /> is odd.

This result and its more recent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b13026093.png" />-version are basic in critical point theory [[#References|[a7]]].

==Other approaches and extensions.==
The Brouwer degree is a very versatile concept which can be defined through techniques of algebraic topology, differential topology or algebraic geometry. For example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b13026094.png" /> is continuous and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b13026095.png" /> is the induced [[Homomorphism|homomorphism]] on the homology groups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b13026096.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b13026097.png" /> (cf. also [[Homology group|Homology group]]), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b13026098.png" /> is isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b13026099.png" /> and hence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b130260100.png" /> becomes multiplication by an integer, which is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b130260101.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b130260102.png" /> is any continuous extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b130260103.png" /> to the closed unit ball <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b130260104.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b130260105.png" />.

In 1995, H. Brézis and L. Nirenberg [[#References|[a10]]], [[#References|[a11]]] defined a Brouwer degree for certain not necessarily continuous mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b130260106.png" /> belonging to a Sobolev or other function space.

Extensions of the Brouwer degree to various classes of mappings between infinite-dimensional spaces are also known. The most fundamental one is the Leray–Schauder degree, defined in 1934 for compact perturbations of the identity defined on the closure of a bounded open subset of a normed vector space (cf. also [[Degree of a mapping|Degree of a mapping]]).

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  K. Borsuk,   "Drei Sätze über die <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b130260107.png" />-dimensionale euklidische Sphäre"  ''Fundam. Math.'' , '''21'''  (1933)  pp. 177–190</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  L.E.J. Brouwer,   "Ueber Abbildungen von Mannigfaltigkeiten"  ''Math. Ann.'' , '''71'''  (1912)  pp. 97–115</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J. Hadamard,   "Sur quelques applications de l'indice de Kronecker"  J. Tannery (ed.) , ''Introduction à la théorie des fonctions d'une variable'' , '''2''' , Hermann  (1910)  pp. 875–915</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  B. Knaster,   C. Kuratowski,   S. Mazurkiewicz,   "Ein Beweis des Fixpunktsatzes für <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130260/b130260108.png" />-dimensionale Simplexe"  ''Fundam. Math.'' , '''14'''  (1929)  pp. 132–137</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  L. Kronecker,   "Ueber Systeme von Funktionen mehrerer Variabeln"  ''Monatsber. Berlin Akad.''  (1869)  pp. 159–193; 688–698</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  J. Leray,   "Topologie des espaces abstraits de M. Banach"  ''C.R. Acad. Sci. Paris'' , '''200'''  (1935)  pp. 1082–1084</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  J. Mawhin,   M. Willem,   "Critical point theory and Hamiltonian systems" , Springer  (1989)</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  M. Nagumo,   "A theory of degree of mapping based on infinitesimal analysis"  ''Amer. J. Math.'' , '''73'''  (1951)  pp. 485–496</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  E. Zeidler,   "Nonlinear functional analysis and its applications" , '''I''' , Springer  (1986)</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  H. Brézis,   L. Nirenberg,   "Degree theory and BMO"  ''Selecta Math.'' , '''1'''  (1995)  pp. 197–263</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  H. Brézis,   L. Nirenberg,   "Degree theory and BMO"  ''Selecta Math.'' , '''2'''  (1996)  pp. 1–60</TD></TR></table>