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''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 $f _ { 0 } , \dots , f _ { n }$ of $n$ real variables such that $0$ is a regular value for $f _ { 0 }$, $K : = f _ { 0 } ^ { - 1 } ( ] - \infty , 0 ] )$ is bounded and the $f_j$ do not vanish simultaneously on $\partial K$. Letting $f = ( f _ { 1 } , \dots , f _ { n } )$, Kronecker showed in 1869 that the number $\chi [ f _ { 0 } , \dots , f _ { n } ]$ defined (in modern notation) by the integral

\begin{equation*} \frac { 1 } { \operatorname{vol} S ^ { n - 1 } } \int _ { \partial K } f ^ { * } \omega, \end{equation*}

where $\omega = \sum _ { j = 1 } ^ { n } ( - 1 ) ^ { j - 1 } \| x \| ^ { - n } x _ { j }\, d x _ { 1 } \wedge \ldots \wedge d x _ { j - 1 } \wedge d x _ { j + 1 } \wedge \ldots \wedge d x _ { n }$, is equal to $\sum _ { x \in f ^{ - 1} ( 0 ) \cap \partial K } \text { sign det } f ^ { \prime } ( x )$, when this sum makes sense, i.e. when the Jacobian of $f$ does not vanish on $f ^ { - 1 } ( 0 )$ (cf. also [[Jacobian|Jacobian]]). The special case when $n = 2$ and $\partial K$ 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 $f$ and more general sets $K$. Hadamard refined Kronecker's analytical approach, but Brouwer created and used new simplicial techniques to define a (global) degree $d [ f , M , N ]$ for continuous mappings $f : M \rightarrow N$ 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 $d [ f / \| f \| , \partial K , S ^ { n - 1 } ]$, or of the (local) Brouwer degree $\operatorname{deg}_{B}[f, \operatorname{int} K, 0]$, defined as follows (cf. also [[Degree of a mapping|Degree of a mapping]]).

If $\Omega \subset \mathbf{R} ^ { n }$ is open and bounded, the Brouwer degree $\operatorname{deg}_B [ f , \Omega , y ]$ of a continuous mapping $f : \overline { \Omega } \subset \mathbf{R} ^ { n } \rightarrow \mathbf{R} ^ { n }$ can be defined for each $y \notin f ( \partial \Omega )$ using an approximation scheme introduced by M. Nagumo [[#References|[a8]]] in 1950. The idea consists in defining it first for $f$ smooth and $y$ a regular value of $f$, through the formula

\begin{equation*} \sum _ { x \in f ^ { - 1 } ( y ) } \operatorname { sign } \operatorname { det } f ^ { \prime } ( x ), \end{equation*}

and then to approximate the continuous function $f$ and the point $y$ 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 $\operatorname{deg}_B [ f , \Omega , y ]$ and being an algebraic count of the number of counter-images of $y$ under $f$ in $\Omega$, which is stable for small perturbations of $f$ and $y$. A similar approach can be used to define $d [ f , M , N ]$ when $M$ and $N$ are oriented boundaryless differentiable manifolds.

==Properties and axiomatic characterization.==
The first basic property of the Brouwer degree is its additivity-excision: if $\Omega _ { 1 } \subset \Omega$ and $\Omega _ { 2 } \subset \Omega$ are disjoint open subsets such that $y \notin f ( \overline { \Omega } \backslash ( \Omega _ { 1 } \cup \Omega _ { 2 } ) )$, then one has $\operatorname { deg } _ { B } [ f , \Omega , y ] = \operatorname { deg } _ { B } [ f , \Omega _ { 1 } , y ] + \operatorname { deg } _ { B } [ f , \Omega _ { 2 } , y ]$.

The second property is its homotopy invariance: let $U \subset \mathbf{R} ^ { n } \times [ 0,1 ]$ be a bounded open set, $U _ { \lambda } = \{ x \in \mathbf{R} ^ { n } : ( x , \lambda ) \in U \}$, let $F : \overline { U } \rightarrow \mathbf{R} ^ { n }$ be continuous, and let $y \notin F ( \partial U )$; then $\operatorname { deg } _ { B } [ F ( ., \lambda ) , U _ { \lambda } , y ]$ is independent of $\lambda$.

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 $\operatorname { deg } _ { B }$ on the set

<table class="eq" style="width:100%;"> <tr><td style="width:94%;text-align:center;" valign="top"><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 $y \in \Omega$, then $\operatorname{deg}_{B} [ l , \Omega , y ] = 1$.

The additivity-excision property implies the existence property: if $\deg_B [ f , \Omega , y ] \neq 0$, then $y \in f ( \Omega )$.

Easy consequences of the homotopy invariance are the equalities $\operatorname { deg } _ { B } [ f , \Omega , y ] = \operatorname { deg } _ { B } [ g , \Omega , y ]$ when $f = g$ on $\partial \Omega$, and $\operatorname { deg } _ { B } [ f , \Omega , y ] = \operatorname { deg } _ { B } [ f , \Omega , z ]$ when $y$ and $z$ belong to the same component $C_i$ of $\mathbf{R} ^ { n } \backslash f ( \partial \Omega )$ (with the common value written $\operatorname { deg } _ { B } [ f , \Omega , C _ { i } ]$).

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 $f : \overline { \Omega } \rightarrow \mathbf{R} ^ { n }$ and $g : \overline { \Delta } \rightarrow {\bf R} ^ { n }$, with $\Delta \supset f ( \overline { \Omega } )$, be continuous functions such that $y \notin g \circ f ( \partial \Omega )$. Denoting by $C_i$ the bounded components of $\Delta \backslash f ( \partial \Omega )$, 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 $K _ { 1 }$ and $K _ { 2 }$ of ${\bf R} ^ { n }$, the sets $\mathbf{R} ^ { n } \backslash K _ { 1 }$ and ${\bf R} ^ { n } \backslash K _ { 2 }$ 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 $B [ R ] \subset \mathbf{R} ^ { n }$ be the closed ball of centre $0$ and radius $R$ and let $g : B [ R ] \rightarrow \mathbf{R} ^ { n }$ be a continuous function such that $g ( \partial B [ R ] ) \subset B[R]$. Then there is at least one $x \in B [ R ]$ such that $g ( x ) = x$.

The special case where $g : B [ R ] \rightarrow B [ R ]$ 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 $\mathbf{Z}_{2}$-symmetry, was conjectured by S.M. Ulam and proved by K. Borsuk in 1933 [[#References|[a1]]]: let $\Omega$ be a bounded open symmetric neighbourhood of the origin in ${\bf R} ^ { n }$ and let $f : \overline { \Omega } \rightarrow \mathbf{R} ^ { n }$ be a continuous odd function such that $0 \notin f ( \partial \Omega )$; then $\text{deg}_B[ f , \Omega , 0 ]$ is odd.

This result and its more recent $S ^ { 1 }$-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 $f : S ^ { n } \rightarrow S ^ { n }$ is continuous and $f ^ { * } : H ^ { * } ( S ^ { n } ) \rightarrow H ^ { * } ( S ^ { n } )$ is the induced [[Homomorphism|homomorphism]] on the homology groups of $S ^ { n }$ over $\mathbf{Z}$ (cf. also [[Homology group|Homology group]]), then $H _ { n } ( S ^ { n } )$ is isomorphic to $\mathbf{Z}$ and hence $f _ { n } ^ { * }$ becomes multiplication by an integer, which is $d [ f , S ^ { n } , S ^ { n } ]$. If $\widetilde { f }$ is any continuous extension of $f$ to the closed unit ball $B ( 1 )$, then $d [ f , S ^ { n } , S ^ { n } ] = \operatorname { deg } _ { B } [ \tilde { f } , B ( 1 ) , 0 ]$.

In 1995, H. Brézis and L. Nirenberg [[#References|[a10]]], [[#References|[a11]]] defined a Brouwer degree for certain not necessarily continuous mappings $f$ 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 $n$-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 $n$-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 {{MR|}} {{ZBL|02.0203.02}} </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 {{MR|}} {{ZBL|0011.16402}} {{ZBL|61.0613.03}} </td></tr><tr><td valign="top">[a7]</td> <td valign="top"> J. Mawhin, M. Willem, "Critical point theory and Hamiltonian systems" , Springer (1989) {{MR|0982267}} {{ZBL|0676.58017}} </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 {{MR|0042696}} {{ZBL|0043.17802}} </td></tr><tr><td valign="top">[a9]</td> <td valign="top"> E. Zeidler, "Nonlinear functional analysis and its applications" , '''I''' , Springer (1986) {{MR|0816732}} {{ZBL|0583.47050}} </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 {{MR|1354598}} {{ZBL|0868.58017}} </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 {{MR|1422201}} {{ZBL|0868.58017}} </td></tr></table>