In
topology
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
, the degree of a
continuous mapping
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in valu ...
between two
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...
oriented manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
s of the same
dimension
In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
is a number that represents the number of times that the
domain
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
**Domain of definition of a partial function
**Natural domain of a partial function
**Domain of holomorphy of a function
* Do ...
manifold wraps around the
range
Range may refer to:
Geography
* Range (geographic), a chain of hills or mountains; a somewhat linear, complex mountainous or hilly area (cordillera, sierra)
** Mountain range, a group of mountains bordered by lowlands
* Range, a term used to i ...
manifold under the mapping. The degree is always an
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
, but may be positive or negative depending on the orientations.
The degree of a map was first defined by
Brouwer Brouwer (also Brouwers and de Brouwer) is a Dutch and Flemish surname. The word ''brouwer'' means 'beer brewer'.
Brouwer
* Adriaen Brouwer (1605–1638), Flemish painter
* Alexander Brouwer (b. 1989), Dutch beach volleyball player
* Andries Bro ...
, who showed that the degree is
homotopy
In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deforma ...
invariant (
invariant
Invariant and invariance may refer to:
Computer science
* Invariant (computer science), an expression whose value doesn't change during program execution
** Loop invariant, a property of a program loop that is true before (and after) each iteratio ...
among homotopies), and used it to prove the
Brouwer fixed point theorem
Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after L. E. J. (Bertus) Brouwer. It states that for any continuous function f mapping a compact convex set to itself there is a point x_0 such that f(x_0)=x_0. The simplest ...
. In modern mathematics, the degree of a map plays an important role in topology and
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
. In
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
, 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
In physics, a topological quantum number (also called topological charge) is any quantity, in a physical theory, that takes on only one of a discrete set of values, due to topological considerations. Most commonly, topological quantum numbers are ...
.
Definitions of the degree
From ''S''''n'' to ''S''''n''
The simplest and most important case is the degree of a
continuous map
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in valu ...
from the
-sphere to itself (in the case
, this is called the
winding number
In mathematics, the winding number or winding index of a closed curve in the plane around a given point is an integer representing the total number of times that curve travels counterclockwise around the point, i.e., the curve's number of turn ...
):
Let
be a continuous map. Then
induces a homomorphism
, where
is the
th
homology group
In mathematics, homology is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, with other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topolog ...
. Considering the fact that
, we see that
must be of the form
for some fixed
.
This
is then called the degree of
.
Between manifolds
Algebraic topology
Let ''X'' and ''Y'' be closed
connected
Connected may refer to:
Film and television
* ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular''
* '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film
* ''Connected'' (2015 TV ...
oriented ''m''-dimensional
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
s. Orientability of a manifold implies that its top
homology group
In mathematics, homology is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, with other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topolog ...
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
In mathematics, the fundamental class is a homology class 'M''associated to a connected orientable compact manifold of dimension ''n'', which corresponds to the generator of the homology group H_n(M,\partial M;\mathbf)\cong\mathbf . The fundam ...
of ''X'', ''Y''). Then the degree of ''f'' is defined to be ''f''
*(
'X''. In other words,
:
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'').
Differential topology
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
In mathematics, a submersion is a differentiable map between differentiable manifolds whose differential is everywhere surjective. This is a basic concept in differential topology. The notion of a submersion is dual to the notion of an immersion ...
of ''f'', consider the finite set
:
By ''p'' being a regular value, in a neighborhood of each ''x''
''i'' the map ''f'' is a local
diffeomorphism
In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable.
Definition
Given two m ...
(it is a
covering map A covering of a topological space X is a continuous map \pi : E \rightarrow X with special properties.
Definition
Let X be a topological space. A covering of X is a continuous map
: \pi : E \rightarrow X
such that there exists a discrete spa ...
). 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
Boundary or Boundaries may refer to:
* Border, in political geography
Entertainment
* ''Boundaries'' (2016 film), a 2016 Canadian film
* ''Boundaries'' (2018 film), a 2018 American-Canadian road trip film
*Boundary (cricket), the edge of the pla ...
but then ''f'' should send the boundary of ''X'' to the boundary of ''Y''.
One can also define degree modulo 2 (deg
2(''f'')) the same way as before but taking the ''fundamental class'' in Z
2 homology. In this case deg
2(''f'') is an element of Z
2 (the
field with two elements), the manifolds need not be orientable and if ''n'' is the number of preimages of ''p'' as before then deg
2(''f'') is ''n'' modulo 2.
Integration of
differential form
In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, ...
s gives a pairing between (C
∞-)
singular homology
In algebraic topology, singular homology refers to the study of a certain set of algebraic invariants of a topological space ''X'', the so-called homology groups H_n(X). Intuitively, singular homology counts, for each dimension ''n'', the ''n''- ...
and
de Rham cohomology:
, where
is a homology class represented by a cycle
and
a closed form representing a de Rham cohomology class. For a smooth map ''f'' : ''X'' →''Y'' between orientable ''m''-manifolds, one has
:
where ''f''
∗ and ''f''
∗ are induced maps on chains and forms respectively. Since ''f''
∗ 'X''= deg ''f'' ·
'Y'' we have
:
for any ''m''-form ''ω'' on ''Y''.
Maps from closed region
If
is a bounded
region
In geography, regions, otherwise referred to as zones, lands or territories, are areas that are broadly divided by physical characteristics (physical geography), human impact characteristics (human geography), and the interaction of humanity and t ...
,
smooth,
a
regular value
In mathematics, a submersion is a differentiable map between differentiable manifolds whose differential is everywhere surjective. This is a basic concept in differential topology. The notion of a submersion is dual to the notion of an immersion ...
of
and
, then the degree
is defined by the formula
:
where
is the
Jacobi matrix of
in
.
This definition of the degree may be naturally extended for non-regular values
such that
where
is a point close to
.
The degree satisfies the following properties:
* If
, then there exists
such that
.
*
for all
.
* Decomposition property:
if
are disjoint parts of
and
.
* ''Homotopy invariance'': If
and
are homotopy equivalent via a homotopy
such that
and
, then
* The function
is locally constant on
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.
Properties
The degree of a map is a
homotopy
In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deforma ...
invariant; moreover for continuous maps from the
sphere
A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
to itself it is a ''complete'' homotopy invariant, i.e. two maps
are homotopic if and only if
.
In other words, degree is an isomorphism between
and
.
Moreover, the
Hopf theorem
The Hopf theorem (named after Heinz Hopf) is a statement in differential topology, saying that the topological degree is the only homotopy invariant of continuous maps to spheres.
Formal statement
Let ''M'' be an ''n''-dimensional compact connec ...
states that for any
-dimensional closed oriented
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
''M'', two maps
are homotopic if and only if
A self-map
of the ''n''-sphere is extendable to a map
from the ''n''-ball to the ''n''-sphere if and only if
. (Here the function ''F'' extends ''f'' in the sense that ''f'' is the restriction of ''F'' to
.)
Calculating the degree
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
, where ''f'' is given in the form of arithmetical expressions.
An implementation of the algorithm is available i
TopDeg - a software tool for computing the degree (LGPL-3).
See also
*
Covering number
In mathematics, a covering number is the number of spherical balls of a given size needed to completely cover a given space, with possible overlaps. Two related concepts are the ''packing number'', the number of disjoint balls that fit in a space ...
, a similarly named term. Note that it does not generalize the winding number but describes covers of a set by balls
*
Density (polytope)
In geometry, the density of a star polyhedron is a generalization of the concept of winding number from two dimensions to higher dimensions,
representing the number of windings of the polyhedron around the center of symmetry of the polyhedron. It ...
, a polyhedral analog
*
Topological degree theory In mathematics, topological degree theory is a generalization of the winding number of a curve in the complex plane. It can be used to estimate the number of solutions of an equation, and is closely connected to fixed-point theory. When one solutio ...
Notes
References
*
*
*
*
External links
* {{springer, title=Brouwer degree, id=p/b130260
Let's get acquainted with the mapping degree, by Rade T. Zivaljevic.
Algebraic topology
Differential topology
Theory of continuous functions