
In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the Borsuk–Ulam theorem states that every
continuous function
In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies there are no abrupt changes in value, known as '' discontinuities''. More preci ...
from an
''n''-sphere into
Euclidean ''n''-space maps some pair of
antipodal point
In mathematics, two points of a sphere (or n-sphere, including a circle) are called antipodal or diametrically opposite if they are the endpoints of a diameter, a straight line segment between two points on a sphere and passing through its cen ...
s to the same point. Here, two points on a sphere are called antipodal if they are in exactly opposite directions from the sphere's center.
Formally: if
is continuous then there exists an
such that:
.
The case
can be illustrated by saying that there always exist a pair of opposite points on the
Earth
Earth is the third planet from the Sun and the only astronomical object known to Planetary habitability, harbor life. This is enabled by Earth being an ocean world, the only one in the Solar System sustaining liquid surface water. Almost all ...
's equator with the same temperature. The same is true for any circle. This assumes the temperature varies continuously in space, which is, however, not always the case.
The case
is often illustrated by saying that at any moment, there is always a pair of antipodal points on the Earth's surface with equal temperatures and equal barometric pressures, assuming that both parameters vary continuously in space.
The Borsuk–Ulam theorem has several equivalent statements in terms of
odd functions. Recall that
is the
''n''-sphere and
is the
''n''-ball:
* If
is a continuous odd function, then there exists an
such that:
.
* If
is a continuous function which is odd on
(the boundary of
), then there exists an
such that:
.
History
According to , the first historical mention of the statement of the Borsuk–Ulam theorem appears in . The first proof was given by , where the formulation of the problem was attributed to
Stanisław Ulam
Stanisław Marcin Ulam ( ; 13 April 1909 – 13 May 1984) was a Polish and American mathematician, nuclear physicist and computer scientist. He participated in the Manhattan Project, originated the History of the Teller–Ulam design, Telle ...
. Since then, many alternative proofs have been found by various authors, as collected by .
Equivalent statements
The following statements are equivalent to the Borsuk–Ulam theorem.
With odd functions
A function
is called ''odd'' (aka ''antipodal'' or ''antipode-preserving'') if for every
,
.
The Borsuk–Ulam theorem is equivalent to each of the following statements:
(1) Each continuous odd function
has a zero.
(2) There is no continuous odd function
.
Here is a proof that the Borsuk-Ulam theorem is equivalent to (1):
(
) If the theorem is correct, then it is specifically correct for odd functions, and for an odd function,
iff
. Hence every odd continuous function has a zero.
(
) For every continuous function
, the following function is continuous and odd:
. If every odd continuous function has a zero, then
has a zero, and therefore,
.
To prove that (1) and (2) are equivalent, we use the following continuous odd maps:
* the obvious inclusion
,
* and the radial projection map
given by
.
The proof now writes itself.
We prove the contrapositive. If there exists a continuous odd function
, then
is a continuous odd function
.
Again we prove the contrapositive. If there exists a continuous odd function
, then
is a continuous odd function
.
Proofs
1-dimensional case
The 1-dimensional case can easily be proved using the
intermediate value theorem
In mathematical analysis, the intermediate value theorem states that if f is a continuous function whose domain contains the interval , then it takes on any given value between f(a) and f(b) at some point within the interval.
This has two imp ...
(IVT).
Let
be the odd real-valued continuous function on a circle defined by
. Pick an arbitrary
. If
then we are done. Otherwise, without loss of generality,
But
Hence, by the IVT, there is a point
at which
.
General case
Algebraic topological proof
Assume that
is an odd continuous function with
(the case
is treated above, the case
can be handled using basic
covering theory). By passing to orbits under the antipodal action, we then get an induced continuous function
between
real projective spaces, which induces an isomorphism on
fundamental groups. By the
Hurewicz theorem, the induced
ring homomorphism
In mathematics, a ring homomorphism is a structure-preserving function between two rings. More explicitly, if ''R'' and ''S'' are rings, then a ring homomorphism is a function that preserves addition, multiplication and multiplicative identity ...
on
cohomology
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed ...
with
coefficients
\mathbb F_2 denotes the GF(2)">field with two elements">GF(2).html" ;"title="here
denotes the GF(2)">field with two elements
:
sends
to
. But then we get that
is sent to
, a contradiction.
[Joseph J. Rotman, ''An Introduction to Algebraic Topology'' (1988) Springer-Verlag ''(See Chapter 12 for a full exposition.)'']
One can also show the stronger statement that any odd map
has odd
degree and then deduce the theorem from this result.
Combinatorial proof
The Borsuk–Ulam theorem can be proved from
Tucker's lemma.
Let be a continuous odd function. Because ''g'' is continuous on a compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact, a type of agreement used by U.S. states
* Blood compact, an ancient ritual of the Philippines
* Compact government, a t ...
domain, it is uniformly continuous. Therefore, for every , there is a such that, for every two points of which are within of each other, their images under ''g'' are within of each other.
Define a triangulation of with edges of length at most . Label each vertex of the triangulation with a label in the following way:
* The absolute value of the label is the ''index'' of the coordinate with the highest absolute value of ''g'': .
* The sign of the label is the sign of ''g'' at the above coordinate, so that: .
Because ''g'' is odd, the labeling is also odd: . Hence, by Tucker's lemma, there are two adjacent vertices with opposite labels. Assume w.l.o.g. that the labels are . By the definition of ''l'', this means that in both and , coordinate #1 is the largest coordinate: in this coordinate is positive while in it is negative. By the construction of the triangulation, the distance between and is at most , so in particular (since and have opposite signs) and so . But since the largest coordinate of is coordinate #1, this means that for each . So , where is some constant depending on and the norm which you have chosen.
The above is true for every ; since is compact there must hence be a point ''u'' in which .
Corollaries
* No subset of is homeomorphic
In mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function betw ...
to
* The ham sandwich theorem: For any compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact, a type of agreement used by U.S. states
* Blood compact, an ancient ritual of the Philippines
* Compact government, a t ...
sets ''A''1, ..., ''An'' in we can always find a hyperplane dividing each of them into two subsets of equal measure.
Equivalent results
Above we showed how to prove the Borsuk–Ulam theorem from Tucker's lemma. The converse is also true: it is possible to prove Tucker's lemma from the Borsuk–Ulam theorem. Therefore, these two theorems are equivalent.
Generalizations
* In the original theorem, the domain of the function ''f'' is the unit ''n''-sphere (the boundary of the unit ''n''-ball). In general, it is true also when the domain of ''f'' is the boundary of any open bounded symmetric subset of containing the origin (Here, symmetric means that if ''x'' is in the subset then -''x'' is also in the subset).
* More generally, if is a compact ''n''-dimensional Riemannian manifold
In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surf ...
, and is continuous, there exists a pair of points ''x'' and ''y'' in such that and ''x'' and ''y'' are joined by a geodesic of length , for any prescribed .
* Consider the function ''A'' which maps a point to its antipodal point: Note that The original theorem claims that there is a point ''x'' in which In general, this is true also for every function ''A'' for which However, in general this is not true for other functions ''A''.
See also
* Topological combinatorics The mathematical discipline of topological combinatorics is the application of topological and algebro-topological methods to solving problems in combinatorics.
History
The discipline of combinatorial topology used combinatorial concepts in topo ...
* Necklace splitting problem
Necklace splitting is a picturesque name given to several related problems in combinatorics and measure theory. Its name and solutions are due to mathematicians Noga Alon and Douglas B. West.
The basic setting involves a necklace with beads of ...
* Ham sandwich theorem
* Kakutani's theorem (geometry)
* Imre Bárány
Notes
References
*
*
*
*
*
External links
*
The Borsuk-Ulam Explorer
An interactive illustration of Borsuk-Ulam Theorem.
{{DEFAULTSORT:Borsuk-Ulam Theorem
Theorems in algebraic topology
Combinatorics
Theory of continuous functions
Theorems in topology