In
mathematics, the Borsuk–Ulam theorem states that every
continuous function from an
''n''-sphere into
Euclidean ''n''-space maps some pair of
antipodal point
In mathematics, antipodal points of a sphere are those diametrically opposite to each other (the specific qualities of such a definition are that a line drawn from the one to the other passes through the center of the sphere so forms a true ...
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's equator with the same temperature. The same is true for any circle. This assumes the temperature varies continuously in space.
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
Stanislaw Ulam
Stanisław Marcin Ulam (; 13 April 1909 – 13 May 1984) was a Polish-American scientist in the fields of mathematics and nuclear physics. He participated in the Manhattan Project, originated the Teller–Ulam design of thermonuclear weapon ...
. 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 the following statement: A continuous odd function from an ''n''-sphere into Euclidean ''n''-space has a zero. PROOF:
* 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,
. Hence the theorem is correct.
With retractions
Define a ''retraction'' as a function
The Borsuk–Ulam theorem is equivalent to the following claim: there is no continuous odd retraction.
Proof: If the theorem is correct, then every continuous odd function from
must include 0 in its range. However,
so there cannot be a continuous odd function whose range is
.
Conversely, if it is incorrect, then there is a continuous odd function
with no zeroes. Then we can construct another odd function
by:
:
since
has no zeroes,
is well-defined and continuous. Thus we have a continuous odd retraction.
Proofs
1-dimensional case
The 1-dimensional case can easily be proved using the
intermediate value theorem (IVT).
Let
be an odd real-valued continuous function on a circle. Pick an arbitrary
. If
then we are done. Otherwise, without loss of generality,
But
Hence, by the IVT, there is a point
between
and
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
In mathematics, the Hurewicz theorem is a basic result of algebraic topology, connecting homotopy theory with homology theory via a map known as the Hurewicz homomorphism. The theorem is named after Witold Hurewicz, and generalizes earlier results ...
, the induced
ring homomorphism 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 viewe ...
with
coefficients
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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
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in Britis ...
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'', 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 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
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in Britis ...
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).
* 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
* Necklace splitting problem
* Ham sandwich theorem
* Kakutani's theorem (geometry) Kakutani's theorem is a result in geometry named after Shizuo Kakutani. It states that every convex body in 3-dimensional space has a circumscribed cube, i.e. a cube all of whose faces touch the body. The result was further generalized by Yamabe ...
* Imre Bárány
Imre Bárány (Mátyásföld, Budapest, 7 December 1947) is a Hungarian mathematician, working in combinatorics and discrete geometry. He works at the Rényi Mathematical Institute of the Hungarian Academy of Sciences, and has a part-time a ...
Notes
References
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External links
*
{{DEFAULTSORT:Borsuk-Ulam Theorem
Algebraic topology
Combinatorics
Theory of continuous functions
Theorems in topology