Borsuk – Ulam Theorem

The Borsuk – Ulam theorem is a classical algebraic topology theorem that states that every continuous function that maps $n$ ${\ displaystyle n}$ $n$ -dimensional sphere in $n$ ${\ displaystyle n}$ $n$ -dimensional Euclidean space for a pair of diametrically opposite points has a common meaning.

Informally, the statement is known as the “temperature and pressure theorem”: at any time on the surface of the Earth ( $\mathbb {S} ^{2}$ ${\ displaystyle \ mathbb {S} ^ {2}}$ ${\ displaystyle \ mathbb {S} ^ {2}}$ ) there are antipodal points with equal temperature and equal pressure ^[1] ; the one-dimensional case is usually illustrated by two diametrically opposite points of the equator ( $\mathbb {S} ^{1}$ ${\ displaystyle \ mathbb {S} ^ {1}}$ ${\ displaystyle \ mathbb {S} ^ {1}}$ ) with equal temperature.

Wording

For continuous function $f:\mathbb {S} ^{n}\to \mathbb {R} ^{n}$ ${\ displaystyle f: \ mathbb {S} ^ {n} \ to \ mathbb {R} ^ {n}}$ $f:\mathbb {S} ^{n}\to \mathbb {R} ^{n}$ where $\mathbb {S} ^{n}$ ${\ displaystyle \ mathbb {S} ^ {n}}$ ${\mathbb {S}}^{n}$ - sphere in $(n+1)$ ${\ displaystyle (n + 1)}$ $(n+1)$ -dimensional Euclidean space , there are two diametrically opposite points $a,-a\in \mathbb {S} ^{n}$ ${\ displaystyle a, -a \ in \ mathbb {S} ^ {n}}$ $a,-a\in \mathbb {S} ^{n}$ , what $f(a)=f(-a)$ ${\ displaystyle f (a) = f (-a)}$ $f(a)=f(-a)$ .

History

The statement was first encountered by Lazar Aronovich Lyusternik and Lev Genrikhovich Shnirelman in 1930 ^[2] ^[3] . The first evidence was published in 1933 by Karol Borsuk , in this article he claims that the wording belongs to Stanislav Ulam .

Variations and generalizations

An equivalent statement is a general zero theorem : every odd (with respect to the diametric opposite) continuous function $g:\mathbb {S} ^{n}\to \mathbb {R} ^{n}$ ${\ displaystyle g: \ mathbb {S} ^ {n} \ to \ mathbb {R} ^ {n}}$ $g:\mathbb {S} ^{n}\to \mathbb {R} ^{n}$ of $n$ ${\ displaystyle n}$ $n$ -dimensional sphere in $n$ ${\ displaystyle n}$ $n$ -dimensional Euclidean space at one of the points $a\in \mathbb {S} ^{n}$ ${\ displaystyle a \ in \ mathbb {S} ^ {n}}$ $a\in \mathbb {S} ^{n}$ vanishes: $g(a)=0$ ${\ displaystyle g (a) = 0}$ $g(a)=0$ . Equivalence is established by introducing a continuous function $f:\mathbb {S} ^{n}\to \mathbb {R} ^{n}$ ${\ displaystyle f: \ mathbb {S} ^ {n} \ to \ mathbb {R} ^ {n}}$ $f:\mathbb {S} ^{n}\to \mathbb {R} ^{n}$ odd function $g(x)=f(x)-f(-x)$ ${\ displaystyle g (x) = f (x) -f (-x)}$ $g(x)=f(x)-f(-x)$ . In the one-dimensional case, the general zero theorem follows directly from the intermediate value theorem ; the general proof uses the (algebraic-topological version), or is deduced from ( combinatorial version; in this case, Tucker's lemma is considered to be a combinatorial analogue of the Borsuk – Ulam theorem).

Abram Ilyich Fet proved this statement not only for the ratio of antipodes, but also for arbitrary involution $n$ ${\ displaystyle n}$ $n$ -dimensional sphere. Namely, for any continuous involution $a\mapsto a^{*}$ ${\ displaystyle a \ mapsto a ^ {*}}$ $a\mapsto a^{*}$ spheres $\mathbb {S} ^{n}$ ${\ displaystyle \ mathbb {S} ^ {n}}$ ${\mathbb {S}}^{n}$ and any continuous function $f:\mathbb {S} ^{n}\to \mathbb {R} ^{n}$ ${\ displaystyle f: \ mathbb {S} ^ {n} \ to \ mathbb {R} ^ {n}}$ $f:\mathbb {S} ^{n}\to \mathbb {R} ^{n}$ there is such a point $a\in \mathbb {S} ^{n}$ ${\ displaystyle a \ in \ mathbb {S} ^ {n}}$ $a\in \mathbb {S} ^{n}$ , what $f(a)=f(a^{*})$ ${\ displaystyle f (a) = f (a ^ {*})}$ $f(a)=f(a^{*})$ .

Notes

↑ O. Ya. Viro, O.A. Ivanov, N. Yu. Netsvetaev, V.M. Kharlamov. Elementary topology
↑ L.A. Lyusternik, L.G. Shnirelman. Topological methods in variational problems // Proceedings of the Institute of Mathematics and Mechanics at Moscow State University (special issue). - 1930.
↑ Jiří Matoušek. Using the Borsuk – Ulam theorem. - Berlin: Springer Verlag, 2003 .-- ISBN 3-540-00362-2 . - DOI : 10.1007 / 978-3-540-76649-0 .

Literature

K. Borsuk Drei Sätze über die n -dimensionale euklidische Sphäre - Fund. Math., 20 (1933), p. 177-190.
Borsuk-Ulam theorem implies the Brouwer fixed point theorem
M. Crane , A. Nudelman . The Borsuk – Ulam theorem, or something about the weather, about a trained horse, and about two-dimensional fields // Quantum . - 1983. - No. 8 . - S. 20-25 .

[1] O. Ya. Viro, O.A. Ivanov, N. Yu. Netsvetaev, V.M. Kharlamov. Elementary topology

[2] L.A. Lyusternik, L.G. Shnirelman. Topological methods in variational problems // Proceedings of the Institute of Mathematics and Mechanics at Moscow State University (special issue). - 1930.

[3] Jiří Matoušek. Using the Borsuk – Ulam theorem. - Berlin: Springer Verlag, 2003 .-- ISBN 3-540-00362-2 . - DOI : 10.1007 / 978-3-540-76649-0 .