Difference between revisions of "Lyusternik-Shnirel'man-Borsuk covering theorem"
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A theorem usually stated as follows: | A theorem usually stated as follows: | ||
| − | 1) Each closed covering | + | 1) Each closed covering $\{ A _ { 1 } , \dots , A _ { n + 1 }\}$ of $S ^ { n }$ contains at least one set $A_i$ with $A _ { i } \cap ( - A _ { i } ) \neq \emptyset$. |
Contrary to the equivalent [[Borsuk–Ulam theorem|Borsuk–Ulam theorem]], it seems to be not common to use the same name also for the following equivalent symmetric versions: | Contrary to the equivalent [[Borsuk–Ulam theorem|Borsuk–Ulam theorem]], it seems to be not common to use the same name also for the following equivalent symmetric versions: | ||
| − | 2) Let | + | 2) Let $A _ { 1 } , \dots , A _ { m } \subset S ^ { n }$ be closed sets with $A _ { i } \cap ( - A _ { i } ) = \emptyset$ $( i = 1 , \dots , m )$. If $\cup _ { i = 1 } ^ { m } A _ { i } \cup ( - A _ { i } ) = S ^ { n }$, then $m \geq n + 1$. |
| − | 3) | + | 3) $\operatorname{cat}_{\mathbf{R} P ^ { n }} \mathbf{R}P^n \geq n + 1$ [[#References|[a2]]]. |
| − | In all these results, the estimates are optimal (in 3), in fact, equality holds). It is worth mentioning that 2) gave the motivation for the notion of the genus of a set symmetric with respect to a free | + | In all these results, the estimates are optimal (in 3), in fact, equality holds). It is worth mentioning that 2) gave the motivation for the notion of the genus of a set symmetric with respect to a free $\mathbf{Z} / 2$-action. |
| − | For other equivalent versions and for generalizations to coverings involving other symmetries (e.g. with respect to free | + | For other equivalent versions and for generalizations to coverings involving other symmetries (e.g. with respect to free $\mathbf{Z} / 2$-actions), cf. [[#References|[a3]]] and the references therein. |
One major field of applications are estimates of the number of critical points of even functionals; this can be used, e.g., in the theory of differential equations. | One major field of applications are estimates of the number of critical points of even functionals; this can be used, e.g., in the theory of differential equations. | ||
====References==== | ====References==== | ||
| − | <table>< | + | <table><tr><td valign="top">[a1]</td> <td valign="top"> K. Borsuk, "Drei Sätze über die $n$—dimensionale Sphäre" ''Fund. Math.'' , '''20''' (1933) pp. 177–190</td></tr><tr><td valign="top">[a2]</td> <td valign="top"> L. Lyusternik, L. Shnirel'man, "Topological methods in variational problems" , Issl. Inst. Mat. Mekh. OMGU (1930) (In Russian)</td></tr><tr><td valign="top">[a3]</td> <td valign="top"> H. Steinlein, "Spheres and symmetry: Borsuk's antipodal theorem" ''Topol. Methods Nonlinear Anal.'' , '''1''' (1993) pp. 15–33</td></tr></table> |
Latest revision as of 16:57, 1 July 2020
A theorem usually stated as follows:
1) Each closed covering $\{ A _ { 1 } , \dots , A _ { n + 1 }\}$ of $S ^ { n }$ contains at least one set $A_i$ with $A _ { i } \cap ( - A _ { i } ) \neq \emptyset$.
Contrary to the equivalent Borsuk–Ulam theorem, it seems to be not common to use the same name also for the following equivalent symmetric versions:
2) Let $A _ { 1 } , \dots , A _ { m } \subset S ^ { n }$ be closed sets with $A _ { i } \cap ( - A _ { i } ) = \emptyset$ $( i = 1 , \dots , m )$. If $\cup _ { i = 1 } ^ { m } A _ { i } \cup ( - A _ { i } ) = S ^ { n }$, then $m \geq n + 1$.
3) $\operatorname{cat}_{\mathbf{R} P ^ { n }} \mathbf{R}P^n \geq n + 1$ [a2].
In all these results, the estimates are optimal (in 3), in fact, equality holds). It is worth mentioning that 2) gave the motivation for the notion of the genus of a set symmetric with respect to a free $\mathbf{Z} / 2$-action.
For other equivalent versions and for generalizations to coverings involving other symmetries (e.g. with respect to free $\mathbf{Z} / 2$-actions), cf. [a3] and the references therein.
One major field of applications are estimates of the number of critical points of even functionals; this can be used, e.g., in the theory of differential equations.
References
| [a1] | K. Borsuk, "Drei Sätze über die $n$—dimensionale Sphäre" Fund. Math. , 20 (1933) pp. 177–190 |
| [a2] | L. Lyusternik, L. Shnirel'man, "Topological methods in variational problems" , Issl. Inst. Mat. Mekh. OMGU (1930) (In Russian) |
| [a3] | H. Steinlein, "Spheres and symmetry: Borsuk's antipodal theorem" Topol. Methods Nonlinear Anal. , 1 (1993) pp. 15–33 |
How to Cite This Entry:
Lyusternik-Shnirel'man-Borsuk covering theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lyusternik-Shnirel%27man-Borsuk_covering_theorem&oldid=18675
Lyusternik-Shnirel'man-Borsuk covering theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lyusternik-Shnirel%27man-Borsuk_covering_theorem&oldid=18675
This article was adapted from an original article by H. Steinlein (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article