Lyusternik-Shnirel'man-Borsuk covering theorem

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