Lusternik–Schnirelmann theorem

This article is about the theorem on open covers of spheres. For the theorem on simple closed geodesics on spheres, see theorem of the three geodesics.

In mathematics, the Lusternik–Schnirelmann theorem, aka Lusternik–Schnirelmann–Borsuk theorem or LSB theorem, says as follows.

If the sphere Sⁿ is covered by n + 1 closed sets, then one of these sets contains a pair (x, −x) of antipodal points.

This implies that the Lusternik–Schnirelmann category of ⁠${\displaystyle \mathbb {RP} ^{n}}$⁠ is at least ⁠${\displaystyle n+2}$⁠ (and it is not difficult to show that ⁠${\displaystyle n+2}$⁠ sets suffice, so the category is exactly ⁠${\displaystyle n+2}$⁠).

It is named after Lazar Lyusternik and Lev Schnirelmann, who published it in 1930.^[1][2][3]

Proof

The theorem can be proved by leveraging the relationship between ${\displaystyle S^{n}}$ and ${\displaystyle \mathbb {RP} ^{n}}$, and the Borsuk-Ulam theorem in higher dimensions.^[4]

Real Projective Space ${\displaystyle \mathbb {RP} ^{n}}$ is defined as the quotient space of ${\displaystyle S^{n}}$ by the antipodal map, where each point ${\displaystyle x\in S^{n}}$ is identified with its antipodal point ${\displaystyle -x\in S^{n}}$. In other words, ${\displaystyle \mathbb {RP} ^{n}=S^{n}/\sim }$, where ${\displaystyle x\sim -x}$.

A pair of antipodal points in ${\displaystyle S^{n}}$ corresponds to a single point in ${\displaystyle \mathbb {RP} ^{n}}$. The concept of "containing a pair of antipodal points" in a subset of ${\displaystyle S^{n}}$ is equivalent to that subset having a non-empty intersection with the pre-image of some point in ${\displaystyle \mathbb {RP} ^{n}}$ under the projection map ${\displaystyle \pi :S^{n}\to \mathbb {RP} ^{n}}$.

The Borsuk-Ulam theorem states that for any continuous map ${\displaystyle f:S^{n}\to \mathbb {R} ^{n}}$, there exists at least one pair of antipodal points ${\displaystyle \{x,-x\}}$ in ${\displaystyle S^{n}}$ such that ${\displaystyle f(x)=f(-x)}$.

Consider ${\displaystyle S^{n}}$ expressed as the union of ${\displaystyle n+1}$ closed sets ${\displaystyle A_{1},A_{2},\ldots ,A_{n+1}}$.

Define distance functions ${\displaystyle d_{i}(x)=\inf _{y\in A_{i}}|x-y|}$ for ${\displaystyle i=1,\ldots ,n}$. These are continuous functions.

Construct a map ${\displaystyle F:S^{n}\to \mathbb {R} ^{n}}$ given by ${\displaystyle F(x)=(d_{1}(x),d_{2}(x),\ldots ,d_{n}(x))}$.

By the Borsuk-Ulam theorem, there exists a pair of antipodal points ${\displaystyle \{x_{0},-x_{0}\}}$ such that ${\displaystyle F(x_{0})=F(-x_{0})}$, meaning ${\displaystyle d_{i}(x_{0})=d_{i}(-x_{0})}$ for all ${\displaystyle i=1,\ldots ,n}$.

If any ${\displaystyle d_{i}(x_{0})=0}$, then ${\displaystyle x_{0}\in A_{i}}$. Since ${\displaystyle d_{i}(x_{0})=d_{i}(-x_{0})=0}$, then ${\displaystyle -x_{0}\in A_{i}}$ as well (because ${\displaystyle A_{i}}$ is closed). Thus, ${\displaystyle A_{i}}$ contains a pair of antipodal points.

If ${\displaystyle d_{i}(x_{0})>0}$ for all ${\displaystyle i=1,\ldots ,n}$, then ${\displaystyle x_{0}\notin A_{i}}$ for ${\displaystyle i=1,\ldots ,n}$. Since ${\displaystyle S^{n}=\bigcup _{j=1}^{n+1}A_{j}}$, it must be that ${\displaystyle x_{0}\in A_{n+1}}$. Similarly, since ${\displaystyle d_{i}(-x_{0})>0}$ for all ${\displaystyle i=1,\ldots ,n}$, then ${\displaystyle -x_{0}\notin A_{i}}$ for ${\displaystyle i=1,\ldots ,n}$, implying ${\displaystyle -x_{0}\in A_{n+1}}$. Therefore, ${\displaystyle A_{n+1}}$ contains a pair of antipodal points.

Equivalent results

There are several fixed-point theorems which come in three equivalent variants: an algebraic topology variant, a combinatorial variant and a set-covering variant. Each variant can be proved separately using totally different arguments, but each variant can also be reduced to the other variants in its row. Additionally, each result in the top row can be deduced from the one below it in the same column.^[5]

Algebraic topology	Combinatorics	Set covering
Brouwer fixed-point theorem	Sperner's lemma	Knaster–Kuratowski–Mazurkiewicz lemma
Borsuk–Ulam theorem	Tucker's lemma	Lusternik–Schnirelmann theorem