# Lusternik-Schnirelmann category

Let $X$ be a topological space. An important topological invariant of $X$ called Lusternik-Schnirelmann category cat is defined as follows:

 ${\rm cat}(X)={\rm min}\{\#(C)\colon\mbox{where C are the coverings of X by% contractible open sets}\}.$

If $X$ is a manifold, ${\rm cat}(X)$ coincides with the minimal number of critical points among all smooth scalars maps $X\to\mathbb{R}$.

This is equivalent to saying that $X$ has a covering $\{U_{s}\}$ such that it is posible to factor homotopically each $U_{s}\lx@stackrel{{\scriptstyle i}}{{\hookrightarrow}}X$ through $U_{s}\lx@stackrel{{\scriptstyle a}}{{\to}}*\lx@stackrel{{\scriptstyle b}}{{\to% }}X$ i.e

 $i\simeq b\circ a.$

This allows us to define another category, e.g.:

We can ask about the minimal number of open sets $U_{s}$ that cover $X$ and are homotopically equivalent to $S^{1}$, say, the inclusion $U_{s}\lx@stackrel{{\scriptstyle i}}{{\hookrightarrow}}X$ and $U_{s}\lx@stackrel{{\scriptstyle a}}{{\to}}S^{1}\lx@stackrel{{\scriptstyle b}}{% {\to}}X$ are $i\simeq b\circ a$.

It is becoming standard to speak of the t-cat of $X$. This is related to the round complexity of the space.

## References

• 1 R.H. Fox, On the Lusternik-Schnirelmann category, Annals of Math. 42 (1941), 333-370.
• 2 F. Takens, The minimal number of critical points of a function on compact manifolds and the Lusternik-Schnirelmann category, Invent. math. 6,(1968), 197-244.
Title Lusternik-Schnirelmann category LusternikSchnirelmannCategory 2013-03-22 15:53:30 2013-03-22 15:53:30 juanman (12619) juanman (12619) 13 juanman (12619) Definition msc 55M30 Topology RoundComplexity