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 thatit 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 $X$ andare 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.