The Lusternik–Schnirelmann category or LScategory of a space $X$ is a homotopy invariant that gives a smallest integer $k$ such that there is an open covering $\{U_i\}_{i \leq i \leq k}$ of $X$, such that all maps $U_i \hookrightarrow X$ are contractible. The invariant has been generalized in several directions including group actions, foliations, study of simplicial complexes and so on.
Motivation
Rather unfortunately named, LSCategory is not a category in a category theoretic sense, rather it is a numerical homotopy invariant which captures complexity of the space. The concept of topological complexity has been a subject of interest for mathematicians for many years. While one classical approach to measuring this complexity is through cohomology and homology theories, one approach that finds its applications in optimal control, statistics and robotics involves Morse theory, focusing on the critical points of differentiable functions on a manifold. However, there exists a more primal measure of topological complexity that dates back to the work of Lusternik and Schnirelmann in the 1930s.
Definition
The LS category of a topological space $X$, denoted as $\text{LScat}(X)$, is defined as the minimal number $\#a$ of elements in an open cover $\{U_y\}$ of $X$ such that each $U_y$ is nullhomotopic in $X$. In other words, each $U_y$ can be contracted to a point within $X$. If a finite cover does not exist, then $\text{LScat}(X) = \infty$. The geometric category of $X$, denoted as $\text{gcat}(X)$, is defined similarly but requires each $U_y$ to be a contractible set. The invariance properties of $\text{LScat}$ and $\text{gcat}$ are wellestablished and can be found in comprehensive introductions like reference.
Examples and NonExamples
For instance, the LS category of a sphere $S^n$ is $\text{LScat}(S^n) = \text{gcat}(S^n) = 2$ for all $n$. A compact surface $S_g$ of genus $g > 0$ has $\text{LScat}(S_g) = \text{gecat}(S_g) = 3$. It is a challenging task to find a space where $\text{LScat}$ and $\text{gcat}$ differ. One simple example is the space $X = S^2 \vee S^1$, which is obtained by gluing together $S^2$ and $S^1$ at a single point. In this case, $\text{gcat}(X) = 3$, but $\text{LScat}(X) = 2$.
Applications and Theorems
Degenerate Morse Theory
The original motivation for investigating LS category was to extend Morse theory to degenerate cases. In standard Morse theory, the Morse inequalities provide a lower bound on the number of critical points for a smooth, nondegenerate functional on a manifold $M$. The LS category offers a similar lower bound but is applicable even when the functional or the manifold is degenerate.
By above, for example, any smooth functional on the 2torus $T^2$ must have at least three critical points. In the case of a Morse function, the smallest number of critical points is greater than $\dim H(T^3,\mathbb{R})=1+2+1=4$.
Given the complexity of computing the LS category for a general space $X$, mathematicians often resort to estimation strategies. One such strategy involves the cup length, denoted as $\text{cup}(X)$, which is the smallest $N$ such that there exist $N$ cohomology classes with nonzero grading and a nonzero cup product. The LS category is bounded as follows: $\text{cup}(X) < \text{LScat}(X)  1 < \dim X$. This bound holds for more general spaces as well, albeit with some modifications.
See Also
