Weak, stable, and ordinary Lusternik-Schnirelmann category of finite spaces

Source abstract

The weak, stable, and ordinary Lusternik--Schnirelmann categories of a finite $T_0$-space $X$ satisfy $\operatorname{cat}_w(X)\leq \operatorname{cat}_s(X)\leq \operatorname{cat}(X)$. We give a general construction answering the simultaneous-strictness question of Cárdenas, Flores, Quintero, and Villar-Liñán. If $P$ is weakly contractible but noncontractible and the deletion of one point makes $P$ contractible, then adjoining $m\geq 2$ incomparable maximal points produces a connected finite space with category triple $(1,2,m)$. Moreover, after any positive number of barycentric subdivisions its ordinary category is equal to $2$. Applying the construction to a nine-point space yields examples on $m+9$ points and, in particular, a twelve-point example for which both inequalities are strict. Using additivity under disjoint unions, we also realize every triple $(a,b,c)$ with $1\leq a<b\leq 2a$ and $c\geq b$, as well as every triple $(a,a,c)$ with $2\leq a\leq c$. We conclude with questions concerning connected realizations and the minimum cardinality of a finite space exhibiting simultaneous strictness.