Estimating the discrete Lusternik-Schnirelmann category

Abstract

Let $K$ be a simplicial complex and suppose that $K$ collapses onto $L$. Define $n$ to be $1$ less than the minimum number of collapsible sets it takes to cover $L$. Then the discrete geometric Lusternik-Schnirelmann category of $K$ is the smallest $n$ taken over all such $L$. In this paper, we give an algorithm which yields an upper bound for the discrete geometric category. We show our algorithm is correct and give several bounds for the discrete geometric category of well-known simplicial complexes. We show that the discrete geometric category of the dunce cap is $2$, implying that the dunce cap is ``further" from being collapsible than Bing's house whose discrete geometric category is $1$.