Estimating the discrete Lusternik–Schnirelmann category

Let K be a simplicial complex and suppose that K collapses onto L. Define n to be 1 minus the minimum number of collapsible sets it takes to cover L. Then the discrete 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 category. We show our algorithm is correct and give several bounds for the discrete category of well-known simplicial complexes. We show that the discrete category of the dunce cap is 2, implying that the dunce cap is “further” from being collapsible than Bing’s house. MSC Classification Primary 55U10, 68Q25; Secondary 57Q15, 55M30, 05E45