Extremal Examples of Collapsible Complexes and Random Discrete Morse Theory

We present extremal constructions connected with the property of simplicial collapsibility. (1) For each d≥ 2 , there are collapsible (and shellable) simplicial d-complexes with only one free face. Also., there are non-evasive d-complexes with only two free faces (both results are optimal in all dimensions). (2) Optimal discrete Morse vectors need not be unique. We explicitly construct a contractible, but non-collapsible 3-dimensional simplicial complex with face vector f= (106 , 596 , 1064 , 573) that admits two distinct optimal discrete Morse vectors, (1, 1, 1, 0) and (1, 0, 1, 1). Indeed, we show that in every dimension d≥ 3 there are contractible, non-collapsible simplicial d-complexes that have (1 , 0 , ⋯ , 0 , 1 , 1 , 0) and (1 , 0 , ⋯ , 0 , 0 , 1 , 1) as distinct optimal discrete Morse vectors. (3) We give a first explicit example of a (non-PL) 5-manifold, with face vector f= (5013 , 72300 , 290944 , 495912, 383136, 110880), that is collapsible but not homeomorphic to a ball. Furthermore., we discuss possible improvements and drawbacks of random approaches to collapsibility and discrete Morse theory. We will introduce randomized versions random-lex-first and random-lex-last of the lex-first and lex-last discrete Morse strategies of Benedetti and Lutz (Exp Math 23(1):66–94, 2014), respectively—and we will see that in many instances the random-lex-last strategy works significantly better than Benedetti–Lutz’s (uniform) random strategy. On the theoretical side, we prove that after repeated barycentric subdivisions, the discrete Morse vectors found by randomized algorithms have, on average, an exponential (in the number of barycentric subdivisions) number of critical cells asymptotically almost surely.