Extremal Examples of Collapsible Complexes and Random Discrete Morse Theory

Abstract

We present extremal constructions connected with the property of simplicial collapsibility. (1) For each $d \ge 2$, 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 $d\ge 3$ there are contractible, non-collapsible simplicial d-complexes that have $(1,0,\dots ,0,1,1,0)$ and $(1,0,\dots ,0,0,1,1)$ 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.

Appendix: Random-Lex-First and Random-Lex-Last Strategies

In [8], the random discrete Morse strategy was used to search for small discrete Morse vectors for various input. As pointed out in the previous section, this random procedure, and also the random versions random-lex-first and random-lex-last of the deterministic strategies lex-first and lex-last of [8], respectively, pick up exponentially many critical cells asymptotically almost surely. Yet, on rather huge examples the heuristics often still are successful in finding good or even optimal discrete Morse vectors—with the new random-lex-last strategy working significantly better than the other implementations; cf. Table 2.

As a (non-trivial) testing ground for discrete Morse (heuristical) algorithms, a library of 45 triangulations is provided in [8]. For 39 out of the 45 examples, optimal discrete Morse vectors were found in [8] with the discrete Morse strategy random or with the lex-first and lex-last strategies. For four of the six open cases,

• hyperbolic_dodecahedral_space, (1,4,4,1),

• non_PL, (1,0,0,2,2,1),

• S2xpoincare, (1,2,3,3,2,1),

• contractible_vertex_homogeneous, (1,0,0,4,8,4,0,0,0,0,0,0),

we miss appropriate lower bounds to show optimality of the found discrete Morse vectors. In the case of the 3-ball triangulation knot, (1,1,1,0) was the best vector obtained in [8], while Lewiner claims in [23, p. 92] to have found (1,0,0,0). For the example triple_trefoil_bsd, (1,1,1,1) was reached in [8].

Table 1 Distribution of discrete Morse vectors in 10,000 rounds

Table 2 Average numbers of critical cells in 10,000 runs

Theorem 5.7

The triangulated 3-sphere example triple_trefoil_bsd has the perfect discrete Morse vector (1,0,0,1) in its discrete Morse spectrum.

We found the discrete Morse vector (1,0,0,1) for the example triple_trefoil_ bsd with both the random-lex-first and the random-lex-last strategy in only 7 and 5 out of 10,000 runs, respectively, which provides us with an interesting benchmark example for which the optimal discrete Morse vector (in bold) is hard to find; see Table 1.

Table 1 displays two further examples, nc_sphere and bing, for which the optimal vector (1, 0, 0, 1) was found way more often with the random-lex-first and the random-lex-last strategies, compared to random-discrete-Morse. Also, for the examples poincare and non_PL (and for other examples as well) we see a substantial simplification of the spectrum when using random-lex-last (and random-lex-first), with the best known discrete Morse vector $(1,0,0,2,2,1)$ for the example non_PL in Table 1 in italic. For the library examples listed in Table 2, we highlighted (in bold), which of the three random strategies yields the smallest average of critical cells, and (in italic), if one of the strategies random-lex-first or random-lex-last falls behind the general strategy random-discrete-Morse. In most cases, the random-lex-last strategy scores best—and even in the case triple_trefoil_bsd where it did not, it revealed the optimum. The last column of Table 2 lists the best known theoretical lower bound for the number of critical cells in a discrete Morse vector. The number is in bold if we can prove that it is actually achievable, i.e., there is a discrete Morse function with that many critical cells.