Cohen–Lenstra Heuristics for Torsion in Homology of Random Complexes

Abstract

We study torsion in homology of the random d-complex Y ∼ Y d (n, p) experimentally. Our experiments suggest that there is almost always a moment in the process where there is an enormous burst of torsion in homology H d−1 (Y ). This moment seems to coincide with the phase transition studied in [1,20,21] , where cycles in H d (Y ) first appear with high probability.Our main study is the limiting distribution on the q-part of the torsion subgroup of H d−1 (Y ) for small primes q. We find strong evidence for a limiting Cohen-Lenstra distribution, where the probability that the q-part is isomorphic to a given q-group H is inversely proportional to the order of the automorphism group |Aut(H)|.We also study the torsion in homology of the uniform random Q-acyclic 2-complex. This model is analogous to a uniform spanning tree on a complete graph, but more complicated topologically since Kalai showed that the expected order of the torsion group is exponentially large in n 2 [14]. We give experimental evidence that in this model also, the torsion is Cohen-Lenstra distributed in the limit.

Small Simplicial Complexes with Prescribed Torsion in Homology

“…This phenomenon of enormous torsion in homology in this random setting has been observed experimentally, for example by [11] and by [6] but the reason it occurs remains unknown. Nevertheless, Table 2 provides examples of randomly generated simplicial complexes with torsion in homology coming from the Linial-Meshulam torsion burst.…”

“…Proof. The proof will be by induction on d. For d = 2 our complex will be the pure simplicial complex on vertex set {1, 2, 3, 4, 5, 6} with orientation induced by the natural ordering on the vertices and top dimensional faces [1,2,6], [1,3,6], [3,5,6], [2,4,6], [2,3,4], [1,3,4], [1,4,5], [1,2,5], and [2,3,5]. This complex is given as Figure 4.…”

“…The torsion burst in the Linial-Meshulam refers to the apparent emergence of torsion in the codimension-1 homology group immediately before the first nontrivial cycle appears in top homology, that is around the threshold found in [2,10]. For example, computational experiments examining the torsion burst in the Linial-Meshulam model in [6] found a 5-dimensional simplicial complex X on 16 vertices with H 4 (X) ∼ = Z 36 ⊕ Z/1,147,712,621,067,945,810,235,354,141,226,409,657,574,376,675Z.…”

“…Nevertheless, Table 2 provides examples of randomly generated simplicial complexes with torsion in homology coming from the Linial-Meshulam torsion burst. For more background on the torsion burst, see [6]. 17 7.521 × 10 82 5 20 2.451 × 10 389 Table 2: Examples of torsion groups in homology from the torsion burst of random complexes While [6,7,9] provide results and examples establishing that small complexes can have large torsion in homology, our purpose here is to answer an inverse question: for a finite abelian group G and dimension d ≥ 2, how many vertices are necessary to construct a d-dimensional simplicial complex X so that H d−1 (X) T is isomorphic to G?…”

“…For more background on the torsion burst, see [6]. 17 7.521 × 10 82 5 20 2.451 × 10 389 Table 2: Examples of torsion groups in homology from the torsion burst of random complexes While [6,7,9] provide results and examples establishing that small complexes can have large torsion in homology, our purpose here is to answer an inverse question: for a finite abelian group G and dimension d ≥ 2, how many vertices are necessary to construct a d-dimensional simplicial complex X so that H d−1 (X) T is isomorphic to G? Towards answering this question, we define for d ≥ 2 and G a finite abelian group, T d (G) as the minimal number of vertices n so that there is a simplicial complex X on n vertices with the torsion part of H d−1 (X) isomorphic to G.…”