Special cycles in independence complexes and superfrustration in some lattices (Q390807)

An \textsl{independent set} in a graph \(G\) is a set of vertices which contains no pair of adjacent vertices and the set of all independent sets in \(G\) form an abstract simplicial complex \(I(G)\), called the \textsl{independence complex} of \(G\). Independent sets appear in certain models of statistical physics where configurations of particles are described by independent sets in a lattice. So, topological features of the independence complex of some periodic lattices translate to physical properties of the associated model and this paper is about the estimation of the sum of the reduced Betti numbers (or \textsl{total Betti number}), denoted \(\text{dim}~\widetilde{H}_*(K)\) or \(\widetilde{\beta}_*(K)\) for a simplicial complex \(K\). The \textsl{superfrustration} phenomenom refers to the exponential growth (in the numbers of vertices of the lattice) of these numbers; on these topics, see the previous work of the author [Extremal problems related to Betti numbers of flag complexes (2013; \url{arXiv:1109.4775})] and [\textit{A. Engström}, Eur. J. Comb. 30, No. 2, 429--438 (2009; Zbl 1157.82007)].NEWLINENEWLINEIn this paper, the author finds a lower bound and an upper bound for \(\text{dim}~\widetilde{H}_*\bigl(I(L),{\mathbb Q}\bigr)\) for some lattices \(L\). Specifically, for the lattice \({\mathbb H}\), called \textsl{hexagonal dimer} or \textsl{Kagome lattice}, he gets : NEWLINE\[NEWLINE1.02^{\text{v}({\mathbb H})}~\approx~ \left(2^{1/36}\right)^{\text{v}({\mathbb H})} ~\leq~ \text{dim}~\widetilde{H}_*\bigl(I({\mathbb H}),{\mathbb Q}\bigr) ~\leq ~ \left(14^{1/36}2^{1/6}\right)^{\text{v}({\mathbb H})}~\approx~ 1.21^{\text{v}({\mathbb H})}NEWLINE\]NEWLINE The lower bound is obtained by constructing linearly independent homology classes. They are induced by cycles corresponding to maximal independent sets (themselves given by dominating transversals of matchings in the lattice). Actually, the author proves that these transversals induce a wedge of spheres in a wedge decomposition of a space homotopy equivalent to \(I({\mathbb H})\).