Certain Homology Cycles of the Independence Complex of Grids

KTH Matematik (Avd) (creator_code:org_t)

2009-09-03
2010
English.
In: Discrete & Computational Geometry. - : Springer Science and Business Media LLC. - 0179-5376 .- 1432-0444. ; 43:4, s. 927-950

Related links:: https://link.springe...; show more...; https://urn.kb.se/re...; https://doi.org/10.1...; show less...

Abstract Subject headings

Let G be an infinite graph such that the automorphism group of G contains a subgroup K congruent to Z(d) with the property that G/K is finite. We examine the homology of the independence complex Sigma(G/I) of G/I for subgroups I of K of full rank, focusing on the case that G is the square, triangular, or hexagonal grid. Specifically, we look for a certain kind of homology cycles that we refer to as "cross-cycles," the rationale for the terminology being that they are fundamental cycles of the boundary complex of some cross-polytope. For the special cases just mentioned, we determine the set Q(G, K) of rational numbers r such that there is a group I with the property that Sigma(G/I) contains cross-cycles of degree exactly r . |G/I| - 1; |G/I| denotes the size of the vertex set of G/I. In each of the three cases, Q( G, K) turns out to be an interval of the form [a, b] boolean AND Q = {r is an element of Q : a <= r <= b}. For example, for the square grid, we obtain the interval [1/5, 1/4] boolean AND Q.

About the subject

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