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Certain Homology Cycles of the Independence Complex of Grids
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
2010 (English)In: Discrete & Computational Geometry, ISSN 0179-5376, E-ISSN 1432-0444, Vol. 43, no 4, 927-950 p.Article in journal (Refereed) Published
Abstract [en]

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.

Place, publisher, year, edition, pages
2010. Vol. 43, no 4, 927-950 p.
Keyword [en]
Grid, Independence complex, Simplicial homology, Tiling, Cross-polytope, morse-theory
National Category
URN: urn:nbn:se:kth:diva-19381DOI: 10.1007/s00454-009-9224-9ISI: 000276424500011ScopusID: 2-s2.0-77952011334OAI: diva2:337428
QC 20100525Available from: 2010-08-05 Created: 2010-08-05 Last updated: 2011-01-13Bibliographically approved

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Jonsson, Jakob
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