CiteExport$(function(){PrimeFaces.cw("TieredMenu","widget_formSmash_upper_j_idt146",{id:"formSmash:upper:j_idt146",widgetVar:"widget_formSmash_upper_j_idt146",autoDisplay:true,overlay:true,my:"left top",at:"left bottom",trigger:"formSmash:upper:exportLink",triggerEvent:"click"});}); $(function(){PrimeFaces.cw("OverlayPanel","widget_formSmash_upper_j_idt147_j_idt149",{id:"formSmash:upper:j_idt147:j_idt149",widgetVar:"widget_formSmash_upper_j_idt147_j_idt149",target:"formSmash:upper:j_idt147:permLink",showEffect:"blind",hideEffect:"fade",my:"right top",at:"right bottom",showCloseIcon:true});});

Certain Homology Cycles of the Independence Complex of GridsPrimeFaces.cw("AccordionPanel","widget_formSmash_some",{id:"formSmash:some",widgetVar:"widget_formSmash_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_all",{id:"formSmash:all",widgetVar:"widget_formSmash_all",multiple:true});
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PrimeFaces.cw("AccordionPanel","widget_formSmash_responsibleOrgs",{id:"formSmash:responsibleOrgs",widgetVar:"widget_formSmash_responsibleOrgs",multiple:true}); 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]

##### 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

Mathematics
##### Identifiers

URN: urn:nbn:se:kth:diva-19381DOI: 10.1007/s00454-009-9224-9ISI: 000276424500011Scopus ID: 2-s2.0-77952011334OAI: oai:DiVA.org:kth-19381DiVA: diva2:337428
#####

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt434",{id:"formSmash:j_idt434",widgetVar:"widget_formSmash_j_idt434",multiple:true});
#####

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt440",{id:"formSmash:j_idt440",widgetVar:"widget_formSmash_j_idt440",multiple:true});
#####

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt446",{id:"formSmash:j_idt446",widgetVar:"widget_formSmash_j_idt446",multiple:true});
##### Note

QC 20100525Available from: 2010-08-05 Created: 2010-08-05 Last updated: 2017-12-12Bibliographically approved

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.

doi
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CiteExport$(function(){PrimeFaces.cw("TieredMenu","widget_formSmash_lower_j_idt1197",{id:"formSmash:lower:j_idt1197",widgetVar:"widget_formSmash_lower_j_idt1197",autoDisplay:true,overlay:true,my:"left top",at:"left bottom",trigger:"formSmash:lower:exportLink",triggerEvent:"click"});}); $(function(){PrimeFaces.cw("OverlayPanel","widget_formSmash_lower_j_idt1198_j_idt1200",{id:"formSmash:lower:j_idt1198:j_idt1200",widgetVar:"widget_formSmash_lower_j_idt1198_j_idt1200",target:"formSmash:lower:j_idt1198:permLink",showEffect:"blind",hideEffect:"fade",my:"right top",at:"right bottom",showCloseIcon:true});});