3-torsion in the Homology of Complexes of Graphs of Bounded Degree
2013 (English)In: Canadian Journal of Mathematics - Journal Canadien de Mathematiques, ISSN 0008-414X, E-ISSN 1496-2479, Vol. 65, no 4, 843-862 p.Article in journal (Refereed) Published
For delta >= 1 and n >= 1, consider the simplicial complex of graphs on n vertices in which each vertex has degree at most delta; we identify a given graph with its edge set and admit one loop at each vertex. This complex is of some importance in the theory of semigroup algebras. When delta = 1, we obtain the matching complex, for which it is known that there is 3-torsion in degree d of the homology whenever (n - 4)/3 <= d <= (n - 6)/2. This paper establishes similar bounds for delta >= 2. Specifically, there is 3-torsion in degree d whenever (3 delta - 1)n - 8/6 <= d <= delta(n - 1) - 4/2. The procedure for detecting torsion is to construct an explicit cycle z that is easily seen to have the property that 3z is a boundary. Defining a homomorphism that sends z to a non-boundary element in the chain complex of a certain matching complex, we obtain that z itself is a non-boundary. In particular, the homology class of z has order 3.
Place, publisher, year, edition, pages
2013. Vol. 65, no 4, 843-862 p.
simplicial complex, simplicial homology, torsion group, vertex degree
IdentifiersURN: urn:nbn:se:kth:diva-127480DOI: 10.4153/CJM-2013-008-4ISI: 000322356800008ScopusID: 2-s2.0-84880016487OAI: oai:DiVA.org:kth-127480DiVA: diva2:645663
FunderSwedish Research Council, 2006-3279
QC 201309052013-09-052013-08-302013-09-05Bibliographically approved