3-torsion in the Homology of Complexes ofGraphs of Bounded Degree

Jonsson, Jakob

Canadian journal of mathematics, Tome 65 (2013) no. 4, pp. 843-862

Résumé

For $\delta \ge 1$ and $n\ge 1$ , 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 $\left( n-4 \right)/3\le d\le \left( n-6 \right)/2$ . This paper establishes similar bounds for $\delta \ge 2$ . Specifically, there is 3-torsion in degree $d$ whenever $$\frac{\left( 3\delta -1 \right)n-8}{6}\le d\le \frac{\delta \left( n-1 \right)-4}{2}.$$ 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.

Détail

DOI : 10.4153/CJM-2013-008-4

Mots-clés : 05E45, 55U10, 05C07, 20K10, simplicial complex, simplicial homology, torsion group, vertex degree