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

  Published:2013-03-20
 Printed: Aug 2013
  • Jakob Jonsson,
    Department of Mathematics, KTH Royal Institute of Technology, Stockholm, Sweden
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Abstract

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 $\frac{n-4}{3} \le d \le \frac{n-6}{2}$. This paper establishes similar bounds for $\delta \ge 2$. Specifically, there is $3$-torsion in degree $d$ whenever $\frac{(3\delta-1)n-8}{6} \le d \le \frac{\delta (n-1) - 4}{2}$. 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 $3$.
Keywords: simplicial complex, simplicial homology, torsion group, vertex degree simplicial complex, simplicial homology, torsion group, vertex degree
MSC Classifications: 05E45, 55U10, 05C07, 20K10 show english descriptions Combinatorial aspects of simplicial complexes
Simplicial sets and complexes
Vertex degrees [See also 05E30]
Torsion groups, primary groups and generalized primary groups
05E45 - Combinatorial aspects of simplicial complexes
55U10 - Simplicial sets and complexes
05C07 - Vertex degrees [See also 05E30]
20K10 - Torsion groups, primary groups and generalized primary groups
 

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