Expand all Collapse all | Results 1 - 1 of 1 |
1. CJM 2013 (vol 65 pp. 843)
3-torsion in the Homology of Complexes of Graphs of Bounded Degree 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 Categories:05E45, 55U10, 05C07, 20K10 |