Let n,d,k,r be related by d=k-1+r,qquad n=2k+3r+1, and let mathsfM_n denote the matching complex of the complete graph on n vertices. 5-torsion conjecture. The group tildeH_d(mathsfM_n;Z) contains nonvanishing 5-torsion whenever lceil frac2n-85rceil≤ d≤lfloor fracn-62rfloor quad Longleftrightarrowquad 3≤ r≤ k+1. The bounds match a previously proved result except for the upper endpoint, which is increased from lfloor(n-7)/2 rfloor to lfloor(n-6)/2 rfloor. The paper presents this as a conjectural extension of the known 5-torsion range, and no resolution is supplied.

The 5-torsion conjecture for matching complexes

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Let $n,d,k,r$ be related by

d=k-1+r,\qquad n=2k+3r+1,

and let $\mathsf{M}_n$ denote the matching complex of the complete graph on $n$ vertices. 5-torsion conjecture. The group $\tilde{H}_d(\mathsf{M}_n;\mathbb{Z})$ contains nonvanishing $5$ -torsion whenever

\left\lceil\frac{2n-8}{5}\right\rceil\leq d\leq\left\lfloor\frac{n-6}{2}\right\rfloor \quad\Longleftrightarrow\quad 3\leq r\leq k+1.

The bounds match a previously proved result except for the upper endpoint, which is increased from $\left\lfloor(n-7)/2\right\rfloor$ to $\left\lfloor(n-6)/2\right\rfloor$ . The paper presents this as a conjectural extension of the known 5-torsion range, and no resolution is supplied.

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Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

The 5-torsion conjecture for matching complexes
Let $n=2k+1+3r$ and $d=k-1+r$ , so that the relevant reduced homology group is $\tilde{H}_{k-1+r}(\mathsf{M}_{2k+1+3r};\mathbb{Z})$ . The 5-torsion conjecture. The group contains elements of order $5$ whenever $r\geq 3$ and $k\geq r-1$ . Equivalently, $\tilde{H}_d(\mathsf{M}_n;\mathbb{Z})$ contains elements of order $5$ whenever
$\frac{2n-8}{5}\leq d\leq\frac{n-6}{2}.$
This extends the ranges where order- $5$ elements are known to occur, but the general assertion is presented as conjectural.
source: Jakob Jonsson, “More Torsion in the Homology of the Matching Complex”, arXiv:1203.5658 (2012).

References

Primary source

Jakob Jonsson, “Exact Sequences for the Homology of the Matching Complex”, arXiv:1203.5641 (2012).

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