MathDB

The 5-torsion conjecture for matching complexes

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

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

and let Mn\mathsf{M}_n denote the matching complex of the complete graph on nn vertices. 5-torsion conjecture. The group H~d(Mn;Z)\tilde{H}_d(\mathsf{M}_n;\mathbb{Z}) contains nonvanishing 55-torsion whenever

⌈2n−85⌉≤d≤⌊n−62⌋⟺3≤r≤k+1.\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 ⌊(n−7)/2⌋\left\lfloor(n-7)/2\right\rfloor to ⌊(n−6)/2⌋\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.

Equivalent formulations 1

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

  1. The 5-torsion conjecture for matching complexes

    Let n=2k+1+3rn=2k+1+3r and d=k−1+rd=k-1+r, so that the relevant reduced homology group is H~k−1+r(M2k+1+3r;Z)\tilde{H}_{k-1+r}(\mathsf{M}_{2k+1+3r};\mathbb{Z}). The 5-torsion conjecture. The group contains elements of order 55 whenever r≥3r\geq 3 and k≥r−1k\geq r-1. Equivalently, H~d(Mn;Z)\tilde{H}_d(\mathsf{M}_n;\mathbb{Z}) contains elements of order 55 whenever

    2n−85≤d≤n−62.\frac{2n-8}{5}\leq d\leq\frac{n-6}{2}.

    This extends the ranges where order-55 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).

Sources & referencesView supporting material

Primary source

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

homologymatching-complexestorsion

Progress summary

Never refreshed

Nothing recorded yet. Refresh searches the literature and the public web for attempts on this problem, and writes the first summary here.

Solutions 0

No solutions have been posted yet.