For every integer n≥ 2, if G_3× n is the 3× n grid graph and M(G_3× n) is its matching complex, then there exist a finite index set J and integers d_j≥ 0 such that M(G_3× n)simeq bigvee_j∈ JS^d_j, where the wedge is taken over spheres of possibly different dimensions.

Wedge-of-spheres conjecture for matching complexes of 3×n grids

For every integer $n\geq 2$ , if $G_{3\times n}$ is the $3\times n$ grid graph and $M(G_{3\times n})$ is its matching complex, then there exist a finite index set $J$ and integers $d_j\geq 0$ such that $M(G_{3\times n})\simeq\bigvee_{j\in J}\mathbb{S}^{d_j}$ , where the wedge is taken over spheres of possibly different dimensions.

References

Primary source

arXiv

Additional references

Homology of matching complexes of 3×n grid graphs — arXiv — Pratiksha Chauhan, Anchal Sharma, Samir Shukla

algebraic-topology topological-combinatorics

Posted 1 week ago · edited 1 week ago

Progress summary

Refreshed 1 week ago

Claimed progress

A new preprint settles the connectivity and first nonzero homology in this family, but the claimed decomposition into spheres remains open.

The conjecture asks whether matching complexes of $3\times n$ grids are wedges of spheres. A September 2026 preprint by Pratiksha Chauhan, Anchal Sharma, and Samir Shukla reports substantial progress, while noting that an earlier proposed homotopy-type determination had a gap.

September 2026 partial result

The preprint reports that for $n\geq 2$ , $\widetilde{H}_i(M(G_{3\times n}))=0$ for $i\leq n-2$ and in the top dimension, while $\widetilde{H}_{n-1}(M(G_{3\times n}))\neq 0$ . It also reports simple connectivity for $n\geq 3$ , hence $(n-2)$ -connectivity, and gives $M(G_{3\times 3})\simeq\bigvee_{5}\mathbb{S}^{2}$ . The all- $n$ wedge-of-spheres assertion remains unresolved.

Current status (as of September 2026): Connectivity and the first nonzero homology are claimed for the $3\times n$ family, but the general wedge-of-spheres conjecture remains open.

Sources

No solutions have been posted yet.