Countable family of torus-grid 3-spheres

Establish the conjectured classification of the specified countable family of Vietoris–Rips complexes VR(T_{n,n},k) as 3-spheres, including the listed candidate parameter pairs, and determine whether these complexes realize the standard genus-one Heegaard decomposition of S^3 by filling the hollow torus with two solid tori.

Background

The paper proves VR(T_{7,7},4)≃S3 and observes homology patterns suggesting infinitely many further 3-sphere examples. The authors list candidate parameter families and conjecture that the resulting complexes are formed by attaching two solid tori to the hollow torus S1×S1, mirroring the standard genus-one Heegaard decomposition of S3.

References

We conjecture that $VR{T_{n,n}{k}$ is homotopy equivalent to a $3$-sphere for a countable family of $(n,k)$ pairs, definitely including $(n,k)=(7,4)$ (see Theorem~\ref{thm:homotopy-through-homology}), and potentially including a subset of \begin{align*} (n,k)\in\ &{(8,5),(9,5)}\cup{(11,6),(12,6)} \ &\cup{(n,7)~|~12\le n\le 15} \ &\cup{(n,8)~|~13\le n\le 19} \ &\cup{(n,9)~|~16\le n\le 21} \ &\cup{(n,10)~|~17\le n\le 24} \ &\cup{(n,11)~|~18\le n\le 26}. \end{align*}

— Vietoris-Rips complexes of torus grids  (2502.07134 - Adams et al., 10 Feb 2025) in Question 8, labeled Question~\ref{ques:3sphere}, Section Conclusion