Homotopy types at scale three for small torus grids

Determine the homotopy types of the Vietoris–Rips complexes VR(T_{5,5},3), VR(T_{6,6},3), and VR(T_{7,7},3), including whether VR(T_{5,5},3) is homotopy equivalent to a wedge of nine 4-spheres.

Background

The paper identifies VR(T_{n,n},3) for n=5,6,7 as among the smallest complexes for which the authors had not established homotopy types. Computations give reduced homology information: beta_4=9 for VR(T_{5,5},3), beta_3=1 and beta_5=12 for VR(T_{6,6},3), and beta_3=1 and beta_4=14 for VR(T_{7,7},3). The question asks for complete homotopy classifications and specifically tests the natural wedge-of-spheres interpretation suggested by the homology of VR(T_{5,5},3).