Representing the β bundles as an infinite fibre that when acts on the geometries been presented as -1 for hyperbolic or saddle curvature and +1 for elliptic or positive curvature with ±1 for both types of curvatures with the exception of 0 curvature Euclidean geometry can cause deformation or suspension η in the complex topological space T^* thereby creating genus g≥1 or with the  R_∩ formalism making the entire manifold M in the form of a suspended disc ∑D_Λ  to collapse as a ring M_R for non–intersection of infinite loops ∮_∞γ as the trivial assumptions for the extreme degrees of freedom.