Download PDFOpen PDF in browserHomotopy Group of Spheres, Hopf Fibrations & Villarceau CirclesEasyChair Preprint 7959, version 210 pages•Date: July 24, 2022AbstractUnlike geometry, spheres in topology have been seen as topological invariants, where their structures are defined as topological spaces. Forgetting, the exact notion of geometry, & the impossibility of embedding one into other, the homotopy relates how one sphere of dimensions can wrap another sphere of dimensions. Here, depending on the pattern, the relation can be of three types, i is equal to n, less than n or greater than n. Each of them has their affine properties & uniqueness that defines homotopy in the mathematical field of algebraic topology. The most important part of homotopy is the Hopf fibrations where i>n & there a special type of mapping and stereographic projection takes place which can be justified by the relation S¹ → S³ → S². S¹ is a 1sphere or a circle which when which exists in the form of points inside the 2sphere, and the mapping, that transforms, the 3sphere to the 2sphere, where each point of 2sphere acts as a circle in 3sphere, generates in turn the third homotopy group of the 2sphere that is, π²(S²) = ℤ, where ℤ ∈ ℝ If we assume that the stereographic projections that is made by the transform mapping S¹ → S³ → S² where the third homotopy groups fiber is a 3dimensional torus of surface area 2πR × 2πr then along with the 2circles, the major and minor there exists also a pair of circles produced by cutting the torus analytically at a certain angle produces a pair of circles called Villarceau circles where they meet all the latitudinal and longitudinal cross sections of the torus at a point of the minor radius being the locus of the torus where the other 3circles intersected and passed through. Keyphrases: Hopf Fibrations, homotopy, topology
