 Homotopy Group of Spheres, Hopf Fibrations & Villarceau Circles

EasyChair Preprint no. 7959, version 2

Versions: 12history
10 pagesDate: July 24, 2022

Abstract

Unlike 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 1-sphere or a circle which when which exists in the form of points inside the 2-sphere, and the mapping, that transforms, the 3-sphere to the 2-sphere, where each point of 2-sphere acts as a circle in 3-sphere, generates in turn the third homotopy group of the 2-sphere 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 3-dimensional torus of surface area 2πR × 2πr then along with the 2-circles, 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 3-circles intersected and passed through.

Keyphrases: homotopy, Hopf Fibrations, topology