Homotopy Groups of Spheres, Hopf Fibrations and Villarceau Circles II

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, and the impossibility of embedding one into another, homotopy theory relates how a sphere of one dimension can wrap around, or map into, a sphere of another dimension. This paper revisits the classical theory of homotopy groups of spheres, providing a detailed exploration of their computation and structure. We place special emphasis on the pivotal role of Hopf fibrations in revealing the higher homotopy groups of spheres, particularly the exotic and fascinating case of π3(S2). Furthermore, we explore the elegant geometric connection to Villarceau circles, demonstrating how these circles on a torus are intimately linked to the Hopf fibration of S3. This work serves as a comprehensive guide, bridging abstract algebraic topology with tangible geometric phenomena. This version expands significantly on the foundational ideas, providing deeper insights and connections to contemporary research, including stable homotopy theory, the Adams conjecture, and generalizations to Calabi-Yau manifolds.