Hopf-Like Fibrations on Calabi-Yau Manifolds

We conducted a comprehensive study of {\em Hopf-like fibrations} in the context of Calabi-Yau (CY) manifolds, exploring fiber bundle structures analogous to the classical Hopf fibrations and their topological implications. In particular, we analyze how Hopf projections emerge in special cases of Calabi-Yau geometry (e.g., in hyperkähler 4-manifolds like Eguchi–Hanson and Taub–NUT spaces) and formulate general criteria for sphere-bundle fibrations in complex Ricci-flat Kähler spaces. We integrate this with a detailed examination of the high homotopy groups $\pi_k(X)$ of the CY manifolds $X$, employing rational homotopy theory, minimal model computations, and known exact sequences. For K3 surfaces (complex 2-dimensional CY) and prototypical CY threefolds (such as the quintic), we compile known results (e.g.\ $\pi_2\cong \mathbb{Z}^{b_2}$ and $\pi_3=\mathbb{Z}^{252}$ for K3) and derive new constraints from bundle constructions. Applications to string theory and M-theory compactifications are discussed, highlighting how such fibration structures influence duality frames, flux configurations, and geometric transitions. The paper is framed in a rigorous mathematical physics context, blending differential geometry, topology, and physical motivation.