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(Mosseri & Dandoloff 2001). Moreover, the Hopf fibration is equivalent to the fiber bundle structure of the Dirac monopole.

THE HOPF INVARIANT ONE PROBLEM. RAFAEL M. SAAVEDRA parallizability of spheres

• A fundamental theorem in K- theory which, in its simplest form, states that for any (compact) space X there exists an isomorphism between the rings K(X)⊗K(S2) and K(X×S2). More generally, if L is a complex vector bundle over X and P(L⊕1) is the projectivization of L⊕1, then the ring K(P(L⊕1)) is a K(X)- algebra with one generator [H] and a unique relation ([H]−[1])([L][H]−[1])=0, where [E] is the image of a vector bundle E in K(X) and H−1 is the Hopf fibration over P(L⊕1). This fact is equivalent to the existence of a Thom isomorphism in K- theory for complex vector bundles. Encyclopedia of Mathematics

In quantum mechanics, the Riemann sphere is known as the Bloch sphere, and the Hopf fibration describes the topological structure of a quantum mechanical two-level system or qubit. Similarly, the topology of a pair of entangled two-level systems is given by the Hopf fibration {$ S^{3}\hookrightarrow S^{7}\to S^{4}$} (4+3=7 sevensome relates knowledge (foursome) and consciousness (three minds))