Hopf Fibrations-Construction and Quaternion Interpretations (I)

Before I begin discussing the Hopf fibration of the 3-spbhere, one of the simplest yet deeply profound example of a non-trivial fiber bundle, I’d like to recall the definition of a fiber bundle.

Let E,B,F represent the entire space, base space and the fiber respectively where E,B are connected. If f:E \mapsto B is a continuous surjection onto the base space, then the structure (E,B,F,f) is said to be a fiber bundle if for every x\in E, there exists a neighborhood U \subset B of f(x) such that there exists a homeomorphism \psi:f^{-1}(U) \mapsto U \times F such that \psi \circ \pi_{U}=f.

What this basically means is that locally, the fiber bundle looks like the product B \times F but globally, it may have different topological properties.

A trivial fiber bundle is a fiber bundle which in which the total space is E=B \times F. In fact, any fiber bundle over a contractible CW Complex is trivial.

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CW Complex Structure of Real Projective Space and Kusner’s Parametrization of Boy’s surface

The real projective space represented by \mathbb{RP}^{n} is the space obtained from \mathbb{R}^{n+1} under the equivalence relation x \sim kx \forall x \in \mathbb{R}^{n+1}. Basically, \mathbb{RP}^{n} is the set of lines which passed through the origin in \mathbb{R}^{n+1}. It can also be understood by identifying antipodal points(points which lie on opposite ends of the diameter) of a unit n-sphere,S^{n}.


One very basic yet deeply interesting example of these spaces is \mathbb{RP}^{2}, known as the real projective plane. While \mathbb{RP}^{0} is a points and \mathbb{RP}^{1} is homeomorphic to a circle with infinity, the real projective plane turns out to be far more interesting indeed. It can’t be embedded in \mathbb{R}^{3} and its immersion/s such as Boy’s surface, Roman surface and Cross Cap have far stranger structures than a mere circle as in the case of \mathbb{RP}^{2}. In fact, I will delineate some of the calculations involved in the Kusner-Bryant 3-dimensional parametrization of Boy’s surface. It’s a little fascinating how much complexity can be added to mathematical structures upon generalization especially in the case of the projective space which I believe have a remarkably simple and ‘non-threatening’ definition.

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