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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