## 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.