# 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.

Before that, I’d like to discuss the simple CW structure of real projective space.

Throughout the post, I’ll let $Top$ represent the category of topological spaces.

Also, $D^{n}$ represents  the $n-ball$.

The essence of the CW structure is the attachment of these $n$-dimensional balls along their boundaries(which are homeomorphic to $S^{n-1}$) to an arbitrary topological space $X$ through an attachment map which I’ll promptly describe below.

Here is a description for the attachment of an $n-cell$ to a topological

Consider some topological space $X$ and let $\phi:S^{n-1} \mapsto X$ be a continuous function from the boundary of the $n-ball$ to the space. $phi$ is called the attachment map.

Now, the boundary inclusion is given by $i_{n}:S^{n-1} \mapsto D^{n}$.

Through this construction, one obtains an adjunction space $X \cup_{\phi} D^{n} \in Top$ which is the pushout of the boundary inclusion of $S^{n-1}$ along $\phi$ . Attaching many such $n-cells$ to a given topological space $X$ or to the empty set is the edifice of the construction of a CW complex. It can understood through the following commutative diagram.

The $n-skeleton$ of a CW complex is the union of all cells whose dimension is less than or equal to $n$.

## CW Structure of $\mathbb{RP}^{n}$

A good way is to build inductively from $\mathbb{RP}^{1}$. If you consider the structure of $\mathbb{RP}^{2}$ through identification of antipodal points on $S^{2}$, you can ‘forget about’ the points on the lower half of the circle as those points are all identifying with their corresponding point on the other half.

But along with the points on the upper plane satisfying $\{(x,y,z)|z \neq 0, x^{2}+y^{2}+z^{2}<1 \}$ there’s the unidentified points lying on the plane $z=0$.

Look at that carefully! That explains all those ridiculous immersions of the real projective plane. But remember the circle, what is that? Oh yes, it is homeomorphic to $\mathbb{RP}^{1}$. So before thinking about immersions or embedding, we can atleast conclude that:

$\mathbb{RP}^{2}=S^{2} \cup \mathbb{RP}^{1}$

If you wish to visualize the identification, think of the ball wrapping twice around the circle in order to connect the antipodes.

Now, with the help of the intuition gained from the structure of $\mathbb{RP}^{2}$, we’re in the position to rigorously discuss the CW structure of real proejctive space and generalize.

If $X=\mathbb{RP}^{n-1}$,

the inclusion map is $i_{n}:S^{n-1}\mapsto D^{n}$

and the attachment map is given by $\phi: S^{n-1} \mapsto S^{n-1}\textbackslash{} \{x \sim -x\} \mapsto \mathbb{RP}^{n-1}$ by definition.

But now remember that $\mathbb{RP}^{0}$ is a point, so building inductively,

$\mathbb{RP}^{n} \cong \bigcup_{i=0}^{n} S^{i}$

So that means that there is one cell in each dimension from $0$ to $n$. A good way to think of real projective space would be to give a basis to $\mathbb{R}^{n+1}$, then the $k-skeleton$ of $\mathbb{RP}^{n}$ lies in the subspace spanned by the first $k+1$ basis vectors and each time a basis vector is added, a new direction is introduced in the CW complex.

## Parametrization of Boy’s surface

Before I jump into the calculations, you can see this video on YouTube which shows how Boy’s surface is constructed.

The first discovered parametrization was given by determined by methods studied by Apery in the 1980s.

$x(u,v)=\frac{\sqrt{2}cos(2u)cos^{2}(v)+cos(u)sin(2v)}{D}$

$y(u,v)=\frac{\sqrt{2}sin(2u)cos^{2}(v)-sin(u)sin(2v)}{D}$

$z(u,v)=\frac{3cos^{2}(v)}{D}$

where $D=2-a\sqrt{2}sin(3u)sin(2v)$ where $0.

A few years after this parametrization was established, Robert Bryant and Rob Kusner discovered a new parametrization of Boy’s surface.

The interesting thing about Boy’s surface is that unlike the Roman surface and Cross Cap, it has no pinch points/singularities except self intersections. Also, it can be split into 3 congruent pieces due to the existence of an axis of discrete rotational symmetry(120 degrees).

The method to prove 3-fold symmetry is quite simple though it requires extnesively laborious calculations. Here, given a point $P(w)$ on Boy’s surface where $w$ is a complex parameter in the unit disk, it suffices to show that a $120^\circ$ rotation of $w$ on the complex plane corresponds to a rotating Boy’s surface by $120^\circ$ on the Z-axis.

So let, $w'=we^{\frac{i 2 \pi}{3}}$ and plug it into the equations. Then, using the identities,

$Re(UV)=Re(U)Re(V)-Im(U)Im(V)$

$Im(UV)=Re(U)Im(V)+Im(U)Re(V)$

$g_{3}$ doesn’t change but both $g_{1},g_{2}$ do. Using these identities repeatedly and comparing the equations of $g_{1}(w')$ and $g_{2}(w')$, we get:

$g_{1}(w')=cos(\frac{-2\pi}{3})g_{1}(w)-sin(\frac{-2\pi}{3})g_{2}(w)$

$g_{2}(w')=sin(\frac{-2\pi}{3})g_{1}(w)+cos(\frac{-2\pi}{3})g_{2}(w)$

$g_{3}(w')=g_{3}(w)$

Writing it in matrix notation,

That’s obviously a rotation matrix($120^\circ$) corresponding to the Z-axis.