How to introduce a CW structure on RP^n?

Question

My first course in topology is going extremely fast, and does not seem like rigorous mathematics. Last lecture, we were given the definition of CW-structures, but did not do any examples. Yet we were assigned us homework problems about them.

Here is the definition (verbatim) he gave: A CW-space $X$ is a Hausdorff space such that $$ X = \bigcup_{i=0}^\infty X_i $$ where $X_0$ is a discrete space, and $X^{i+1}$ is obtained by attaching the disjoint union of $i+1$ disks to $X^i$ along a continuous map.

I have no intuition about what this is supposed to mean, or how to use it. There is no reference book for the course, and all the resources I have found are at the advanced graduate level, so they use terminology that I am unfamiliar with.

One problem I am supposed to solve is: Give a CW-structure on $\mathbb{S}^2$ that corresponds to a CW-structure on $\mathbb{RP}^2$. To my understanding, I need to introduce a CW-structure on both of these spaces, then send the structure on $\mathbb{S}^2$ through the quotient map to $\mathbb{RP}^2$, and verify this is also a CW-structure.

Next, I am supposed to introduce a CW-structure on $\mathbb{RP}^n$. The definition seems like an inductive construction, so once I understand the definition, and how to introduce a CW-structure on $\mathbb{RP}^2$, this problem should become much easier.

Please help - I have no idea what I got myself into by taking this course.

Edit: Here is the picture I think I am supposed to draw? Comment: A "proof by pictures" is not proof to me. This does not seem like mathematics. I want to go back to analysis - epsilon was my friend.

Answer 1

To help get you started, here is a CW structure on $\mathbb{RP}^2$ and one that matches on $S^2$. Start with $X_0=\lbrace -1,1\rbrace$ in the complex plane. Then let $X_1=\lbrace \partial\mathbb{D}^+,\partial\mathbb{D}^-\rbrace$ be the boundary circle of the unit disk split into two at the points in $X_0$. These are our one dimensional disks. Then choose $X_2=\mathbb{D}$. So far we have a CW-complex that is the disk, so we have neither of the spaces we are after. To get $\mathbb{RP}^2$ we need to quotient the disk. Sketch this on some paper. Orient the components of $X_1$ going counter-clockwise. Now glue just these two pieces together, respecting orientation. The resulting manifold is $\mathbb{RP}^2$.

Notice that a sphere is the same thing as two disks sewn together. So double the construction that we had at the beginning, getting two CW complexes for disks, then glue the top boundaries and the bottom boundaries together (respectively).

Edit: Every sphere $n-$sphere can be made this way; glue two $n-$disks together along their boundaries.

Answer 2

For $n \geq 1$ observe that we can form the push out diagram

$ \require{AMScd}\begin{CD} S^n @>{\pi}>> \mathbb{R}\mathbb{P}^n\\ @VVV @VVV \\ D^{n+1} @>{}>> \mathbb{R}\mathbb{P}^{n+1} \end{CD}$

where $\pi \colon S^n \rightarrow \mathbb{R}\mathbb{P}^n$ is the projection if we consider $\mathbb{R}\mathbb{P}^n$ as a sphere with identified antipodal points and the left vertical map is the inclusion of a sphere in a disk as boundary.

This let us define inductively a CW complex structure on $\mathbb{R}\mathbb{P}^n$.

EDIT: I will explain better my answer. First let us start with the CW complex structure on $D^n$: as Prototank explained you have such a structure on $S^1$ by taking two points and gluing two segments (which are just copies of $D^1$) to these the endpoints. You you consider the usual embedding of $S^1$ in $\mathbb{R}^2$ you can take the two points to be $(1,0)$ and $(-1,0)$ and the two segments to be the upper and lower emispheres. Notice that these two points can be identified with $S^0$ which is the intersection of $S^1$ with the line $y=0$.

Now we can use this top construct a CW complex on $S^2$: consider $S^1 \subset S^2$ the equator for $z=0$, we want this to be the $1$-skeleton of the sphere. So we attach to this two disks $D^2$ by gluing their boundaries $\partial D^2=S^1$ along the copy of $S^1$ we start with. In this way clearly we get a sphere $S^2$ as we wanted.

Now you can inductively construct a CW complex structure on $S^n$ for any $n$: you just attach two $n$-disks $D^n$ along their boundaries to a copy of $S^{n-1}$ which by induction has already a CW-structure. You will see that the $S^{n-1}$ is just the equator obtained intersecting the $n$-sphere in $\mathbb{R}^{n+1}$ with the plane $x_{n+1}=0$ and the two disks form the two hemispheres.

Now we want to induce a CW structure on $\mathbb{R}\mathbb{P}^n$: by definition this space is just the sphere $S^n$ modulo the identification of the antipodal points. We can reformulate this as follows: we can define on $S^n$ an action of $C_2$ (the $2$-cyclic group) by sending an element $x$ to its antipodal $-x$. The key property of this action is that it respects the CW complex structure we defined above on the sphere. Consider the case of $S^1$: this has two $0$-cells given by the points $(1,0)$ and $(-1,0)$ and two $1$-cells given by the two hemispheres. The action we defined maps one of the $1$-cells to the other, moreover it maps the interior of the $1$-cell to the interior of the other cell. Similarly it maps the two points to each other (since they are points the statement about the interior is trivial). You can see that this property holds for every $S^n$ for its $k$-cells for $0 \leq k \leq n$.

Now there is a important theorem that says in this situation the quotient $S^n/C_2=\mathbb{R}\mathbb{P}^n$ inherits a CW structure from the original space in which a $k$-cell is given by the $k$-cells identified by the group action. This is the CW complex structure we are interested on the projective space.

So we have established that the CW-structure $\mathbb{R}\mathbb{P}^n$ is the quotient of the one on $S^n$: the latter consists of attaching two $n$-disks to a copy of $S^{n-1}$. Now when we identify the antipodal points the two hemispheres become the same $n$-cell. What about the equator? This is a copy of $S^{n-1}$ in which we are identifying the antipodal point: by definition this is just $\mathbb{R}\mathbb{P}^{n-1}$. Therefore we can describe the CW-structure on $\mathbb{R}\mathbb{P}^n$ alternatively as gluing an $n$-cell on a copy of $\mathbb{R}\mathbb{P}^{n-1}$.

This is what the pushout I wrote was refering to: the pushout is a categorical construction (see here https://en.wikipedia.org/wiki/Pushout_(category_theory)) which in the category of topological spaces reduces to attaching the bottom left and upper right corners along the maps included in the diagram. The result of the gluing is the space in the lower right corner. In our case we are attaching the $n$-disk to $\mathbb{R}\mathbb{P}^n$ to get $\mathbb{R}\mathbb{P}^{n+1}$ as I described above.