How to visualize topological differences between $\mathbb{R}P^{2n}$ and $\mathbb{C}P^n$

Question

I never stopped to really understand the topology of $\mathbb{R}P^{2n}$ and $\mathbb{C}P^n$. In my Algebraic Topology class, we calculated the homology of them, and they are significantly different. I have some intuition on why they are different (considering you are taking lines with complex coefficients, you must have a whole different topology, since you are allowing rotations etc) but they are all vague. Therefore, my request is: please, help me to visualize why the difference between their topology is so significant.

Answer 1

One way to see the difference is via their standard CW complex structures.

Real projective space $\mathbb{RP}^n$ can be defined as $S^n/\sim$ where $x \sim y$ if and only if $y = \pm x$.

Let $U_{\pm} = S^n\cap\{x \in \mathbb{R}^{n+1} \mid \pm x_{n+1} > 0\}$ and $E = S^n\cap\{x \in \mathbb{R}^{n+1} \mid x_{n+1} = 0\}$; $U_+$, $U_-$ are the upper and lower hemispheres of $S^n$ respectively, and $E$ is the equator.

In the quotient, $U_-$ is identified with the $U_+$, whereas $E$ can be identified with $S^{n-1}/\sim$ via the map $[(x_1, \dots, x_n, 0)] \mapsto [(x_1, \dots, x_n)]$. Now note that $U_+$ is diffeomorphic to the unit $n$-disc via the map $(x_1, \dots, x_{n+1}) \mapsto (x_1, \dots, x_n)$ and $S^{n-1}/\sim$ is just $\mathbb{RP}^{n-1}$. Therefore, $\mathbb{RP}^n = e^n\cup\mathbb{RP}^{n-1}$ where $e^n$ is used to denote an $n$-cell (i.e. an $n$-dimensional disc). We can repeat this observation to obtain

$$\mathbb{RP}^n = e^n\cup\mathbb{RP}^{n-1} = e^n\cup e^{n-1}\cup\mathbb{RP}^{n-2} = \dots = e^n\cup e^{n-1}\cup\dots\cup e^1\cup e^0.$$

So $\mathbb{RP}^n$ has a CW complex which consists of a single cell in each dimension between $0$ and $n$ (inclusive).

The standard CW complex for $\mathbb{CP}^n$ consists of a single cell in every even dimension between $0$ and $2n$ (inclusive), and no cells in any odd dimension; i.e. $\mathbb{CP}^n = e^{2n}\cup e^{2n-2}\cup\dots\cup e^2\cup e^0$. One can obtain this cell decomposition in a similar way to its real counterpart.

So $\mathbb{RP}^{2n}$ has a single cell in every dimension between $0$ and $2n$ (inclusive) whereas $\mathbb{CP}^n$ has a single cell is every even dimension between $0$ and $2n$ (inclusive) and no other cells. This hopefully give you some insight into how the two spaces differ topologically.

It is also worth noting that these CW complexes allow you to compute their homology groups relatively quickly using cellular homology. This calculation will explain why their homology groups are significantly different, as you have already observed.