Is there a CW Complex Structure on $\mathbb{R}P^2$ with 5 0-Cells , 5 1-Cells and 5 2-Cells?

Question

Is there a CW Complex Structure on $\mathbb{R}P^2$ with 5 0-Cells , 5 1-Cells and 5 2-Cells?

Euler Characteristic of $\mathbb{R}P^2$ is $\chi(\mathbb{R}P^2)=1$

Since it has 1 0-Cell,1 1-Cell and 1 2-Cell.

How can I use this information to solve the problem?Is there any other method to solve this?

Any hint will be appreciated.

Answer 1

No: consider Hatcher's Theorem $2.44$ and the definition right before of the Euler characteristic. On the one hand $\chi(\Bbb R P^2)$ is the alternating sum of the number of $n$-cells, so $\chi(\Bbb R P^2)=5-5+5=5$. On the other hand $\chi(\Bbb R P^2)=1$ using Theorem $2.44$.