# Generator of the fundamental group of $\mathbb RP^{2}$.

Take a closed hemisphere and identify the antipodal points on the equator ,we get $\mathbb RP^{2}$ and inside $\mathbb RP^{2}$ we have copy of $\mathbb RP^{1}$.So, what will be the induced map on fundamental group induced by inclusion?

I think ,if we know which homotopy class of loop in $\mathbb RP^{2}$ is generator,then some conclusion can be said easily.

-
Draw a line of longitude on the hemisphere. Notice that since you're identifying points on the equator, this is actually a closed curve. That is the generator of $\pi_1$. – Jason DeVito Dec 16 '12 at 19:34
@JasonDeVito: Thanks a lot. – Shraddha Srivastava Dec 16 '12 at 19:55

If you draw a line of longitude on the hemisphere, this is actually a closed curve due to the identification you make on the equator. It will generate $\pi_1(\mathbb{R}P^2$.
The easiest way, I think, is to note that it lifts to a non-closed curve in $S^2$. – Jason DeVito Feb 26 '13 at 18:06