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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.

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    @JasonDeVito: Thanks a lot.2012-12-16

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I guess I might as well answer here to keep this from being bumped:

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 $\pi_1(\mathbb{R}P^2$.

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    The easiest way, I think, is to note that it lifts to a non-closed curve in $S^2$.2013-02-26