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Let $X$ be the quotient space of $S^2$ under the identifications $x\sim −x$ for $x$ in equator $S^1$. I want to compute the fundamental group and homology groups $H_i(X)$. I also want to repeat this exercise for $S^3$ with antipodal points of the equatorial $S^2$ contained in $S^3$ identified.

Yikes, thanks in advance for any help. :P

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(Just to make a useless comment) This space has a standard name that you will find out at some point if you haven't already. –  Michael Hardy Dec 14 '11 at 19:22
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Hint: You can decompose $X$ as the union of the upper and lower hemispheres (with antipodal points on the equator identified), each of which is homeomorphic to $B^2$ with antipodal points on its boundary identified. Thus each of the components in the decomposition is $RP^2$. Their intersection is the equator with antipodal points identified, which is homeomorphic to $S^1$. Then use Seifert-van Kampen to get the fundamental group, and Mayer-Vietoris to get the homology groups. An analogous decomposition works for the second part, see if you can figure it out for yourself.

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Thanks very much Brandon! :). Very helpful. –  Anton Dec 14 '11 at 23:16
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The simplest way would be using cellular homology. Just check.

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