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