Let X be the quotient space of $S^2$ under the identifications $x\sim-x$ for $x$ in the equator $S^1$. Compute the homology groups $H_i(X)$. Do the same for $S^3$ with antipodal points of the equator $S^2 \subset S^3$ identified.
This is probably related to cellular homology. Thanks.
Best Answer
Let $X=S^2/\mathord\sim$ and,letting $S^2_+\subseteq S^2$ be the upper closed hemisphere in the sphere, let $X_+=S^2_+/\mathord\sim$ be the quotient of $S^2_+$ by the restricted equivalence relation. Now consider the long exact sequence in reduced homology for the pair $(X,X_+)$.
Using excision &c, show that the relative homology of $(X,X_+)$ is the same as that of the result of collapsing $X_+$ to a point, so that you get a $2$-sphere. On the other hand, $X_+$ is a projective plane, so you also know its homology. Now use the long exact sequence.