Let $X$ be the quotient space of $S^2$ under the identifications $x\sim-x$ for $x$ in the equator $S^1$. I want to compute the homology groups $H_n(X)$. I've seen this but didn't really understand.
The quotient space $X$ will look like this, isn't this space homeomorphic to the wedge of two $S^2$'s? If this is the case, then it is easy to compute the homologies; they are $0$ for $n\not=2$ and $\Bbb{Z}\bigoplus\Bbb{Z}$ for $n=2$. But this shouldn't be that easy, there is something wrong I guess.
Best Answer
Let's do this directly from the definitions of cellular homology. We'll call your space $X$.
The chain groups are:
We have a sequence $$ C_2(X)\stackrel{\phi}{\rightarrow}C_1(X)\stackrel{\psi}{\rightarrow} C_0(X)\rightarrow 0$$
Let's work out what $\phi$ and $\psi$ do: