I have the following problem:
Let $A = S^1 \times S^1$ and $B = S^1 \vee S^1 \vee S^2$. Compute their universal coverings. Prove that $A$ and $B$ have isomorphic homology groups for any $n \in \mathbb{Z}_{\geq 0}$, but that their universal coverings do not.
I haven't had much practice, so I was wondering if my reasoning is sound. Also, there are some questions mixed in (bolded). Anyway, my work:
$A$ has universal cover since the product of covers is a cover (so given the universal cover of $S^1$, I get a cover of $S^1 \times S^1$). But $\mathbb{R^2}$ is contractible, so I know this is the universal cover of $A$. For $B$, I have an idea of what the covering space is (I think its exactly the same as the universal cover for $S^1 \vee S^1$, except a copy of $S^2$ is attached to each node in the graph). I'm not entirely sure if this is correct, but we tried some simple examples of this in class (computing universal covers of small wedge products), and this is what I thought to do; is there a reference, other than Hatcher, that explains why I can construct the universal cover in this manner?
The homology of $\mathbb{R}^2$ is $\mathbb{Z}$ for $n=0$ and is trivial for $n \in \mathbb{N}$ ; in particular $H_2(\mathbb{R^2})=0$. On the other, let $U_B$ be the universal cover of $B$. Since the universal cover of $S^1 \vee S^1$ is contractible, it follows that $U_B$ deformation retracts onto (and is homotopy equivalent to) the wedge of countably many copies of $S^2$. Thus, by additivity of homology over wedge sums, $H_n(U_B) = \oplus_{\mathbb{N}} H_n(S^2)$; in particular, $H_2(U_B)=\oplus_{\mathbb{N}}\mathbb{Z} \neq 0$, so in dimension 2, $A$ and $B$ have non-isomorphic homology groups.
Again, using additivity of homology of wedge sums, the homology groups of $B$ are given by $\mathbb{Z} \oplus \mathbb{Z} \oplus \mathbb{Z}$ for $n=0$; $\mathbb{Z} \oplus \mathbb{Z}$ for $n=1$; $\mathbb{Z}$ for $n=2$; and is trivial for $n \geq 3$. My main problem is that I'm not sure how to compute the homology groups for a torus. Working only from the Eilenberg-Steenrod axioms, is there a way to compute the homology groups for a torus?
Thanks in advance for any help!