I'm trying to compute the homology groups of $A=S^1\times(S^1 \vee S^1)$ with integer coefficients using Kunneth Formula. I have a few questions
Here's the formula (omitting coefficients): $H_k(X \times Y)\cong \oplus_{i+j=k} H_i(X) \otimes H_j(Y) \oplus_{i+j=k-1} \mbox{Tor}(H_i(X),H_j(Y))$.
Should I always include the case when $i=j=0$? How would the Tor part of the formula work for $k=0$?
I know $H_1(S^1) \cong \mathbb{Z} \cong H_0(S^1) \cong H_0(S^1 \vee S^1)$ and $H_1(S^1 \vee S^1) \cong \mathbb{Z} \times \mathbb{Z}$.
By the formula, $H_1(A) \cong H_0(S^1) \otimes H_1(S^1 \vee S^1) \oplus H_1(S^1) \otimes H_0(S^1 \vee S^1) \oplus \mbox{Tor}(H_0(S^1),H_1(S^1 \vee S^1) \cong \mathbb{Z} \otimes (\mathbb{Z} \times \mathbb{Z}) \oplus \mathbb{Z} \otimes \mathbb{Z} \cong \mathbb{Z} \otimes (\mathbb{Z} \times \mathbb{Z}) \oplus \mathbb{Z}$
Have I made any mistake in my calculation? Can it be simplified further?
I saw online that $\mbox{Tor}(\mathbb{Z},\mathbb{Z})=0$, but why is this true? How exactly do I calculate $\mbox{Tor}(G,H)$ for any two groups $G$ and $H$.
Also, how do I know if the space $A$ is path connected, i.e. $H_0(A) \cong \mathbb{Z}$?
Any help is appreciated!