I am trying to compute the homology of $S^1\times \mathbb{RP}^2$ for all $n\geq 0$. I read online that Kunneth's Theorem about homology of product spaces might be useful, but unfortunately, we haven't covered that in my course, so I think I should not use it. I tried to picture the space as follows
I want to use a CW complex structure to compute the homology. My idea is to assign a single 0-cell to a point in the exterior of the solid "torus", then two 1-cells, each a loop of the torus (meridian and longitude), then one 2-cell representing the surface of the torus, and finally a 3-cell representing the "filling" of the torus. However, I have been having trouble computing the degrees of the attaching maps composed with the projections. For example, I want to compute
$\partial_2:\mathbb{Z}e_1^2 \to \mathbb{Z}e^1_1 \oplus \mathbb{Z}e^1_2$
where I know that
$\partial_2(e^2_1) = $deg$(p_1\circ \phi)e_1^1 + $deg$(p_2\circ \phi)e^1_2$
where $e^k_i$ represents the ith k-cell and $p_i: X = S^1\times \mathbb{RP}^2 \to X/(X- e^1_i)$. Also $\phi:S^1\to X$ is the attaching map. I believe in this case deg$(p_1\circ \phi)=2$ because of the $\mathbb{RP}^2$ condition on the boundary, and deg$(p_2\circ \phi)=1$ since it is basically the identity. This would give me (I believe) that $H_1(X) = \mathbb{Z}/2$. However, I am not sure about this, and I don't know how to procede with computing $\partial_3:\mathbb{Z}e^3_1\to \mathbb{Z}e^2_1$, so in $\partial_3 = $deg$(p_1\circ \phi)e^2_1$, I can't figure out what the degree is supposed to be.
I would appreciate it if someone could guide me on this path or tell me if this is a wrong way of thinking about this problem. Thank you.