Give an example of two 3-manifolds with different second de rham groups

Question

I am asked to construct two 3-manifolds $M_1$,$M_2$ both covered by two open sets $U$,$V$ (different for each manifold) s.t. the intersection is diffeomorphic to $\mathbb S^1 \times\mathbb R^2$ but the second de Rham group of $M_1$ and $M_2$ is trivial and not trivial respectively. My thought so far is to take $M_1=\mathbb S^2 \times\mathbb R$ and $M_2=\mathbb S^1 \times\mathbb R^2$ but I can't work out the part about the intersection.

Answer 1

For $M_2$, finding $U$ and $V$ is pretty obvious so I guess you need help for $M_1$. For this, consider a "latitude map" $z : \mathbb{S}^2 \rightarrow [-1,1]$ sending the south pole on $-1$, the north pole on $1$ and any point of the equator on $0$. Then $U = \{z < 1/2\} \times \mathbb{R} \subset M_1$ and $V = \{z > -1/2\} \times \mathbb{R}$ fit because $U \cap V = \{-1/2 < z < 1/2\} \times \mathbb{R}$ and $\{-1/2 < z < 1/2\} \subset \mathbb{S}^2$ is a strip around the equator, thus homeomorphic to $\mathbb{S}^1 \times \mathbb{R}$.

I suggest you make a draw of what happens on $\mathbb{S}^2$, it is pretty obvious when you visualize it.