I am trying to visualize the simplest 3-manifolds: $T^3$, $S^3$, $S^2 \times S^1$, $\mathbb{R}P^3$, $S^2 \times I$, and so on.
I am used to thinking about $T^2$, $S^1 \times I$, $S^2$, and $\mathbb{R}P^2$ in terms of the fundamental square. $T^2$ is the square with opposite edges identified, $S^1 \times I$ is gotten by chopping the square in half and leaving the boundary, $S^2$ is gotten from $S^1 \times I$ by shrinking the boundary to a point, and $\mathbb{R}P^2$ is gotten by chopping $S^2$ in half and identifying opposite edges of the new boundary.
Can someone describe an analogous procedure to me for 3-manifolds whose fundamental domain is the cube?
Thanks