So suppose you take an $S^2$, then you put an $S^1$ fiber over it which degenerates by smoothly shrinking to a point at its poles. What is the topology of this space in more familiar terms (assuming some nicer representation is possible)?
To be clear, it is almost $S^2 \times S^1$, except at two points where the $S^1$ degenerates to a point, so it's $S^2 \times S^1$ quotiented by the action of the $S^1$ at two points, $(x,\theta)\sim(x,\theta')$ for $x$ equal to each pole. You might think of this in terms of having to specify two longitudes at every latitude.
Your space is the suspension of the 2-dimensional torus.