As in the title, is there a homotopical equivalence between $S^1 \times S^1 / S^1 \times \{ 1 \}$ and $S^2 \wedge S^1$, i.e. the torus with a single circle contracted and the wedge some of a circle and a sphere?
I was given a hint to choose the contracted circle to be the inner horizontal one in the torus, but I still dont see it at all.
If you follow the hint, the space you get is a sphere where the north and south pole have been squished together (i.e. they meet inside the sphere). If you instead let the contracted circle be a vertical one, what you get is a sphere with the north and south poles glued together externally.
Either way, the space you get is relatively easily proven to be homotopy equivalent to a sphere where the north and south poles are connected by a line segment (either internally as a diameter, or externally, like the magnet field lines we see on drawings of the Earth), which again isn't difficult to prove is homotopy equivalent to the wedge product of the sphere with a circle.