This is for a homework problem:
Take the unit sphere $\mathbb{S}^2$ and join the north and south poles with a line segment. Is the resulting space homotopy equivalent to a surface?
Intuitively, I think the answer is no, because I can't think of any way to deform the poles to become locally Euclidean. I've noted that the fundamental group is $\mathbb{Z}$ and that the space can be deformed into a sphere with the poles identified, or into a torus with a disc in the middle. But I don't know how to use any of this.