I asked this question previously on stackexchange (http://math.stackexchange.com/questions/18668/mapping-from-x-to-s4) but could not get a solution. Any help is appreciated.
The question is (Topology II: homotopy and homology: classical manifolds):
Show that the quotient space $X = S^2 \times S^2 / [(x_1,x_2) \sim (Rx_1,Rx_2)]$ where R is the reflection in the equatorial plane, is homeomorphic to $S^4$.
I am still in the process of learning topology and I really don't think I can prove this result. I apologize in advance if this is trivial, but it will be of great help if someone could provide a homeomorphism. I would really like to use the mapping of $X$ to $S^4$ in my work.
I did manage to get a mapping from $S^1 \times S^2 / [(x_1, x_2) \sim (Rx_1,Rx_2)]$ to $S^3$, but I am having trouble extending the result to $S^2 \times S^2$