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I am trying to understand (simplify) the following space

$Z = S^2 \times S^2 / E$

Let $(a_1 , a_2 , a_3) \in S^2$ and $(n_1 , n_2 , n_3) \in S^2$. Here ${a_1}^2 + {a_2}^2 + {a_3}^2 = 1$ and ${n_1}^2 + {n_2}^2 + {n_3}^2 = 1$.

The equivalence relations $E$ are the following

1) $[(a_1, a_2, a_3);(n_1, n_2, n_3)] \sim [(-a_1, -a_2, -a_3);(n_1, n_2, n_3)]$

2) $[(a_1, a_2, a_3);(n_1, n_2, n_3)] \sim [(a_1, a_2, a_3); (\pi; a_1,a_2,a_3)*(-n_1, -n_2, -n_3)]$

where $(\pi; a_1,a_2,a_3)$ is a rotation of angle $\pi$ along the axis $(a_1, a_2, a_3)$. So the point $(\pi; a_1,a_2,a_3)*(-n_1, -n_2, -n_3)$ is the mirror image of $(n_1, n_2, n_3)$ in the plane whose normal is $(a_1, a_2, a_3)$.

The objective is to simplify the equivalence relation (2) by a homeomorphism of $S^2 \times S^2$. For example, the approach I am trying is to rotate the second $S^2$ such that the mirror plane becomes horizontal (XY plane). Since the normal of the mirror plane is $(a_1, a_2, a_3)$, the amount by which the second $S^2$ has to be rotated depends on the $(a_1, a_2, a_3)$ it corresponds to. And for this transformation to be a homeomorphism the rotation operations should vary continuously. The transformation should not complicate the first equivalence relation; one way this condition could be imposed is by ensuring that the rotation operation corresponding to $(a_1, a_2, a_3)$ and $(-a_1, -a_2, -a_3)$ are be the same. I am not able to come up with such a transformation. All my attempts to simplify (2) end up complicating (1).

To be more specific, I want to get a homeomorphism between the space $Z$ and the following space:

$Z^' = S^2 \times S^2 / E^' $

The equivalence relations $E^'$ are the following

1) $[(a_1, a_2, a_3);(n_1, n_2, n_3)] \sim [(-a_1, -a_2, -a_3);(n_1, n_2, n_3)]$

2) $[(a_1, a_2, a_3);(n_1, n_2, n_3)] \sim [(a_1, a_2, a_3); (n_1, n_2, -n_3)]$

Any ideas are appreciated. Please let me know if you have any questions.

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    No, I am graduate student in Materials Science department and it is a part of my project. Just to give a brief background: The space belongs to a product space of rotations and normal vectors $(SO(3) \times S^2)$. The space $S^2 \times S^2$ I mentioned above is a subspace of this space and my final objective is to be able to do statistical analysis on this space.2010-11-11

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This isn't quite what you asked for, but notice that (1) simply gives us $\mathbb{RP}^2 \times S^2$. I'm not sure I understand your notation for (2); it seems like it doesn't depend at all on $\pi$. If it's true that you're just reflecting $\vec{n}$ through the plane perpendicular to $\vec{a}$, then you're just getting a copy of $D^2$ over every point of $\mathbb{RP}^2$. This is a fiber bundle, and if you think through it I think you'll find that it's the disk bundle of the tangent bundle $T\mathbb{RP}^2$.

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    Gotcha. Yeah Henle will tell you some stuff, but imho if you want to get to fancier things you may want to switch over to Hatcher (or something comparable). You should probably skip Ch. 0 though, at least on a first read, and if you mainly want to know about (co)homology you can actually probably safely start with Ch. 2...2010-11-13