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I have a question concerning an exercises from a text call Topology and Groupoid authored by Ronald Brown

The question is as follows:

Let $E^2 = \{(x, y) \in \mathbb R^2 : x^2 + y^2 \leq 1\}$. The space $S^1 \times E^2$ is called the solid torus. Prove that the 3-sphere $S^3 = \{(x_1, x_2, x_3 , x_4) \in \mathbb R^4 : (x_1)^2+(x_2)^2+(x_3)^2+(x_4)^2 = 1\}$ is the union of two spaces each homeomorphic to a solid torus and with intersection homeomorphic to a torus [Consider the subspaces of $S^3$ given by $(x_1)^2 + (x_2)^2 \leq (x_3)^2 + (x_4)^2$ and by $(x_1)^2 + (x_2)^2 \geq (x_3)^2 + (x_4)^2$]

I am not certain I understand the hint from the square bracket in geometric terms.

From what I understand of how the 3-sphere can be constructed, one takes two 2-spheres and superimposes the boundary of one on top of the other and then glues both boundaries together.

The two 2-sphere can be represented as $S^3_+ = \{(x_1, x_2, x_3 , x_4) \in\mathbb R^4 : (x_1)^2+(x_2)^2+(x_3)^2 = 1, (x_4)^2 \geq 0\}$ and $S^3_- = \{(x_1, x_2, x_3 , x_4) \in\mathbb R^4 : (x_1)^2+(x_2)^2+(x_3)^2 = 1, (x_4)^2 \leq 0\}$

Is the question asking me to show that both $S^3_+$ and $S^3_-$ are individually homeomorphic to the solid torus and $S^3_+ \cap S^3$ is homeomorphic to the torus? If so how does the hint become relevant?

Thanks in advance

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    You are talking about the usual construction of $S^{3}$ the one asked is bit unusual. In general there are many ways to triangulate a surface (and here break up a solid into polyherdral parts: essentially a CW complex). If you keep track of the smaller parts and their intersections you can decompose the solid into more manageable parts. The question is showing the sphere is two solid tori.2012-09-14
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    @s.b. it is actually quite usual in the correct circles! :-) Just picture the usual solid torus in $\mathbb R^3$ and look at its complement: it is also a solid torus minus a point, which is the one you would add in compactifying $\mathbb R^3$ to get a sphere.2012-09-14
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    @MarianoSuárez-Alvarez, I recall when I was first learning this stuff, this question stumped me for some time. Where as when you look at any sphere (after learning some basic general topology) the idea of breaking it up into two hem-spheres seems much more natural.2012-09-15

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