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