What is a surjective continuous function from a 3-torus onto a cube? What are some examples and could I see some illustrations? Are they topologically equivalent to a 3-torus obtained by embedding a cube in 3-dimensional space or higher, and physically gluing the three pairs of opposite faces of the cube?
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For example (representing the $3$-torus as $0 \le x, y, z \le 1$ with $0$ and $1$ identified), $f(x,y,z) = [4 x(1-x), 4 y(1-y), 4 z(1-z)]$. Note that most points of the cube have $8$ inverse images in the $3$-torus.