Is there a 3-sheeted covering map of torus S^1 × S^1?

Question

Basically, I'm having trouble solving this question. Does anyone know how to approach this?

Answer 1

The torus is the quotient of $\mathbb{R}\times \mathbb{R}$ by $f(x,y)=(x+1,y), g(x,y)=(x,y+1)$ consider

the quotient of $\mathbb{R}^2$ by $u(x,y)=(x+{1\over 3},y), v(x,y)=(x,y+1)$ is a $3$-cover of the torus, in fact this cover is isomorphic to the torus itself.

Answer 2

Yes. The map $m_3 : S^1 \to S^1, m_3(z) = z^3$, is a $3$-sheeted covering. Hence $m_3 \times id$ is also a $3$-sheeted covering. Note that products of two coverings maps are coverings maps, and the number of sheets of such product maps is the product of the number of sheets of the two factors.