Finding sheet number of torus using universal cover

Question

I have a question from my lecture notes that I need clearing up: Given a covering $p: S^1\times S^1 \rightarrow S^1\times S^1$ by $p(z,w)=(z^a w^b,z^c w^d), a,b,c,d\in\mathbb{Z}$, we want to find the sheet number of the covering. So we consider universal coverings of $S^1\times S^1$ to obtain a lift of $p$ from $\mathbb{R}^2$ to $\mathbb{R}^2$. We call the lift $L$ say, which always exist because $\pi(\mathbb{R}^2)=1$. \begin{matrix} \mathbb{R}^2 & \rightarrow & \mathbb{R}^2 \\ \downarrow & & \downarrow \\ S^1\times S^1 & \rightarrow & S^1\times S^1 \end{matrix} Now, after some work, we get that $L(x,y)=(ax+by+Q,cx+dy+R)$ for some integers $Q,R$. How do we find the sheet number from here? I think that it maybe the number of lattice points in the parallelogram obtained from $L$, but why? Thanks for the help in advance.

Answer 1

Think about the linear map $f\colon \mathbb R^2\to\mathbb R^2$ given by $f(x,y)=(ax+by,cx+dy)=A(x,y)$. When $a,b,c,d\in\mathbb Z$, why does $ \det A$ tell us the sheet number (counting signs) of the induced map $\mathbb R^2/\mathbb Z^2\to\mathbb R^2/\mathbb Z^2$?