= Equivalence of connected double covers of the torus
Any two connected degree-two coverings of the torus fit into a commuting square with homeomorphisms of their total and base spaces. They correspond to index-two subgroups of $\mathbb Z^2$, which are kernels of the three nonzero maps $\mathbb Z^2\to\mathbb Z/2\mathbb Z$. Reduction modulo two shows that $GL_2(\mathbb Z)$ acts transitively on these subgroups; the <classification of connected covering spaces> and the <lifting criterion for a covering space> then supply the homeomorphism of total spaces.
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