On the <torus> with basepoint $(1,1)$ take $p(z,w)=(z^2,w)$ and $p'(z,w)=(z,w^2)$. Both are <connected> two-sheeted coverings. Their induced subgroups of $\pi_1(S^1\times S^1)=\mathbb Z^2$ are $2\mathbb Z\times\mathbb Z$ and $\mathbb Z\times2\mathbb Z$, which are different. Equivalence over the fixed base would force equal based subgroups. Thus \b[the two degree-two covers are inequivalent].

Solution

Past exam of the topology and Groups and Algebraic Topology course of the University of Oxford / 2007 / A / 4 / i / Solution

 Solution (source code)