Question

Let T=S1×S1T=S^{1} \times S^{1}, the torus. There is an isomorphism of π1(T,b0×b0)\pi_{1}\left(T, b_{0} \times b_{0}\right) with Z×Z\mathbb{Z} \times \mathbb{Z} induced by projections of T onto its two factors. (a) Find a covering space of T corresponding to the subgroup of Z×Z\mathbb{Z} \times \mathbb{Z} generated by the element m×0m \times 0, where m is a positive integer. (b) Find a covering space of T corresponding to the trivial subgroup of Z×Z\mathbb{Z} \times \mathbb{Z}. (c) Find a covering space of T corresponding to the subgroup of Z×Z\mathbb{Z} \times \mathbb{Z} generated by m×0m \times 0 and 0×n0 \times n where m and n are positive integers.