Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 24 / 1 / c / Solution

Use the quotient lens spaces and set $\zeta =e^{2\pi i/7}$. Let $g_j(z_1,z_2)=(\zeta z_1,{\zeta }^j{z}_2)$. The mapis a homeomorphism and satisfiesbecause $3\cdot 2\equiv -1(\mathrm{mod}7)$. Since $2$ is invertible modulo $7$, $g_3^2$ generates the entire target deck group. Hence $F$ maps cyclic orbits bijectively to cyclic orbits, as does its inverse, and descends to a homeomorphism:This is an explicit map, so no classification theorem for lens spaces is required. It reverses orientation: complex conjugation in one coordinate reverses the ambient real orientation, while swapping the two complex coordinates preserves it.