Fixed-point-free self-maps homotopic to the identity on products of surfaces ($(g_1-1)(g_2-1)=0$)

Such a self-map of ${\Sigma }_{g_1}\times {\Sigma }_{g_2}$ exists exactly when one factor is a torus. Necessity follows because its Lefschetz number equals the Euler characteristic $4(1-g_1)(1-g_2)$ and must vanish. Sufficiency is translation by a nonzero point in the torus factor, times the identity on the other factor. The condition does not determine the product $g_1{g}_2$, since the other genus can be arbitrary.