For the product $S_1\times S_2$ of any two connected compact hyperbolic surfaces $S_1$ and $S_2$, we give a finite bound $\mathcal{ B}$ such that for any self-homeomorphism $f$ of $S_1\times S_2$ and any fixed point class $F$ of $f$, the index $|ind(f, F)|\leq \mathcal{ B}$, which is an affirmative answer for a special case of a question asked by Boju Jiang. Moreover, we also give bounds for the Lefschetz number $L(f)$ and the Nielsen number $N(f)$ of the homeomorphism $f$.
Categories: math.GT