Given the topological connected and compact surfaces $S_1$ and $S_2$ is it true that $$ \chi(S_1)=\chi(S_2)=-18 \implies S_1, S_2 \mbox{ homeomorphic}?$$
I know that the converse is always true, but in this case I don't know how to proceed.
Have you any book references which explain this or any help on how to justify the correctness or not of the statement?