I know that the fundamental group of the double torus is $\pi_1(M)=\langle a,b,c,d;a^{-1}b^{-1}abc^{-1}d^{-1}cd\rangle$.
How can I calculate its center subgroup $C$? Is $C$ trivial?
Let $p$ be the quotient map from $\pi_1(M)$ to $H_1(M)$, maybe it's easy to prove that $p(C)$ is $0$ in $H_1(M).$ That will also solve my problem.
Thanks, Yan