If \(n > 1\), show that the fundamental group of the \(n\)-fold torus is not abelian. [Hint: Let \(G\) be the free group on the set \(\ts{\alpha_1, \beta_1, \ldots, \alpha_n, \beta_n}\); let \(F\) be the free group on the set \(\ts{\gamma, \delta}\). Consider the homomorphism of \(G\) onto \(F\) that sends \(\alpha_1\) and \(\beta_1\) to \(\gamma\) and all other \(\alpha_i\) and \(\beta_i\) to \(\delta\).]
Solution
For the \(n\)-fold torus, \[ \pi_1(T^{\# n})\cong \left\langle \alpha_1,\beta_1,\ldots,\alpha_n,\beta_n\ \middle|\ \prod_{i=1}^n[\alpha_i,\beta_i]=1\right\rangle. \] Let \(F=F(\gamma,\delta)\) be the free group on two generators. Define a homomorphism from the free group on the surface generators by \[ \alpha_1,\beta_1\mapsto\gamma, \qquad \alpha_i,\beta_i\mapsto\delta\quad(i\ge2). \] The defining relator maps to \[ [\gamma,\gamma]\,[\delta,\delta]^{\,n-1}=1, \] so the map factors through \(\pi_1(T^{\# n})\). Since \(n>1\), both \(\gamma\) and \(\delta\) occur in the image, so the induced homomorphism onto \(F(\gamma,\delta)\) is surjective. A quotient of an abelian group is abelian, while \(F(\gamma,\delta)\) is nonabelian. Hence \(\pi_1(T^{\# n})\) is nonabelian.