Let \(X = S^1\times S^1\) and \(\gamma\) a loop based at \(x_0\). What is the induced map \(\gamma_\sharp\)?
Solution
1. The basepoint-change automorphism associated to the loop \(\gamma\) is conjugation by \([\gamma]\):\[ \gamma_\sharp([\alpha])=[\gamma]^{-1}[\alpha][\gamma] \](up to the opposite conjugation convention for basepoint change).
Changing the basepoint along a path \(\gamma\) sends a loop \(\alpha\) to the concatenation \(\gamma^{-1}*\alpha*\gamma\) under this convention. For a loop beginning and ending at the same basepoint, this is an automorphism of the same fundamental group.
2. Since\[ \pi_1(T^2,x_0)\cong\mathbb Z^2 \]is abelian, this conjugation is trivial.
For any \(g,h\) in an abelian group, \(g^{-1}hg=h\).
3. Therefore\[ \boxed{\gamma_\sharp=\operatorname{id}_{\mathbb Z^2}.} \]
Apply 2 to every element of \(\pi_1(T^2,x_0)\).