Suppose \(f_t: X \to X\) is a homotopy such that \(f_0\) and \(f_1\) are each the identity map. Use Lemma 1.19 to show that for any \(x_0 \in X\), the loop \(f_t(x_0)\) represents an element of the center of \(\pi_1(X, x_0)\). [One can interpret the result as saying that a loop represents an element of the center of \(\pi_1(X)\) if it extends to a loop of maps \(X \to X\).]
Solution
Fix \(x_0\in X\) and define \[ h(t)=f_t(x_0). \]
1. The path \(h\) is a loop based at \(x_0\).
Since \[ f_0=f_1=\operatorname{id}_X, \] one has \[ h(0)=f_0(x_0)=x_0=f_1(x_0)=h(1). \]
2. Lemma 1.19 gives\[ (f_0)_*=\beta_h\,(f_1)_*. \]
Lemma 1.19 states that for a homotopy \(f_t:X\to X\) and the basepoint track \(h(t)=f_t(x_0)\), the induced maps satisfy exactly this relation after the appropriate change of basepoint by \(h\). Here both endpoint maps take \(x_0\) to \(x_0\), so all three maps act on \[ \pi_1(X,x_0). \]
3. Therefore\[ \beta_h=\operatorname{id}_{\pi_1(X,x_0)}. \]
Both \(f_0\) and \(f_1\) are the identity map, so \[ (f_0)_*=(f_1)_*=\operatorname{id}_{\pi_1(X,x_0)}. \] Substituting into 2 gives \[ \operatorname{id}=\beta_h\operatorname{id}=\beta_h. \]
4. For every \([\gamma]\in\pi_1(X,x_0)\),\[ [h][\gamma][h]^{-1}=[\gamma]. \]
By the definition of the change-of-basepoint automorphism associated to the loop \(h\), \[ \beta_h([\gamma])=[h\cdot\gamma\cdot\bar h] =[h][\gamma][h]^{-1}. \] Now use 3.
5. Hence \([h]\) lies in the center of \(\pi_1(X,x_0)\).
Multiplying the equality in 4 on the right by \([h]\) gives \[ [h][\gamma]=[\gamma][h] \] for every \([\gamma]\in\pi_1(X,x_0)\). This is precisely the condition that \([h]\) be central.