The Fund. Groups of a Path-Con. Topological Space are Isomorphic

The Fundamental Groups of a Path Connected Topological Space are Isomorphic

Theorem 1: Let $X$ be a topological space. If $X$ is path connected then $\pi_1(X, x_1)$ is isomorphic to $\pi_1(X, x_2)$ for all $x_1, x_2 \in X$.
  • Proof: If $x_1 = x_2$ the theorem holds trivially. Assume instead that $x_1 \neq x_2$. Since $X$ is path connected there exists a path $\gamma : I \to X$ such that $\gamma(0) = x_2$ and $\gamma (1) = x_1$. Define a function $\phi : \pi_1(X, x_1) \to \pi_2(X, x_2)$ for all homotopy classes $[\alpha] \in \pi_1(X, x_1)$ by:
\begin{align} \quad \phi ([\alpha]) = [\gamma \alpha \gamma^{-1}] \end{align}
  • We need to show that $\phi$ is well-defined. Suppose that $\alpha \simeq \alpha'$. Then we must show that $[\gamma \alpha \gamma^{-1}] = [\gamma \alpha' \gamma^{-1}]$. But we know this holds since $[\alpha] = [\alpha']$.
  • We now show that $\phi$ is a homomorphism. Let $[\alpha], [\beta] \in \pi_1(X, x_1)$. Then:
\begin{align} \quad \phi([\alpha][\beta]) &= \phi ([\alpha\beta]) \\ &= [\gamma\alpha\beta\gamma^{-1}] \\ &= [\gamma\alpha c_x \beta \gamma^{-1}] \\ &= [\gamma \alpha \gamma^{-1} \gamma \beta \gamma^{-1}] \\ &= [\gamma \alpha \gamma^{-1}][\gamma \beta \gamma^{-1}] \\ &= \phi([\alpha]) \phi ([\beta]) \end{align}
  • So indeed, $\phi$ is a homomorphism. Lastly, to show that $\phi$ is bijection, we define an inverse function $\psi : \pi_1(X, x_2) \to \pi_1(X, x_1)$ for all $[\beta] \in \pi_1(X, x_2)$ by:
\begin{align} \quad \psi ([\beta]) = [\gamma^{-1} \beta \gamma] \end{align}
  • Then clearly $\psi \circ \phi = \mathrm{id}_{\pi_1(X, x_1)}$ and $\phi \circ \psi = \mathrm{id}_{\pi_1(X, x_2)}$. So $\phi$ is an isomorphism and:
\begin{align} \quad \pi_1(X, x_1) \cong \pi_1(X, x_2) \quad \blacksquare \end{align}
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