The Fundamental Group - Main Theorem

The central theorem concerning the fundamental group establishes its invariance under homotopy equivalence, confirming its status as a topological invariant.

Proof sketch: Given a loop $\gamma$ at $x_0$, we construct a homotopy between $f \circ \gamma$ and $g \circ \gamma$ by $K(s,t) = H(\gamma(s), t)$. Since $H(x_0, t) = y_0$ for all $t$, we have $K(0,t) = K(1,t) = y_0$, so $K$ is a path homotopy. Thus $[f \circ \gamma] = [g \circ \gamma]$, giving $f_*([\gamma]) = g_*([\gamma])$.

This theorem immediately yields powerful consequences. The most important is that homotopy equivalent spaces have isomorphic fundamental groups.

Proof: Since $g \circ f \simeq \text{id}_X$ and $f \circ g \simeq \text{id}_Y$, we have by homotopy invariance: $g_* \circ f_* = (g \circ f)_* = (\text{id}_X)_* = \text{id}_{\pi_1(X,x_0)}$ $f_* \circ g_* = (f \circ g)_* = (\text{id}_Y)_* = \text{id}_{\pi_1(Y,f(x_0))}$ Therefore $f_*$ is an isomorphism with inverse $g_*$.

This explains why removing a point from $\mathbb{R}^2$ creates a "hole" detected by $\pi_1$: any loop around the origin cannot be contracted to a point without passing through the removed point.

Proof: Let $c : X \to \{*\}$ be the constant map and $i : \{*\} \to X$ the inclusion. If $X$ is contractible, then $i \circ c \simeq \text{id}_X$, so $c_* : \pi_1(X, x_0) \to \pi_1(\{*\}) = \{e\}$ is surjective and $i_* : \{e\} \to \pi_1(X, x_0)$ is injective with $i_* \circ c_* = \text{id}$. This forces $\pi_1(X, x_0) = \{e\}$.

These theorems establish the fundamental group as a homotopy invariant: it depends only on the homotopy type of the space, not on its specific geometric realization. This makes $\pi_1$ a powerful tool for showing two spaces are not homotopy equivalent by computing their fundamental groups and showing they are not isomorphic.