Fundamental Groups Under Homeomorphisms On Topological Space
Fundamental Groups under Homeomorphisms on Topological Spaces
We now look at a very important theorem regarding the fundamental groups of topological spaces.
| Theorem 1: Let $X$ and $Y$ be path-connected topological spaces. If $X$ and $Y$ are homotopically equivalent then $\pi_1(X, x) \cong \pi_1(Y, y)$ for every $x \in X$ and for every $y \in Y$. |
Recall from the Homeomorphic Topological Spaces are Homotopically Equivalent page that if two spaces are homeomorphic then they are also homotopically equivalent. Therefore, if $X$ and $Y$ are homeomorphic path-connected topological spaces then for every $x \in X$ and for every $y \in Y$:
(1)\begin{align} \quad \pi_1(X, x) \cong \pi_1(Y, y) \end{align}
Thus, the fundamental group of a space is a topological property. This theorem is extremely powerful for showing that two topological spaces are not homeomorphic.
| Theorem 2: The closed unit disk $D^2$ is not homeomorphic to the sphere $S^2$. |
- Proof: Suppose instead that $D^2$ is homeomorphic to $S^2$ and let $f : D^2 \to S^2$ be a homeomorphism. Let $x^* \in D^2$. Then $S^2 \setminus \{ x^* \}$ must be homeomorphic to $D^2 \setminus \{ f(x^*) \}$.
- Consider the fundamental group of $D^2 \setminus \{ x^* \}$. The circle is a deformation retract of $D^2 \setminus \{ x \}$ and so:
\begin{align} \quad \pi_1(D^2 \setminus \{ x^* \}, x) \cong \mathbb{Z} \end{align}
- Now consider the fundamental group of [[$ S^2 \setminus \{ f(x^*) \} $. The space can be deformed into a disk as illustrated below:
- Therefore:
\begin{align} \quad \pi_1(S^2 \setminus \{ f(x^*) \}, x) \cong \{ 1 \} \end{align}
- But these fundamental groups are different which contradict theorem 2. Therefore the assumption that $D^2$ was homeomorphic to $S^2$ was false. So $D^2$ is not homeomorphic to $S^2$. $\blacksquare$