|
|
$\pi_1(S^1) ≅ \pi_0(\Omega S^1)$ follows from the adjunction between the suspension functor $\Sigma$ and the loop space functor $\Omega$, which yields the general equivalence of pointed homotopy classes of maps $[\Sigma A, B]_* \cong [A, \Omega B]_*$. Setting $A = S^0$ and $B = S^1$ gives $\pi_1(S^1) = [S^1, S^1]_* \cong [S^0, \Omega S^1]_* \cong \pi_0(\Omega S^1)$.
$\pi_0(\Omega S^1) ≅ \pi_0(\mathbb{Z})$ follows from the fact that the loop space $\Omega S^1$ is homotopy equivalent to the discrete space $\mathbb{Z}$.
View $S^1$ as the unit circle in the complex plane, with basepoint $1$. The loop space $\Omega S^1$ consists of all continuous maps $\gamma: [0,1] \to S^1$ such that $\gamma(0) = \gamma(1) = 1$, equipped with the compact-open topology (where two loops are "close" if their images are uniformly close on [0,1]).
The universal cover of $S^1$ is the projection map $p: \mathbb{R} \to S^1$ given by $p(x) = e^{2\pi i x}$. For any based loop $\gamma$ in $S^1$, there is a unique lift to a path $\tilde{\gamma}: [0,1] \to \mathbb{R}$ with $\tilde{\gamma}(0) = 0$ and $p \circ \tilde{\gamma} = \gamma$. The endpoint $\tilde{\gamma}(1)$ must be an integer $n$, called the winding number of $\gamma$ (positive for counterclockwise windings, negative for clockwise).
This lifting gives a homeomorphism: $\Omega S^1$ is homeomorphic to the disjoint union over $n \in \mathbb{Z}$ of the spaces $P_n$, where each $P_n$ is the set of all continuous paths $f: [0,1] \to \mathbb{R}$ with $f(0) = 0$ and $f(1) = n$, again with the compact-open topology.
Now, each $P_n$ is contractible. To see this, note that $P_n$ is convex: for any two paths $f, g \in P_n$, the straight-line combination $h_t = (1-t)f + t g$ (for $t \in [0,1]$) is a continuous homotopy within $P_n$ connecting $f$ to $g$, since it preserves the endpoints and remains continuous. You can explicitly contract any $f \in P_n$ to the straight-line path $\ell(u) = n u$ (from 0 to $n$) via the homotopy $h_t(u) = (1-t) f(u) + t \cdot n u$.
Moreover, the $P_n$ are exactly the path components of $\Omega S^1$. A continuous path in $\Omega S^1$ (i.e., a homotopy of loops) lifts to a continuous family of paths in $\mathbb{R}$ starting at 0, and the endpoints form a continuous map to $\mathbb{R}$. But since the endpoints land in $\mathbb{Z} \subset \mathbb{R}$ and $\mathbb{Z}$ is discrete, this endpoint map must be constant. Thus, you can't connect loops with different winding numbers via a path in $\Omega S^1$.
Since each path component $P_n$ is contractible (homotopy equivalent to a single point) and there is one for each integer $n$, the entire space $\Omega S^1$ is homotopy equivalent to the disjoint union of $\mathbb{Z}$ many points, i.e., $\mathbb{Z}$ with the discrete topology. |
|
