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How do we formally prove that the fundamental group of any Lie group is always commutative?

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As Vahid says, it is true for any topological group. Here is a proof. I'm sure there are nicer, more conceptual ones out there, but here goes.

Let $G$ be your topological group. Take two loops $\sigma$ and $\gamma$ in $G$, based at the identity of $G$, which we will denote by $e$. Let $\sigma \cdot \gamma$ be the concatenation of the two loops. This is given by $$ (\sigma \cdot \gamma) (t) = \begin{cases} \sigma(2t) & \quad \text{ if } 0 \le t \le 1/2 \\\ \gamma(2t-1) &\quad \text{ if } 1/2 \le t \le 1 \end{cases} $$ (Sorry, couldn't manage to format that any better. Feel free to edit if you know how to put a nice brace bracket to the left of that definition.)

The idea is this. We will show that $\sigma \cdot \gamma$ is homotopic to to the loop given by the pointwise product of $\sigma$ and $\gamma$. Let's call that loop $\rho$, so $$ \rho(t) = \sigma(t)\gamma(t).$$

Now define an auxiliary function $P : [0,1] \times [0,1] \to G$ by $$ P(s,t) = \begin{cases} \sigma\left( \frac{2t}{1+s} \right) & \quad \text{ if } 0 \le t \le \frac{1+s}{2} \\\ e &\quad \text{ if } \frac{1+s}{2} \le t \le 1 \end{cases}$$

At $s=0$, this function does the whole loop $\sigma$ as $t$ goes from $0$ to $1/2$, then sits at $e$. In other words, at $s=0$ this is the first half of the loop $\sigma \cdot \gamma$. As $s$ gets larger, $P$ does the whole loop $\sigma$ as $t$ goes from $0$ to $\frac{1+s}{2}$. At $s=1$, $P$ does the loop $\sigma$ at normal speed.

Then similarly define a function $Q : [0,1] \times [0,1] \to G$ by $$ Q(s,t) = \begin{cases} e & \quad \text{ if } 0 \le t \le \frac{1-s}{2} \\\ \gamma \left( \frac{2t-1+s}{1+s} \right) &\quad \text{ if } \frac{1-s}{2} \le t \le 1 \end{cases}$$

At $s=0$ this is just the second half of the loop $\sigma\cdot\gamma$, while at $s=1$ it is exactly the loop $\gamma$.

So finally, define $$ H(s,t) = P(s,t) \cdot Q(s,t). $$ At $s=0$ this is $\sigma \cdot \gamma$, while at $s=1$ it is the pointwise product loop $\rho$. $H$ is clearly continuous, and $H(s,0) = e = H(s,1)$ for all $s$, so this is a homotopy of loops between $\sigma \cdot \gamma$ and $\rho$.

Now we can redo that process and show that $\rho$ is homotopic to the other concatenation $\gamma \cdot \sigma$. So this shows that $\pi_1(G)$ is abelian.

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One-sentence explanation: because the fact that a topological group $G$ is a group object in topological spaces makes its fundamental group $\pi_1(G)$ a group object in groups, and this is an abelian group.

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    $\begingroup$ It's certainly the nicest proof of this result I've ever seen! $\endgroup$
    – Stefano V.
    May 25 '12 at 10:30
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    $\begingroup$ This is terrific. $\endgroup$ May 25 '12 at 12:11
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    $\begingroup$ "...by the Eckmann-Hilton argument." $\endgroup$ May 25 '12 at 12:20
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    $\begingroup$ the point is that the group law in the fundamental group of a topological group is a group homomorphism; a group whose law is a homomorphism is easily checked to abelian (I knew this argument but not with the interpretation by group objects). $\endgroup$
    – YCor
    May 25 '12 at 13:18
  • $\begingroup$ Nice! I knew there was some clever way of doing it. $\endgroup$
    – MTS
    May 25 '12 at 14:58
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Geometric proof: A connected Lie group $G$ is homotopy equivalent to a maximal compact subgroup, so we may assume $G$ is compact. Being compact, $G$ admits a bi-invariant Riemannian metric with respect to which it is a symmetric space, the symmetry $s$ at the identity being just the inversion map. Now a homotopy class in $\pi_1(G,1)$ can be represented by a closed geodesic $\gamma$ (of minimal length in its homotopy class, by a shortening process). Since the differential of $s$ at $1$ is minus identity, $s$ sends $\gamma$ to itself parametrized backwards. It follows that the homomorphism induced by $s$ on the $\pi_1$-level is inversion. However, the inversion map in a group is a homomorphism if and only if the group is Abelian.

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    $\begingroup$ +1 That's a very cool argument! $\endgroup$
    – Igor Rivin
    May 25 '12 at 14:49
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It is actually true for all topological groups. Topological groups possess a structure which makes them H-spaces and fundamental group of every H-space is abelian. The formulation and the proof is given in Algebraic Topology, Homotopy and Homology, by Switzer Pages 14-16.

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