Commutativity of the fundamental group of any Lie Group - MathOverflow [closed]

Commutativity of the fundamental group of any Lie Group

2012-05-25T04:30:32Z

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

Answer by Vahid Shirbisheh for Commutativity of the fundamental group of any Lie Group

2012-05-25T04:54:59Z

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.

Answer by MTS for Commutativity of the fundamental group of any Lie Group

2012-05-25T05:48:32Z

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.

Answer by Ryan Reich for Commutativity of the fundamental group of any Lie Group

2012-05-25T07:14:26Z

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.

Answer by Claudio Gorodski for Commutativity of the fundamental group of any Lie Group

2012-05-25T13:06:12Z

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.