Fundamental group of $GL^{+}_2(\mathbb{R})$

Question

I know that fundamental group of $GL^{+}_2(\mathbb{R})$ is isomorphic to $\mathbb{Z}$. It's written everywhere. I don't know how to proceed to prove it, any hint ?

Thanking you.

Answer 1

The way to do this in general is to learn how to decompose Lie groups.

The Iwasawa decomposition (I think) here is $GL^+(2,\Bbb R)=KAN$ where

• $K$ is the subgroup $SO(2)$,
• $A$ is the subgroup of diagonal matrices with positive entries,
• $N$ is the subgroup of unitriangular matrices.

To see this, observe how $SO(2)$ acts transitively on circles in the plane, so given any column vector there exists a rotation taking it to the positive $x$-axis. So given any matrix in $GL^+(2,\Bbb R)$ there is a rotation matrix we can apply to it on the left to make its first column have positive first entry and zero second entry, which forces the lower right corner to be positive too. Thus the group elements may be decomposed as rotation matrices times upper triangular matrices. Then the subgroup of upper triangular matrices with positive diagonal entries can further be decomposed as $AN$.

Indeed, every $g\in GL^+(2,\Bbb R)$ is uniquely expressible as $g=kan$ with $k\in K,a\in A,n\in N$ and there is a diffeomorphism $K\times A\times N\to G$ given by $(k,a,n)\mapsto kan$. Now observe

• $K\cong \Bbb S^1$
• $A\cong \Bbb R^2$
• $N\cong\Bbb R^1$

To see $A$, notice it is $\Bbb R^{>0}\times\Bbb R^{>0}$, and $\Bbb R^{>0}\cong\Bbb R$ via $\log$ and $\exp$.