A About calculating a fundamental group

[aalma]

What is the way to compute $\pi_1(PGL_2(R))$?
Is it related to defining an action of $PGL_2(R)$ on $S^3$?

it would be helpful if you can provide me with relevant information regarding this

 

[Infrared]

There is a bundle $\mathbb{R}^\times\to GL_n(\mathbb{R})\to PGL_n(\mathbb{R})$ where $\mathbb{R}^\times$ is the subgroup of nonzero scalar matrices. The identity component of the fiber is contractible, so $PGL_n(\mathbb{R})$ and $GL_n(\mathbb{R})$ has the same homotopy groups in positive degrees, and also the identity component of $GL_n(\mathbb{R})$ is homotopy equivalent to $SO(n;\mathbb{R})$ by performing Gram-Schmidt on the columns.

So in this case, $\pi_1(PGL_2(\mathbb{R}))\cong\pi_1(SO(2;\mathbb{R}))\cong \pi_1(S^1)\cong\mathbb{Z}.$

 

[aalma]

Thanks:)
Is the idea here to move from the fibration you first mentioned to a long exact sequence, knowing that $\pi_0(GL_2(R))=\pi_0(SO_2(R))$?
When saying "The identity component of the fiber is contractible" to what fiber are you referring and then you mean that $\pi_1(GL_2(R))=\pi_1(PGL_2(R))$?