The Low Dimensional Homology of Projective Linear Group of Rank Two

In this article we study the low dimensional homology of the projective linear group $\textrm{PGL}_2(A)$ over a $\textrm{GE}_2$-ring $A$. In particular, we prove a Bloch-Wigner type exact sequence over local domains. As applications we prove that $H_2(\textrm{PGL}_2(A),\mathbb{Z}\left[\frac{1}{2}\right])\simeq K_2(A)\left[\frac{1}{2}\right]$ and $H_3(\textrm{PGL}_2(A),\mathbb{Z}\left[\frac{1}{2}\right])\simeq K_3^{\textrm{ind}}(A)\left[\frac{1}{2}\right]$.