I am reading the Fulton& Harris "Representation Theory:A First Course".In Page 136 they gave a group $$ G_0 = \left\{ \begin{pmatrix} a & b \\ 0 & 1 \end{pmatrix}: a \neq 0 \right\} \subset GL_2(\mathbb{C}) $$ and they said "topologically this group is homeomorphic to $\mathbb{C}\times \mathbb{C}^*$", could I ask that how to prove this? Thanks!

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  • 4
    $\begingroup$ Can you think of a continuous map from $G_0$ to $\mathbb{C} x \mathbb{C}^\times$? $\endgroup$ May 14 at 3:36
  • 5
    $\begingroup$ I mean, $a$ and $b$ are right there, and one of them isn't allowed to be zero... $\endgroup$ May 14 at 4:10
  • $\begingroup$ But I thought as @StefanDawydiak said, this homeomorphism needed to be proved by constructing a continuous map, or maybe prove the fundamental group of $G_0$ is isomorphic as the fundamental group of $\mathbb{C}\times \mathbb{C}^*$... $\endgroup$
    – Snow
    May 14 at 7:08
  • 2
    $\begingroup$ Proving the fundamental groups are isomorphic is not enough, but this question is much simpler than that. I think you are overthinking this. Elements of $G_0$ are given by two complex numbers, one nonzero. Where should the map you want to construct send these two numbers? As Jonathan says, this map has a very simple definition. $\endgroup$ May 14 at 11:54
  • $\begingroup$ I see. Thank you very much! $\endgroup$
    – Snow
    May 15 at 5:15


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