Undergrad Proving SL_2(C) Homeomorphic to SU(2)xT & Simple Connectedness

[aalma]

Using the QR decomposition (the complex version) I want to prove that $SL_2(C)$ is homeomorphic to the product $SU(2) × T$ where $T$ is the set of upper-triangular 2×2-complex matrices with real positive entries at the diagonal. Deduce that $SL(2, C)$ is simply-connect.

So, I can define a map $SU(2)×T–>SL(2,C)$ given by: $(u,t) –> ut$ where $u \in SU(2)$ and $t \in T$ (from QR decomposition we have that each $A$ in $GL(2,C)$ can be written as $ut$ where $u \in U(n)$ and $t \in T$ ($T$ mentioned above)).
We have that the intersection $SU(2)\cap T=1$.
This map is injective:
if $u_1t_1=u_2t_2$ then this gives
$u_2^{-1}u_1=t_2t_1^{-1} \in SU(2)\cap T$ so $u_1=u_2$ and $t_1=t_2$.
Can I say that this map is surjective by the QR decomposition?
why is this map and its inverse contiuous?

Thanks for help!

 

[fresh_42]

Where does the determinant of $t$ go to in case it is not $(a,a^{-1})?$

Maybe one of these mappings here serves you better:
https://www.physicsforums.com/insig...c-part/#3-The-3-Sphere-or-Glome-or-SU2mathbbC