Vtyjkyuk

Is the following map a quotient map?
$$f:GL_2(Bbb R) to Bbb C setminus text0$$
$$
beginpmatrix
a & b\
c & d\
end pmatrix
mapsto fracaiota+bciota+d $$

I wanted a quotient map from $SL_2(Bbb R)$ to $mathscr H$(the upper half plane), if I could show $f$ is a quotient map then somehow the restriction to $SL_2(Bbb R)$ could be shown to be a quotient map.

Expanding my (possibly somewhat misguided) comment to an argument showing that $f:gammamapsto gammacdotiota$ is a quotient map from $SL_2(BbbR)$ to the upper half plane $H$.
First let's look at the following subgroup
$$
G=leftleft(beginarrayccy^1/2&xy^-1/2\0&y^-1/2endarrayright)mid x,yinBbbR,y>0rightle SL_2(BbbR)
$$
consisting of all the upper triangular matrices in $SL_2(BbbR)$ with positive diagonal entries.
Call the described matrix $gamma(x,y)$. We immediately see that $f(gamma(x,y))=x+iyin H$. In other words the restriction of $f$ to the subgroup $G$ is a bijection.

We also see easily that the stabilizer of $iota$ in $SL_2(BbbR)$ consists of the matrices of plane rotations w.r.t. the usual basis:
$$
Stab_SL_2(iota)=K=leftleft(beginarraycccostheta&-sintheta\ sintheta&costhetaendarrayright)mid thetainBbbRright.
$$

We also see that $G$ is a closed subgroup of $SL_2(BbbR)$. Futhermore, if we identify the elements of $G$ with the corresponding points in $H$, we get a homeomorphism.
Putting all the pieces together we see that:

• If $Csubseteq H$ is closed, then the set $tildeC:=f^-1(C)cap
  G$ is closed as a subset of $SL_2(BbbR)$.


• $f^-1(C)=tildeCK=ckmid cintildeC, kin K$ is a closed subset of $SL_2(BbbR)$ as a product of a closed and a compact subset.


• If $D$ is not a closed subset of $H$ then $f^-1(D)cap G$ is not a closed subset of $G$ either (homeomorphism!). Consequently $f^-1(D)$ cannot be a closed subset of $SL_2$.


• So $Csubseteq H$ is closed if and only if $f^-1(C)subseteq SL_2(BbbR)$ is closed proving that $f$ is a (topological) quotient map.

I'm sure that there exists a suitable more general result you can use. Say, about a topological group acting transitively on a space such that all the point stabilizers are compact? Or some such. I plead ignorance at this point, and look forward to be educated.