Covering space of $mathrm{SL}_2(mathbb{R})$
How to show that $mathbb{R} times (0,+infty) times mathbb{R}$ is a universal cover of $mathrm{SL}_2(mathbb{R})$ by Iwasawa Decomposition?
My attempt By Iwasawa Decomposition, for any $A in mathrm{SL}_2(mathbb{R})$, we have
$$
A =
begin{pmatrix}
cos theta & -sin theta \
sin theta & cos theta
end{pmatrix}
begin{pmatrix}
r & 0 \
0 & frac 1r
end{pmatrix}
begin{pmatrix}
1 & x \
0 & 1
end{pmatrix}.
$$
Thus we get a map $varphi: mathbb{R} times (0,+infty) times mathbb{R} to mathrm{SL}_2(mathbb{R})$. I think it's sufficient to show that $mathbb{R}$ is a universal cover of the subspace
$$
left{
begin{pmatrix}
cos theta & -sin theta \
sin theta & cos theta
end{pmatrix}
:
theta in mathbb{R}
right}.
$$
But I don't know how to show it rigorously. Can you give me some hints?
abstract-algebra algebraic-topology
asked Nov 26 at 14:58
Kai Xing
464
add a comment |
How to show that $mathbb{R} times (0,+infty) times mathbb{R}$ is a universal cover of $mathrm{SL}_2(mathbb{R})$ by Iwasawa Decomposition?
My attempt By Iwasawa Decomposition, for any $A in mathrm{SL}_2(mathbb{R})$, we have
$$
A =
begin{pmatrix}
cos theta & -sin theta \
sin theta & cos theta
end{pmatrix}
begin{pmatrix}
r & 0 \
0 & frac 1r
end{pmatrix}
begin{pmatrix}
1 & x \
0 & 1
end{pmatrix}.
$$
Thus we get a map $varphi: mathbb{R} times (0,+infty) times mathbb{R} to mathrm{SL}_2(mathbb{R})$. I think it's sufficient to show that $mathbb{R}$ is a universal cover of the subspace
$$
left{
begin{pmatrix}
cos theta & -sin theta \
sin theta & cos theta
end{pmatrix}
:
theta in mathbb{R}
right}.
$$
But I don't know how to show it rigorously. Can you give me some hints?
abstract-algebra algebraic-topology
asked Nov 26 at 14:58
Kai Xing
464
1
I don't understand why this was downvoted.
– Shaun
Nov 26 at 15:22
5
the subspace you mention is homeomorphic to a circle (basically, the parametrisation you gave factors into the homeomorphism). Hope that helps.
– user120527
Nov 26 at 15:27
@user120527 Thanks! It's a good explaination.
– Kai Xing
Nov 27 at 1:49
add a comment |
How to show that $mathbb{R} times (0,+infty) times mathbb{R}$ is a universal cover of $mathrm{SL}_2(mathbb{R})$ by Iwasawa Decomposition?
My attempt By Iwasawa Decomposition, for any $A in mathrm{SL}_2(mathbb{R})$, we have
$$
A =
begin{pmatrix}
cos theta & -sin theta \
sin theta & cos theta
end{pmatrix}
begin{pmatrix}
r & 0 \
0 & frac 1r
end{pmatrix}
begin{pmatrix}
1 & x \
0 & 1
end{pmatrix}.
$$
Thus we get a map $varphi: mathbb{R} times (0,+infty) times mathbb{R} to mathrm{SL}_2(mathbb{R})$. I think it's sufficient to show that $mathbb{R}$ is a universal cover of the subspace
$$
left{
begin{pmatrix}
cos theta & -sin theta \
sin theta & cos theta
end{pmatrix}
:
theta in mathbb{R}
right}.
$$
But I don't know how to show it rigorously. Can you give me some hints?
abstract-algebra algebraic-topology
asked Nov 26 at 14:58
Kai Xing
464
How to show that $mathbb{R} times (0,+infty) times mathbb{R}$ is a universal cover of $mathrm{SL}_2(mathbb{R})$ by Iwasawa Decomposition?
My attempt By Iwasawa Decomposition, for any $A in mathrm{SL}_2(mathbb{R})$, we have
$$
A =
begin{pmatrix}
cos theta & -sin theta \
sin theta & cos theta
end{pmatrix}
begin{pmatrix}
r & 0 \
0 & frac 1r
end{pmatrix}
begin{pmatrix}
1 & x \
0 & 1
end{pmatrix}.
$$
Thus we get a map $varphi: mathbb{R} times (0,+infty) times mathbb{R} to mathrm{SL}_2(mathbb{R})$. I think it's sufficient to show that $mathbb{R}$ is a universal cover of the subspace
$$
left{
begin{pmatrix}
cos theta & -sin theta \
sin theta & cos theta
end{pmatrix}
:
theta in mathbb{R}
right}.
$$
But I don't know how to show it rigorously. Can you give me some hints?
abstract-algebra algebraic-topology
abstract-algebra algebraic-topology
asked Nov 26 at 14:58
Kai Xing
464
asked Nov 26 at 14:58
Kai Xing
464
edited Nov 26 at 15:13
asked Nov 26 at 14:58
Kai Xing
464
asked Nov 26 at 14:58
Kai Xing
464
asked Nov 26 at 14:58
Kai Xing
464
464
1
I don't understand why this was downvoted.
– Shaun
Nov 26 at 15:22
5
the subspace you mention is homeomorphic to a circle (basically, the parametrisation you gave factors into the homeomorphism). Hope that helps.
– user120527
Nov 26 at 15:27
@user120527 Thanks! It's a good explaination.
– Kai Xing
Nov 27 at 1:49
add a comment |
1
I don't understand why this was downvoted.
– Shaun
Nov 26 at 15:22
5
the subspace you mention is homeomorphic to a circle (basically, the parametrisation you gave factors into the homeomorphism). Hope that helps.
– user120527
Nov 26 at 15:27
@user120527 Thanks! It's a good explaination.
– Kai Xing
Nov 27 at 1:49
1
1
I don't understand why this was downvoted.
– Shaun
Nov 26 at 15:22
I don't understand why this was downvoted.
– Shaun
Nov 26 at 15:22
5
5
the subspace you mention is homeomorphic to a circle (basically, the parametrisation you gave factors into the homeomorphism). Hope that helps.
– user120527
Nov 26 at 15:27
the subspace you mention is homeomorphic to a circle (basically, the parametrisation you gave factors into the homeomorphism). Hope that helps.
– user120527
Nov 26 at 15:27
@user120527 Thanks! It's a good explaination.
– Kai Xing
Nov 27 at 1:49
@user120527 Thanks! It's a good explaination.
– Kai Xing
Nov 27 at 1:49
add a comment |
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1
I don't understand why this was downvoted.
– Shaun
Nov 26 at 15:22
5
the subspace you mention is homeomorphic to a circle (basically, the parametrisation you gave factors into the homeomorphism). Hope that helps.
– user120527
Nov 26 at 15:27
@user120527 Thanks! It's a good explaination.
– Kai Xing
Nov 27 at 1:49