5
$\begingroup$

I´m thinking about universal covering spaces. I´ve seen a lot of examples and authors ever say "the sphere $S^n$ is the universal covering space of the $n$-dimensional projective space $\mathbb{R}P^n$ for $n \geq 1$.

So my question is: and what about the real projective line $\mathbb{R}P^1$? Has it universal covering space?

Thanks!

$\endgroup$
5
$\begingroup$

$\mathbb{R}P^1$ is homeomorphic to $\mathbb{S}^1$. To see this, first note that more generaly $\mathbb{R}P^n\simeq\mathbb{S}^n/_{\pm id}$ and in the case $n=1$ you have $ \mathbb{S}^1/_{\pm id}\simeq \mathbb{S}^1$ (just factorize the map $z\mapsto z^2$).

From here you can conclude.

$\endgroup$
  • $\begingroup$ Sure! I had the homeomorphism but I missed it! Lol. Thanks! :D $\endgroup$ – user183002 Apr 20 at 17:19
4
$\begingroup$

The real projective line is just a circle, so the universal covering space is the real line.

$\endgroup$
  • $\begingroup$ I don´t agree. Even it´s not a circumference. It´s a quotient of a circumference, yeah? $\endgroup$ – user183002 Apr 20 at 17:10
  • 1
    $\begingroup$ @user A cheap way to see it is that the unique closed one dimensional manifold is a circle. There's nothing else it can be. $\endgroup$ – Matt Samuel Apr 20 at 17:15
  • $\begingroup$ Important to note is that it isn’t the universal cover. $\endgroup$ – Connor Malin Apr 20 at 18:47
  • $\begingroup$ What I mean is that $S^1 \rightarrow \mathbb{R}P^1$ is a covering space, but not a universal one. $\endgroup$ – Connor Malin Apr 20 at 19:29
  • 1
    $\begingroup$ @Connor I agree. Note my answer says the real line is the universal cover. $\endgroup$ – Matt Samuel Apr 20 at 19:31

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.