Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

How can I prove that $p:S^1\rightarrow S^1$, $z\mapsto z^2$ is a covering map? Please help. I was not able to prove it by applying definition of covering space.

share|improve this question
add comment

2 Answers 2

up vote 3 down vote accepted

Here's a different approach using some differential topology. If $f:M\to N$ is a surjective smooth map between manifolds and $df$ is everywhere an isomorphism, then $f$ is a covering map.*

Here, $z\mapsto z^2$ is a surjective map of one-manifolds and the differential is multiplication by $2$, in particular an isomorphism of each tangent space onto its image. Hence the map is a cover.

*The proof uses the inverse function theorem and is not difficult, so I'll leave it to you as an exercise. You can find it in standard differential topology textbooks like Guillemin-Pollack.

share|improve this answer
    
Nice. Besides leaving the proof of the fact that proves what the OP asked, can you perhaps give some links to sites/books where this result can be found? –  DonAntonio Feb 28 '13 at 12:00
    
@DonAntonio Edited as per request. Guillemin-Pollack is (I think) a standard reference for this sort of thing. –  Neal Feb 28 '13 at 15:28
add comment

You can prove it by checking that the map is continuous, surjective, and that the 4 open semicircles are evenly covered.

enter image description here


LaTeX code for picture:

\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{arrows}
\begin{document}
\newcommand{\opensemicircle}[2]{  
    \draw #1 circle (1);
    \draw[ultra thick,red,(-)] #1 ++ (#2:1) arc (#2:#2+180:1);
    }
\begin{tikzpicture}
\opensemicircle{(0,3)}{0}
\opensemicircle{(3,3)}{90}
\opensemicircle{(0,0)}{180}
\opensemicircle{(3,0)}{270}
\end{tikzpicture}
\end{document}
share|improve this answer
2  
Nice picture! How did you draw it? –  user27126 Feb 28 '13 at 5:49
1  
@Sanchez: Thanks! I added the LaTeX code I used to generate it. –  Zev Chonoles Feb 28 '13 at 6:21
1  
Thats really awsome! –  Bunuelian Trick Feb 28 '13 at 6:22
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

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