"You can also get a cyclic group of order p by attaching a disk to a circle by wrapping it around the circle p times (the fact that the fundamental group is Z/pZ follows from Van-Kampen’s theorem). " But I can't understand what's the figure looks like about" attaching a disk to a circle by wrapping it around the circle p times". Thank you.
Asked
Active
Viewed 983 times
2
-
For $p\ge2$, you obtain a manifold that cannot be embedded into $\mathbb{R}^3$, and thus is hard to visualize. It is called the $p$-fold dunce cap: http://en.wikipedia.org/wiki/Dunce_hat_(topology) – Daniel Robert-Nicoud Aug 16 '14 at 21:37
-
2For p > 2 it's not really a manifold. – jxnh Aug 16 '14 at 21:39
1 Answers
3
Consider the map $f_p\colon S^1\to S^1$ given by $f(e^{i\theta})=e^{pi\theta}$. This 'wraps' the circle around itself $p$ times. Now glue the disk $D^2$ to $S^1$ on it's boundary via the map $f_p$, that is, glue the point $e^{i\theta}$ in $\partial D^2$ to the point $e^{pi\theta}$ in $S^1$.
Dan Rust
- 30,108
-
Thank you very much, but could you tell me what are the generator and the relation of this fundamental group? – 6666 Aug 16 '14 at 22:53
-
A single loop going around the circle that the disk is glued onto will generate the fundamental group. If you're familiar with the usual $n$-gon with side-matchings method of forming spaces, then it is equivalent to the $p$-gon with edges labelled $\underbrace{aa\ldots aa}_{p\mbox{ times}}$. So a generator is given by $a$ and the relation would then be $a^p=e$. – Dan Rust Aug 16 '14 at 23:10
-
-
Yes, let $U$ be the space with a point removed somewhere on the interior of the disk, and let $V$ be a small open disk around that point. – Dan Rust Aug 17 '14 at 10:14
-
-
@Joseph put the basepoint inside the intersection of $U$ and $V$. The space is connected so this won't change the fundamental group. – Dan Rust Aug 17 '14 at 19:35
-
-
-
@DanRust: Your image link expired (I came here hoping to find a duplicate). – Lee Mosher Mar 23 '24 at 13:19
