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I was wondering if anyone visiting would be up for solving the following interesting little exercise out of Fulton's Algebraic Topology: A First Course.

Let $G$ a group act on a set $Y$. Say that $G$ acts evenly on $Y$ if any point in $Y$ has a nbhd. $V$ such that $g \cdot V = \{g \cdot y : y \in V\}$ (the image of $V$ under the group action) and $h \cdot V$ are disjoint for any distinct elements $g$, $h$ in $G$. Prove that the following actions are even:

(1) $\mathbb{Z}$ acts on $\mathbb{R}$ by translation: $n \cdot r = r+n$.

(2) $\mathbb{Z}$ acts on the right half plane (RHP) in $\mathbb{R^2}$ so that the polar coordinate covering map $$p: \{(r, \theta) \in \mathbb{R^2}: r>0\} \rightarrow \mathbb{R^2}\setminus \{(0,0)\}$$ given by $p(r,\theta) = (r \cos \theta, r \sin \theta),$ is the quotient of the RHP by this $\mathbb{Z}$-action.

(3) The cyclic group $G = \mu_n$ of $n$th roots of unity in $\mathbb{C}$ acts by multiplication on $S^1$ [regarded as the unit circle in the complex plane].

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up vote 4 down vote accepted

For (1): given $r\in\mathbb{R}$, let $V_r=(r-\frac{1}{2},r+\frac{1}{2})$. Then $$n\cdot V_r=(r+n-\tfrac{1}{2},r+n+\tfrac{1}{2}), \quad m\cdot V_r=(r+m-\tfrac{1}{2},r+m+\tfrac{1}{2})$$ intersect if and only if $r+m-\frac{1}{2}<r+n+\frac{1}{2}$ and $r+n-\frac{1}{2}<r+m+\frac{1}{2}$, i.e. $m-n<1$ and $n-m<1$, hence $|n-m|<1$. Since $n,m\in\mathbb{Z}$ this forces $n=m$.

I'm not sure I understand the statement of problem 2. Am I correct that the action is intended to be $n\cdot (r,\theta)=(r,\theta+2\pi n)$?

If so, then given $p=(r,\theta)\in\text{RHP}$, let $V_p=B_\delta(p)=\left\{(a,b)\in\text{RHP}\mid \sqrt{(b-\theta)^2+(a-r)^2} < \delta \right\}$ where $\delta=\min\{r,\pi\}$. For example, if $\pi\leq r$, then we get a picture like this:

enter image description here

the point $p=(r,\theta)$ is marked, and the circles represent $V_p$ and a few of its translates. If we had $r<\pi$, i.e. the point $p$ was less than $\pi$ away from the $y$-axis, we'd have to shrink the circles to have radius $r$ (or less) to ensure they don't leave our set, $\text{RHP}$.

For (3): Given $p=(\cos(\theta),\sin(\theta))\in S^1$, let $V_p=\{(\cos(\psi),\sin(\psi))\in S^1\mid \theta-\frac{2\pi}{n}<\psi<\theta+\frac{2\pi}{n}\}$. I'll let you show that $\zeta\cdot V_p$ and $\omega\cdot V_p$ are disjoint for any distinct $\zeta,\omega\in\mu_n$.

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Thanks for responding! I think the action you note for problem (2) is the right one (and I don't recall Fulton making the group action explicit when he discusses the polar coordinate map in Chapter 11, but I will check). Again thanks. – Vulcan Jan 31 '12 at 2:53
@Robert: No problem, glad to help! I've included a choice of $V_p$ for (2), assuming our interpretation of the problem is right. – Zev Chonoles Jan 31 '12 at 3:28

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