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Let $Circ$ be the topological group

$(\{z\in \mathbb{C} : \overline{z}\cdot z = 1\},\cdot , \{U\in 2^{\{z\in \mathbb{C} : \; \overline{z}\cdot z \, = \, 1\}} : \{z\in \mathbb{C} : \overline{z}\cdot z = 1\}-U $ is closed in $ \mathbb{C}\})$.


Let $(G,\star,T)$ be a topological group such that $(G,T)$ is homeomorphic to $(\{z\in \mathbb{C} : \overline{z}\cdot z = 1\}, \{U\in 2^{\{z\in \mathbb{C} : \; \overline{z}\cdot z \, = \, 1\}} : \{z\in \mathbb{C} : \overline{z}\cdot z = 1\}-U $ is closed in $ \mathbb{C}\})$.
Does it follow that

  1. $(G,\star)$ is isomorphic to $(\{z\in \mathbb{C} : \overline{z}\cdot z = 1\},\cdot)$?
  2. $(G,\star,T)$ is homeomorphically isomorphic to $Circ$?
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19  
What a complicated way of saying "with its usual topology"! –  Mariano Suárez-Alvarez Oct 28 '10 at 5:06
3  
-1 for the complicated formulation. –  Qfwfq Oct 28 '10 at 9:34
    
Note that it's not true for $Circ \times Z_2$, which has three group structures, as pointed out to me by Greg Kuperberg. –  Allen Knutson Oct 28 '10 at 18:28
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2 Answers

up vote 14 down vote accepted

It is a theorem of Scheinberg that two compact connected abelian groups are topologically isomorphic whenever they are homeomorphic.

Now, by the solution of Hilbert's 5th problem, a topological group $G$ homeomorphic to $S^1$ is a Lie group, which is necessarily of dimension $1$. It follows that $G$ is abelian. Afirmative answers to your two questions follow.

More generally, since A compact solvable Lie group is abelian, the same reasoning shows that a topological group homeomorphic to a 2-torus is isomorphic to a 2-torus

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Hilbert's 5th is a heavy stuff: there is an elementary proof that takes a fraction of time it would cost to do Hilbert's 5th... I will have a go. –  Bugs Bunny Oct 28 '10 at 9:00
    
OK, so what is an example of two locally compact (not discrete) groups that are homeomorphic, but not isomorphic as topological groups? By this, they are not compact connected abelian; but can they have two of those three properties? –  Gerald Edgar Oct 28 '10 at 18:13
    
@Gerald: a countable product of copies of the discrete group $\mathbb Z_2$ is homemorphic to a countable product of copies of the discrete group $\mathbb Z_3$, but they are not isomorphic as groups. So the groups can be compact, non-discrete and abelian. –  Mariano Suárez-Alvarez Oct 28 '10 at 21:42
    
If now you do direct productsof those two groups with a discrete $\mathbb Z$, I thing you have an example of what you want. –  Mariano Suárez-Alvarez Oct 28 '10 at 22:25
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Here is also a direct argument (which only works in dimension $1$): The universal cover of $G$ is a group homeomorphic to ${\mathbb R}$. Now, it is enough to show that the only topological group homeomorphic to ${\mathbb R}$ is the additive group of reals. To show this, first note that each right or left translation $$ \lambda_g: x \mapsto g \cdot x , \quad \rho_g: x \mapsto x \cdot g $$ is continuous and bijective, hence it has to be monotone. Now, the ones that are increasing form a subgroup of index at most $2$, and since ${\mathbb R}$ is connected, it follows that every left or right translation is strictly increasing. Also $\lambda_g$ has no fixed point, which implies that either $\lambda_g(x)>x$ or $\lambda_g (x)<x $ for all $x \in R$. Now, this allows you to define a total order on $G$: [ f

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