19
votes
Accepted
Does $\Bbb Z[X]$ determine $X$?
The answer is no.
If $X$ is a simplicial set then $\mathbb Z[|X|]\cong|\mathbb Z[X]|$. That is, your topological abelian group freely generated by the realization of $X$ is homeomorphic to the ...
14
votes
Accepted
Are there extremally disconnected groups?
Yes: every locally compact ED group is discrete.
Indeed, for topological groups, ED passes to quotients and open subgroups.
Let by contradiction $G$ be a non-discrete ED locally compact group. Passing ...
11
votes
Accepted
Homotopic but not equivariantly homotopic maps
For any $G$-space the $G$-equivariant maps $[EG,X]_G$, also known as the homotopy fixed points $X^{hG}$, are a Borel homotopy invariant of $X$ meaning that it is an invariant of $G$-equivariant maps ...
9
votes
Groups with no (proper) closed subgroups?
The only discrete groups without any non-trivial proper closed subgroups are the cyclic groups of prime order. For the non-discrete case, we have a characterization of such subgroups.
Theorem: Let $G$ ...
9
votes
Homotopic but not equivariantly homotopic maps
$\newcommand{\RP}{\mathbb{RP}}$Connor Malin's answer is excellent. Derived from that, here is a small example: Let $G = C_2$, the group with two elements, let $X = S^1$ with antipodal action, and let $...
7
votes
Accepted
Extreme amenability of topological groups and invariant means
The action $G\curvearrowright\beta G$ is continuous iff $G$ is discrete, so for nondiscrete groups it is not true that $G$ is extremely amenable iff this action has a fixed point.
What one should look ...
7
votes
Accepted
Non-continuous group homomorphism from p-adic field to C*
Since $\mathbf C^\times$ is a divisible group, Zorn’s lemma tells us for every abelian group $G$ and subgroup $H$ that each group homomorphism $H\to \mathbf C^\times$ extends (somehow) to a group ...
6
votes
Accepted
Is every contractible homogeneous space of a connected Lie group homeomorphic to a Euclidean space?
The main references would be Theorem 1.2 of Sergey Antonyan's paper and Theorem 32.5 in Markus Stroppel's book.
Topological group $G$ is called almost connected if the factor group $G/G_0$ of $G$ ...
6
votes
Accepted
Topologies that turn the real numbers into a compact Hausdorff topological group
A compact abelian group $A$ is the same as the Pontryagin dual of a discrete abelian group $B$. The group $A$ is divisible ($nA=A$ for all $n\ge 1$) if and if $B$ is torsion-free, and $A$ is torsion-...
6
votes
Accepted
Does every (Abelian) Polish group have a nontrivial locally compact subgroup?
No. Here's a counterexample.
Every 3-adic integer $x\in\mathbb Z_3$ has a unique representation as $x=\sum_{n\geq 0} a_n 3^n$ with balanced ternary digits $a_n\in\{-1,0,1\}.$ Define $f(x)=\sum_{n\geq ...
5
votes
Accepted
$\operatorname{Spaces}/BG$ $\sim$ $\operatorname{Spaces}^G$ $\sim$ $??(\Omega G)$
If $A$ is a braided ∞-group, the delooping $\def\B{{\sf B}}\B A$ is an ∞-group.
Consider the ∞-category of spaces equipped with an action of the ∞-group $\B A$.
Since $\B Ω G≃G$, this ∞-category is ...
4
votes
Accepted
Examples of non-discrete, cocompact subgroups
You can find many such examples among groups acting on trees.
Let $T$ be a $k$-regular tree and let $G$ be the subgroup of $\operatorname{Aut}(T)$ of automorphisms stabilizing each of the 2 parts of ...
4
votes
Is this submonoid of the isometry group on $\Bbb Q_2$ closed to inverses?
Unfortunately I lack the reputation to comment, and the question is extremely hard to parse so I can’t be all that confident (silver lining: more space to ramble). To me it appears that if you fix up ...
4
votes
Does $\Bbb Z[X]$ determine $X$?
As folks have correctly pointed out, my previous positive answer was not right. (And I have deleted it.) And Simon Henry observes that one can't even recover a finite set $S$ from $\mathbb Z[S]$ in a ...
4
votes
Accepted
Is there a purely topological definition of $\text{Spin}(p,q)$?
Here is a copy of my answer at MathStackExchange.
Almost nobody does this. There is a discussion in the Wikipedia article on spinor groups but it has too many mistakes and too few references. The only ...
4
votes
Accepted
Classifying space of centralizer
$\newcommand{\S}{\mathcal S}$
The answer is yes, always. Let's compute $\Omega(map(BG,BH),Bf)$.
The idea is to realize the map $* \xrightarrow{Bf} map(BG,BH)$ as a map of the form $\Gamma(BG,-)$ where ...
4
votes
Accepted
In a topological group, is $G/A\to G/B$ a covering map if $A$ is open in $B$?
Let $C$ be the intersection of the conjugates $bAb^{-1}$. Assume the additional hypothesis that $C$ is open in $B$. This holds, for example, when $B$ is locally connected.
The set $B\backslash C$, ...
3
votes
Compact subgroups of a linear group over non-Archimedean local field
The group $G={\rm GL}(n,F)$ acts transitively on $\mathcal O$-lattices of $F^n$. The stabilizer of the standard lattice $L_o = {\mathcal O}^n$ is $K={\rm GL}({\mathcal O})$. It is an open compact ...
3
votes
Is every contractible homogeneous space of a connected Lie group homeomorphic to a Euclidean space?
I don't have an answer. Note that $H$ need not be compact.
For example, $G$ could be the universal covering of $SL_2\mathbb R$, and
$H$ could be a $1$-dimensional closed subgroup.
Here are some more ...
3
votes
Accepted
Does every locally compact group G contain a maximal open subgroup P which is a pro-Lie group?
After thinking about the problem for some time, I came up with a counter-example, so as YCor wrote in his comment, it is indeed true that there is not always a maximal open pro-Lie subgroup.
Let $A$ ...
3
votes
Accepted
Is every compact quasisimple group a Lie group?
The group $G/Z(G)$ being simple, is a Lie group by Peter-Weyl. Hence it is either finite or connected. If it is finite, $G$ has center of finite index, hence has a finite derived subgroup. Since $G$ ...
2
votes
Willis theory for discrete groups?
One situation where you can apply tdlc group theory in a nontrivial way to discrete groups is if the group $G$ that you are interested in has a commensurated subgroup $H$ (meaning, all conjugates of $...
2
votes
orbits in locally compact group
These are exactly
(a) all locally compact groups in which every element is torsion, and
(b) the circle group.
(a) All groups in which every element is torsion work. This involves many discrete groups (...
2
votes
Accepted
Open conjugacy classes in a second countable profinite group
Yes, there exist profinite groups $G$ with a conjugacy class of empty interior and consisting of elements of finite order, generating $G$ as an abstract group.
Let $H$ be a nonabelian finite simple ...
2
votes
Accepted
Uniqueness of left-invariant Borel probability measure on compact groups
Uniqueness does hold, as a direct consequence of Halmos's theorem that Haar measure is completion regular.
Here are pointers to relevant parts of Fremlin's book. "Completion regular" is ...
2
votes
Is there a purely topological definition of $\text{Spin}(p,q)$?
What about the complex point of view?
The complex analogue of $SO(p,q)$ (complex matrices preserving a nondegenerate bilinear form and having determinant $1$) depends only on $p+q$, since all complex ...
1
vote
Uniqueness of left-invariant Borel probability measure on compact groups
[This was too long to be a comment, but is basically just a recontextualization of the other answer.]
If you use Baire measures rather than Borel measures (which is probably the more natural setting ...
1
vote
Parametrization of topological algebraic objects
Please regard this as a tentative answer since I am not sure that it corresponds to the spirit of your query. Many of the situations that you use to illustrate your query can be coded by using the ...
1
vote
Accepted
Continuity of central character
Let $K\subset G$ be a compact open subgroup such that $V^K\ne0$. Now $Z$ acts on $V^K$ by the central character $\omega_\pi$, and the action is trivial on $Z\cap K$. Thus $\omega_\pi$ is trivial on ...
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