Questions tagged [topological-groups]
A topological group is a group $G$ together with a topology on the elements of $G$ such that the group operation and group inverse function are both continuous (with respect to the topology).
756 questions
67
votes
1
answer
7k
views
Why can't a nonabelian group be 75% abelian?
This question asks for intuition, not a proof.
An earlier question,
Measures of non-abelian-ness
was thoroughly answered by Arturo Magidin.
A paper by Gustafson1
proves that, for a nonabelian group,
...
- 151k
53
votes
6
answers
6k
views
Does homeomorphic and isomorphic always imply homeomorphically isomorphic?
Let $(G,\cdot,T)$ and $(H,\star,S)$ be topological groups such that
$(G,T)$ is homeomorphic to $(H,S)$ and $(G,\cdot)$ is isomorphic to $(H,\star)$.
Does it follow that $(G,\cdot,T)$ and $(H,\star,S)...
user5810
51
votes
5
answers
9k
views
Fundamental group as topological group
Background
Let $(X,x)$ be a pointed topological space. Then the fundamental group $\pi_1(X,x)$ becomes a topological space: Endow the set of maps $S^1 \to X$ with the compact-open topology, endow the ...
- 63.1k
43
votes
7
answers
8k
views
Why are free groups residually finite?
Why is it that every nontrivial word in a free group (it's easy to reduce to the case of, say, two generators) has a nontrivial image in some finite group? Equivalently, why is the natural map from a ...
- 2,356
36
votes
4
answers
5k
views
Compact open topology on $\mathrm{Homeo}(X)$
Let $X$ and $Y$ be topological spaces. Define the compact open topology on the set $\mathrm{M}(X,Y)$ of continuous maps from $X$ to $Y$ via the subbase $[K,O]$ of all maps $f:X\rightarrow Y$ s.t. $f(K)...
- 2,751
32
votes
3
answers
3k
views
morphism from a compact group to Z ?
I wonder if it there exists a topological compact group $G$ (by compact, I mean Hausdorff and quasi-compact) and a non-zero group morphism
$\phi : G \to \mathbb{Z}$ (without assuming any topological ...
- 343
32
votes
1
answer
2k
views
A group allowing exactly 7 group topologies
Is there a group $G$ allowing exactly 7 group topologies on $G$: $\mathcal T_{\text{trivial}}, \mathcal T_{\text{discrete}}, \mathcal T_1, \mathcal T_2,\mathcal T_3,\mathcal T_4, \mathcal T_5$ with
$$...
- 1,145
31
votes
2
answers
3k
views
Is a normed space which is homeomorphic to a Banach space complete?
I have a normed space $(E,||\cdot||)$ which is homeomorphic (as a topological space) to a Banach space $F$.
Does this imply that $(E,||\cdot||)$ is also a Banach space?
I think I read something ...
- 495
30
votes
2
answers
2k
views
Is every connected subgroup of a Euclidean space closed?
The question listed above (in the context of the complex numbers, but it is a reasonable question to ask in any dimension) was asked by a student in my complex analysis class, and I did not have an ...
- 114k
27
votes
1
answer
840
views
Can closed compacts in a topological group behave "paradoxically" with respect to unions, intersections, and one-sided translations?
Consider two closed compacts $A$ and $B$ in a topological group $\Gamma$. Let $A'$ be a left translation of $A$ and $B'$ a left translation of $B$:
$A' = aA$,
$B' = bB$.
Suppose it is known that $A'\...
- 1,481
24
votes
4
answers
7k
views
Compact open topology
What is the intuition behind using compact open topology for eg. in the case of Pontryagin dual ?
- 1,209
24
votes
1
answer
2k
views
Which principlal bundles are locally trivial?
If $H$ is a closed subgroup of a topological group $G$, then the orbit map $G\to G/H$ is a principal bundle, yet somewhat surprisingly, it need not be locally trivial.
In the wikipedia article on ...
- 29.1k
23
votes
5
answers
4k
views
Fundamental groups of topological groups.
Let $G$ be a topological group, and $\pi_1(G,e)$ its fundamental group at the identity. If $G$ is the trivial group then $G \cong \pi_1(G,e)$ as abstract groups. My question is:
If $G$ is a non-...
- 273
23
votes
0
answers
1k
views
Do all possible trees arise as orbit trees of some permutation groups?
I.Motivation from descriptive set theory
(Contains some quotes from Maciej Malicki's paper.)
The classical theorem of Birkhoff-Kakutani implies that every metrizable topological group G admits a ...
- 181
21
votes
3
answers
1k
views
Is a reductive adelic group a Type I group?
I foresee that to experts of automorphic forms this question will sound unimportant or useless or even not worthy of an answer; but none of these are going to stop me from asking it!
The question is ...
- 955
21
votes
0
answers
732
views
Closed connected additive subgroups of the Hilbert space
It is a classical result that a closed and connected additive subgroup of $\mathbb{R}^n$ is necessarily a linear subspace. However, this is no longer true in infinite dimension: a very easy example is ...
- 60.5k
20
votes
4
answers
3k
views
Find a "natural" group that contains the quotient of the infinite symmetric group by the alternating subgroup
Let $S_\infty$ the group of permutations of $\mathbb{N}$. It can be shown that there is no homomorphism $S_\infty \to \mathbf{Z}/2$ extending the sign on the finite symmetric groups. Is it possible to ...
- 63.1k
20
votes
2
answers
2k
views
Is every compact topological ring a profinite ring?
There are a lot of compact (Hausdorff) groups, whereas every compact field is finite. What about rings? Is there a classification theorem for compact rings? If you take a cofiltered limit of finite ...
- 2,200
20
votes
2
answers
2k
views
Is every topological (resp. Lie-) group the isometrygroup of a metric space (resp. Riemannian manifold)?
The isometry group of a metric space is a topological group (with the compact open topology). The isometry group of a Riemann Manifold is a Liegroup. (Thm. of Steenrod-Myers)
So, is every topological ...
- 2,974
20
votes
0
answers
333
views
Existence of orthonormal basis for $L^2(G)$ in $C_c(G)$
Remark: I cross-posted this question on MSE and added a bounty to it.
Suppose that $G$ is a locally compact (Hausdorff) group endowed with the Haar measure. It is well-known that the compactly ...
- 321
19
votes
9
answers
6k
views
Haar measure on a quotient, References for
I remember reading Weil's "Basic Number Theory" and giving up after a while. Now I find myself thinking of it (thanks to some comments by Ben Linowitz).
Right from the very beginning, Weil ...
- 7,442
19
votes
2
answers
2k
views
Without choice, can every homomorphism from a profinite group to a finite group be continuous?
In ZFC, some homomorphisms from profinite groups to finite groups are discontinuous. For instance, see the examples in this question. However, all three constructions given use consequences of the ...
- 148k
19
votes
2
answers
566
views
Ostrowski's Theorem for topological rings?
Ostrowski's theorem classifies all absolute values on a number field $K$.
Questions:
More generally, can one classify all Hausdorff topologies on $K$ making $K$ into a topological field?
In ...
- 63.9k
19
votes
2
answers
1k
views
$2$-categorical structure in Grothendieck's Galois Theory
Grothendieck's Galois Theory, as developed in SGA I, V.4, or very gently in Lenstra's notes, establishes an equivalence between profinite groups and Galois categories. We can put this into the ...
- 63.1k
19
votes
0
answers
703
views
The cofinality of $(\mathbb{N}^\kappa,\le)$ for uncountable $\kappa$?
For a partially ordered set $P$, a set $A\subseteq P$ is cofinal if for each element of $P$ there is a larger element in $A$. The cofinality of $P$, ${\rm cof}(P)$, is the minimal cardinality of a ...
- 3,104
18
votes
2
answers
1k
views
Anything between vector bundles and sphere bundles?
There are two extremities: on the "easy end" one has vector bundles which are classified by maps to the (more or less) well understood spaces like Grassmanians; on the "hard end" there are spherical ...
- 18.4k
18
votes
1
answer
1k
views
Are there non-reflexive abelian topological groups isomorphic to their second dual?
I posted the following question in a comment at
Are there non-reflexive vector spaces isomorphic to their bi-dual? and it got one upvote, but it didn't get an answer, so I'll post it as an ...
- 50.6k
18
votes
0
answers
373
views
Can Rep(G) tell us whether G is discrete?
Given a locally compact group $G$, let $$\mathrm{Rep}(G)$$ be its category of unitary representations.
The objects of that category are strongly continuous unitary representations of $G$ on Hilbert ...
- 43.2k
17
votes
6
answers
6k
views
What is a good book on topological groups?
I am looking for a good book on Topological Groups. I have read Pontryagin myself, and I looked some other in the library but they all seem to go in length into some esoteric topics.
I would love ...
17
votes
1
answer
1k
views
Does ZF prove that topological groups are completely regular?
Let $\mathbf{G} = \langle G,\cdot,\mathcal{T}\;\rangle$ be a topological group. Let $\mathbf{e}$ be the identity element of $\langle G,\cdot \rangle$.
Assume $\{\mathbf{e}\}$ is closed in $\langle G,\...
user5810
17
votes
0
answers
1k
views
What groups are Lie groups?
We know how to tell if a topological group is a Lie group: this was famously asked by Hilbert and answered gloriously by Gleason, Montgomery and Zippin in the 50s (a locally compact topological group ...
- 47.3k
16
votes
1
answer
481
views
Where can I learn more about the topology on $\mathbb{R}$ induced by the map $\mathbb{R} \to \prod_{a>0} (\mathbb{R}/a\mathbb{Z})$?
Consider the (continuous, injective, abelian group homomorphism) map $\Phi \colon \mathbb{R} \to \prod_{a>0} (\mathbb{R}/a\mathbb{Z})$ (where the target is given the product topology) taking $x\in \...
- 32.5k
16
votes
2
answers
1k
views
rationalization of classifying spaces
This question is probably trivial for anyone who is more familiar with rational homotopy theory than me, but anyway:
Let $G$ be a simply-connected topological group. In particular, it is an $H$-...
- 7,637
16
votes
1
answer
1k
views
A possible mistake in Walter Rudin, "Fourier analysis on groups"
I have the following lemma 4.2.4 on page 80 in the book (we have locally compact abelian topological groups $G_1, G_2$ and their duals $\Gamma_1, \Gamma_2$):
Suppose $E$ is a coset in $\Gamma_2$ ...
- 556
15
votes
3
answers
3k
views
Are measurable homomorphisms $ (\Bbb{C},+) \to (\Bbb{C},+) $ or $ (\Bbb{C},+) \to (\Bbb{C},*) $ continuous, and do they admit an explicit description?
I am interested in generalizations of the following fact (known as automatic continuity, as pointed out below). I am especially looking for references to papers dating back to 1920’s. I feel that ...
- 1,299
15
votes
2
answers
1k
views
Does $\Bbb Z[X]$ determine $X$?
For a Hausdorff space $X$, consider the free abelian group $\mathbb{Z}[X]$ generated by $X$. Equip it with the finest topology which makes the map $X\to\mathbb{Z}[X]$, $x\mapsto [x]$ continuous and ...
user473423
15
votes
3
answers
3k
views
Why is the dual of a torus the same as its fundamental group?
The set of continuous homomorphisms from a torus ${\mathbb T}^n = ({\mathbb R}/{\mathbb Z})^n \to {\mathbb R}/{\mathbb Z}$ can be identified with ${\mathbb Z}^n$ if we assign to each $k = (k_1, \ldots ...
- 2,243
15
votes
1
answer
986
views
Is a left topological group which is a manifold a topological group?
Let $G$ be a left topological group, i.e. a topological space with group operation such that left multiplication $L_g : x \mapsto gx$ is continuous (but right multiplication and inversion are not ...
- 1,269
15
votes
1
answer
498
views
For what LCH groups is the Haar measure $\mu(U x U)$ bounded?
Let $G$ be a locally compact Hausdorff (LCH) topological group with left Haar measure $\mu$. Given a compact unit neighborhood $U$, consider the function
$$
\Phi: \quad G \to (0,\infty), \quad x \...
- 734
14
votes
3
answers
1k
views
Countable subgroups of compact groups
What is known about countable subgroups of compact groups? More precisely, what countable groups can be embedded into compact groups (I mean just an injective homomorphism, I don't consider any ...
- 1,318
14
votes
3
answers
1k
views
Examples of locally compact groups that do not admit enough finite dimensional representations
I apologize in advance if this is well-known, but I can't seem to find the answer in the literature. Let me be precise about my question. I am looking for concrete examples of locally compact ...
- 1,586
14
votes
2
answers
1k
views
Baire Category Theorem for complete uniform spaces
The version of the Baire Category Theorem I have in mind is the statement that a countable intersection of dense open subsets of a complete metric space is dense. The question is: is it likewise ...
- 1,270
14
votes
1
answer
940
views
Contractible topological groups
Does there exist a Hausdorff topological group which is contractible and of finite covering dimension but which is not homeomorphic to $\mathbb{R}^n$ for some $n$?
- 141
14
votes
1
answer
1k
views
Distributivity of group topologies on $\Bbb Z$
Let $\mathcal L$ be the set of all group topologies on $\Bbb Z$.
It is known that $(\mathcal L,\subseteq)$ is a modular complete lattice [1].
Is $(\mathcal L,\subseteq)$ distributive?
$$~$$
[1] ...
- 1,145
14
votes
1
answer
295
views
Is $Alt_\omega$ a dense subgroup of a non-discrete locally compact topological group?
Let $S_\omega$ be the group of bijections of the countable ordinal $\omega:=\{0,1,2,\dots\}$ and $Alt_\omega$ be the subgroup of $S_\omega$ consisting of even permutations of $\omega$ (i.e., the ...
- 41.8k
14
votes
1
answer
581
views
How flexible is the infinite-dimensional torus?
Let $\mathbb T=\mathbb R/\mathbb Z$ be the circle group and $\mathbb T^\omega$ be the infinite-dimensional torus, considered as an abelian compact topological group.
Problem 1. Is it true that for ...
- 41.8k
14
votes
1
answer
452
views
In what topological abelian groups does convergence to zero imply summability?
(This question has been on math.SE for over a week and has not gotten any answers.)
Let $G$ be a (T$_0$) topological abelian group, and let $0$ be its identity element.
Assume that for all index ...
user5810
14
votes
0
answers
255
views
Is the group $\operatorname{Diff}^1_0(\mathbb R^d)$ connected?
Is the group
$$ \operatorname{Diff}^1_0(\mathbb R^d) = \operatorname{Diff}^1(\mathbb R^d) \cap \big(\operatorname{Id}_{\mathbb R^d} + C^1_0(\mathbb R^d,\mathbb R^d)\big) $$
connected? Here
$$ C^1_0(\...
- 570
13
votes
2
answers
514
views
subsets of groups which have to be closed no matter what
One example of a subset of a group $G$ which has to be closed in any topology on $G$ compatible with the group operations is a centraliser. Are there any other interesting examples?
- 2,125
13
votes
0
answers
421
views
A meager subgroup of the real line, which cannot be covered by countably many closed subsets of measure zero?
Is there a ZFC-example of a subgroup $H$ of the real line $\mathbb R$ such $H$ is meager, has zero Lebesgue measure, but cannot be covered by countably many closed subsets of measure zero in $\mathbb ...
- 41.8k