# Tagged Questions

A topological group is a group endowed with a topology such that the group operation and inversion are continuous maps. They are useful in various areas of mathematics. Every topological vector space is a topological group. Locally compact groups are important in harmonic analysis.

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### Is there a contractible space with a free circle action?

Question in title. Seems no to me (some vague intuition here about contracting orbits to a fixed point), but I can't prove it. I'd prefer to be wrong. (I'm curious because I am thinking about group ...
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### Is there a topology such that $(\Bbb R, +, \mathcal T)$ is a compact Hausdorff topological group?

I already know that this is impossible for $(\Bbb Q, +, \mathcal T)$ to be a compact Hausdorff topological group (notice that the trivial topology does not work because it is not Hausdorff). Indeed, ...
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### Reference request for "isomorphism upto compact kernel /cokernel “

Let $A$, $B$ be abelian topological groups with a map $f :A \to B$. Assume also that the kernel and cokernel of this map are compact. Then we call f an isomorphism upto compactness. Now let $A, B, C$...
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### An exact sequence of compact topological groups.

Let $A, B, C$ be abelian topological groups such that we have the following exact sequence : $$0\to A \to B \to C \to 0.$$ Assume also that A, C are compact and all the maps are open. Then it's it ...
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### About an article regarding free groups

I am currently reading this paper https://blms.oxfordjournals.org/content/35/5/624.abstract and I have some difficulties to understand two steps of the proof of the main theorem. Let $G$ be a non ...
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### Let $G$ a simple connected topological group and $H$ a normal discrete subgroup, then $\pi_1(G/H,e) = H.$

I know that $G$ is a covering space for $G/H$ and there is a injection between the fundamental group of $G$ and $G/H.$ How to proceed to show that $\pi_1(G/H,e) = H?$.
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### How theorem 3.1 is a consequence of lemmas 3.2-3.4?

In the article "Tight wavelet frames on local fields", the author states that "Theorem 3.1 is an easy consequence of lemmas 3.2-3.4". How?
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### Does every locally compact group $G$ have a nontrivial homomorphism into $\mathbb{R}$?

Does every locally compact group (second countable and Hausdorff) topological group $G$ that is not compact have a nontrivial continuous homomorphism into $\mathbb{R}$? Obviously for compact groups ...
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### Connectedness of punctured $G$

If I take a connected topological group $G$ and I remove the indentity from $G$, when I can say that $G-\lbrace 1 \rbrace$ is connected? Any suggestion or reference is appreciated.
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### Fundamental groups of path connected subspaces

Does every path connected subspace of $\mathbb{R}^2$ have a fundamental group the trivial or an infinity group? For example, for convex subspaces we know that, but if we take only path connected ...
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### Topology of bundle maps in Atiyah-Singer IV

I'm trying to read "The index of elliptic operators IV" by Atiyah and Singer, and I do not understand why the topology on $\mathrm{Diff}(X,E)$ on page 123 is constructed in such a peculiar way. Is ...
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### Lie group structures inside of Clifford algebras

I am reading a text by Jean Gallier on Clifford algebras, Pin and Spin groups. I have a problem with one little innocent-looking paragraph establishing Pin and Spin as Lie groups on the page 37. I don'...
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### Closed subgroup implies open?

If $H$ is a closed subgroup of a topological group, $H$ is also open?, I know that an open subgroup of a topological group is also closed, but the converse is true? if isn't, wich could be a ...
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### Show that $\operatorname{Sp}(n)=\{A \in M_n(\mathbb{H}) \mid AA^*=I=A^*A\}$ is a compact group

Let $M_n(\mathbb{H})$ be the set of all $n \times n$ matrices with entries in the quaternions $\mathbb{H}$. For $A=(a_{ij} )$ let $A^*=(a^*_{ij} )$ be the matrix with $a^*_{ij}=\bar{a}_{ij}$, ...
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### Let $G$ be a Lie group and $H$ be a closed subgroup of $G$. Show that if $H$ and homogeneous space $\frac{G}{H}$ are connected, then $G$ is connected.

Let $G$ ge a Lie group and $H$ be a closed subgroup of $G$. Show that if $H$ and homogenuous space $\frac{G}{H}$ are connected, then $G$ is connected. Remark: For this proof I use the following ...
### Proving that $B:=\{f(x)\in C[a,b]:f(a)=0\}$ is close set
Let $B$ be a group of all the continuous functions in the interval $[a,b]$ such that $f(a)=0$. Prove that $A$ is close group in the metric space $C[a,b]$ My attempt: Metric space $C[a,b]$ defined ...
For a set $S$ and some function $f: S \rightarrow S$ and $a\in S$, $f$ is continuous at $a$ under a topology $N$ if for all neighborhoods $N_1(f(a))$ there exists a neighborhood $N_2(a)$ such that \$f(...