Tagged Questions

2
votes
1answer
34 views

Function spaces and transitive group actions

Note: this question is really a subquestion of this one, but I decided to ask it separately since it seems it might be attacked first. Let $B$ be a topological space and $G$ a topological group ...
3
votes
1answer
60 views

Infinite products of a (finite) group

So I'm having a little trouble understanding the concept of infinite (cartesian) products of a group -- specifically, my notes (and, of course, homework questions) have concepts of, say ...
4
votes
1answer
22 views

Proving that the Flag Variety $Fl(n;m_1,m_2)$ is connected.

I wish to prove that the flag variety $Fl(n;m_1,m_2) = \{ W_1 \subset W_2 \subset V | dimW_i = m_i \}$, for $0 \le m_1 \le m_2 \le n$ where V is an n-dimensional vector space over $\mathbb{C}$ and ...
1
vote
1answer
39 views

All the compact covering spaces of torus.

I know the covering spaces of the of a torus $T^2$ are homeomorphic to $T^2,S^1\times\mathbb{R},\mathbb{R}^2$. I am interested in finding all of the covers with covering space $T^2$. The subgroups of ...
1
vote
1answer
47 views

An example for a non-precompact minimal topological group.

Do you have an example of a non-precompact minimal topological group? A topological group $(G,\mathcal T)$ is said to be minimal iff it is Hausdorff and for any compatible Hausdorff topology ...
3
votes
2answers
49 views

Group, metric, completion

Let $G$ be a group, $(G, \rho)$ - metric space, $p: G \rightarrow \mathbb{R}_+$ such that $p(x)=0 \iff x=e_G, \ \ p(x^{-1})=p(x), \ \ p(xy)\le p(x)+p(y), \ \ p(xy)=p(yx)$ Now let $\rho ...
3
votes
1answer
83 views

Proof that $U(n)$ is connected

I'm trying to prove that $U(n)= \{ X\in Mat_n(\mathbb{C})|X^T\bar{X}=I\}$ is connected, but most of the proof comes down to proving that $SU(n)= \{ X\in Mat_n(\mathbb{C})|X^T\bar{X}=I $ and $ ...
9
votes
2answers
115 views

Representation theorems for groups

There are two baffling representation theorems for groups: Every group is isomorphic to the automorphism group of some graph. (see Frucht's theorem) Every group is isomorphic to the fundamental ...
3
votes
3answers
129 views

what are all the open subgroups of $(\mathbb{R},+)$

I am not able to find out what are all the open subgroups of $(\mathbb{R},+)$, open as a set in usual topology and also subgroup.
3
votes
1answer
49 views

Continuity of the canonical map

I have canonical map $\phi$, I think that it is continuous, but I can not prove it. $$\phi:G/G^x\longrightarrow G(x)$$ $G$ is tological group. $G^x=\{g\in G\mid gx=x\}$ (Isotropy Group). ...
4
votes
1answer
76 views

Action of a subgroup of finite index on a tree induced by an action of a group on a tree

Let $G$ be a group wich acts on a tree $\Gamma$. Then $U$ acts on $\Gamma$ for every $U\leq G$. Question: Why does the following hold? If $|G:U|<\infty$. Then the minimal $U$-invariant subtree ...
2
votes
1answer
132 views

proof by nuke of the fact that fundamental group of topological group is abelian

"The fundamental group of a topological group is abelian". does this problem admit a proof by nuke. This is inspired by a a question in mathoverflow. The usual proof is by a Eckmann-Hilton ...
0
votes
0answers
45 views

subgroups of matrix lie group

please just confirm me, for 1.3 (a) is wrong as inverse doesnt exists, for (b) and (c) they forms subgroups as they are invertible. for 1.4 (a) is $SL_n(\mathbb{C})$ is normal clearly, set of all ...
2
votes
1answer
82 views

discrete subgroup of locally compact abelian group

Let $G$ be a locally compact abelian infinite group but non-compact. In some paper, the author claims that the dual group $\widehat{G}$ contains an infinite discrete group $K$. What do you think ...
1
vote
2answers
44 views

clopen subgroup of a topological group

could any one just give hint for this one? $G$ be a topological group such that $\forall x\in G, x\mapsto xy$ Homeomorphism, $H$ is a open subgroup of $G$, we need to prove $H$ is also closed
0
votes
0answers
28 views

How to relate $S^1$ (SO(2)) to a stretched version of $S^1$ ($SO(2) \subset SO(3)$)?

I want to decompose the quaternion parameterization of $SO\left(3\right)$ into the subspace of $SO\left(2\right)$ defined by choosing the z-axis as the rotation axis, and find a relationship between ...
8
votes
2answers
251 views

Why isn't there interest in nontrivial, nondiscrete topologies on finite groups?

A topology on a group is required to be compatible with the group structure (multiplication must be a continuous map $G\times G\to G$ and inversion must be continuous). I've only ever seen the ...
4
votes
1answer
90 views

Start with a topological group, take the meet of the two uniformities, and take the topology. Is the result again a topological group?

And what else can be said, if so? In more detail: Say $(G,\mathscr{T})$ is a topological group. It has a left uniformity $\mathscr{L}$ and a right uniformity $\mathscr{R}$. (It also has a two-sided ...
7
votes
2answers
363 views

Normal subgroups of free groups: finitely generated $\implies$ finite index.

I am looking at what should be a simple exercise in geometric group theory. I have reduced the problem to just completing an exercise from Hatcher, Section 1.B page 87: 7. If $F$ is a finitely ...
4
votes
2answers
231 views

Why metrizable group requires continuity of inverse?

A metrizable group is a metric space $(G,d)$ with a binary operation $\cdot$ such $(G,(\cdot))$ is a group and maps $(\cdot):G\times G\to G$ and $f:G\to G$ given by $(\cdot)(x,y)=xy$ and $f(x)=x^{-1}$ ...
2
votes
2answers
59 views

what does linear type mean?

What does it mean when we say a topological group $\Gamma$ has linear type? Is it an algebraic property or a topology property? I wonder if anyone could give some references.
2
votes
0answers
41 views

JSJ-decompositions of groups and 3-manifolds: a reference request

I am, for whatever reason, interested in learning about the JSJ-decomposition of groups. Having asked around a bit, it was suggested I first learn about what is happening in the manifolds and then ...
3
votes
1answer
150 views

Is it true that fundamental group of a manifold with boundary can't be simple? If so, why?

Is it true that fundamental group of a manifold with boundary cannot be simple? I think I read that during a hurried research run through the basement of Geisel library, but didn't mark the source. ...
5
votes
3answers
99 views

How much does $\operatorname{Aut}(H_1(S))$ determine a homeomorphism $S \to S$?

Let $S$ be an orientable compact surface. A homeomorphism $f: S \to S$ induces an isomorphism $f_{*}: H_1(S) \to H_1(S)$. How much can we say the converse? Namely, if we are given an element of ...
2
votes
1answer
130 views

Yet another question on Group actions and $G$-coverings!

I was wondering if anyone visiting could help me figure out how to prove the following exercises from Ch.11 of Fulton's Algebraic Topology: A First Course. (1) Show that any two-sheeted covering has ...
0
votes
0answers
123 views

A question connecting group actions and the identification of the resulting quotient spaces

I was wondering if anyone visiting would be up for walking through/solving the following related exercises from Ch.11 of Fulton's Algebraic Topology: A First Course. Let $H = Homeo(\mathbb{R^2}, ...
7
votes
1answer
186 views

Is a topological group action continuous if and only if all the stabilizers are open?

Let $G$ be a topological group and $(X,\mu)$ be a $G$-set, i.e. $\mu$ defines an action $X \times G \rightarrow X$. Is it then true that $\mu$ is continuous if and only if for every $x \in X$ the ...
12
votes
2answers
340 views

$(x,y)\to xy$ continuous but $x\to x^{-1}$ not

In the definition of topological groups we impose both $(x,y)\to xy$ and $x\to x^{-1}$ to be continuous. However, I cannot find an example where the first condition holds but the second fails. Is ...
0
votes
1answer
44 views

Finding a homeomorphic image of a subset defined by a group action

let $X$ be a top space and $G$ a group acting on $X$. Consider $$ F=\bigl\{(x,gx)\in X\times X\mid x\in X, g\in G\bigr\}\ $$ i want to write an homeomorphic image of $F$. for example take $G=\mathbb ...
3
votes
1answer
352 views

Are orbits equal if and only if stabilizers are conjugate?

Let $X$ be a topological space and $G$ a group acting on $X$. Do we have this property: $$\operatorname{orb}(x)=\operatorname{orb}(y)\iff\operatorname{stab}(x)\sim \operatorname{stab}(y)\qquad ?$$ ...
3
votes
0answers
43 views

expression of a $G$-space in terms of stablizer subgroups

Given a topological space $X$ and a group $G$ acting on $X$, define $$F=\{(x,y)\in X\times X \mid \operatorname{orb}(x)\cap \operatorname{orb}(y)=\emptyset\}.$$ I want to write $F$ in terms of ...
7
votes
1answer
131 views

“indecomposability” in theory of groups and topological spaces

One can define the notion of "indecomposable" in many of the categories that mathematicians think about. However there's no real reason, as far as I can see, to expect it to behave at all well. Here ...
3
votes
2answers
126 views

The fundamental group of $K_{3,3}$ — relationship between its generators and embedding into manifolds

So I've been reading this wonderful PDF textbook on algebraic topology: http://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf In particular, I'm very interested in the chapter on graphs. This ...
0
votes
1answer
137 views

writing a subgroup of the affine group as a semidirect product

I read the following: The group $\mathbb Z +...+\mathbb Z=\mathbb Z^n$ of covering translations and $S_n$ acting on $\mathbb R^n$ by permuting coordinates, both lie in $aff(\mathbb R^n)$ the group ...
1
vote
1answer
130 views

induced action on quotient space

Let $X$ be a topological space on which a group $G$ acts . let $N$ and $K$ be subgroups of $G$. under what condition we have an induced action of $K$ on $X/N$? My guess: if $N$ is normalized by ...
1
vote
1answer
251 views

Why the discrete subgroup of $R^n$ has finite balls?

Consider a discrete subgroup $G$ of $R^n$. Then let we have an open ball $B\subset G$ (topology in $G$ is induced from $R^n$). Why is $|B|<\infty$? For example I can take ...
2
votes
0answers
126 views

Property (T) for groups vs top

I have encountered two properties in different areas of math. One is the property (T) of groups and the other is the property (T) of topologies. What is the connection between these two ? Thank you. ...
2
votes
1answer
148 views

topologies on $C(X,Y)$

Let $X,Y$ be two topological spaces. One can define various topologies on $C(X,Y),$ the space of continuous functions from $X$ to $Y.$ Let us take the group $G=Hom(X),$ the homeomorphisms of $X.$ Let ...
0
votes
1answer
119 views

injectivity of a group action

The action of a group $G$ on $X$ is always "injective" in the following sense: if $x\not = y$ then $\forall g\in G$, $gx\not = gy$ indeed if $gx=gy$ then ...
1
vote
1answer
146 views

induced homomorphism from a group action

let $X$ be a topological space on which a group $G$ acts. 1) is it true that this action always induces an homomorphism $G\rightarrow Aut(X)$? My guess is no. because i think the induced ...
2
votes
1answer
95 views

Neighborhood of a quotient by the symmetric group

Let $X$ be a topological space and $S_3$ the symmetric group acting on $X^3$ by permuting coordinates. Let $\pi:X^3\rightarrow X^3/S_3$. Denote $[x,y,z]=\pi(x,y,z)$. Let $U_x$ be the neighborhood of ...
3
votes
1answer
126 views

Image of a neighborhood by a quotient map

let $X$ be a topological space. let $G$ be a group acting on $X$. let $x\in X$ with neighborhood $U_x$. let $\pi:X\rightarrow X/G$ the quotient map, we denote $[x]=\pi(x)$ 1) is $\pi(U_x)$ a ...
1
vote
1answer
70 views

quotient of a hyperplane by the action of cyclic group

let $H=\{(x,y,-x-y)\in \mathbb C^3\}$ and let $S^3$ the unit sphere in $H$. Why the following is true : The linear action of $\mathbb Z_3$ on $S^3$ is free and $H/\mathbb Z_3=C(M)$ the cone on ...
1
vote
0answers
128 views

how to factor a map by a group action

Let $X$ and $Y$ be topological spaces and a surjective map $f:X\rightarrow Y $. Suppose that a group $G$ acts on $X$. and let $\pi:X\rightarrow X/G$ be the quotient map. 1) Under what conditions $f$ ...
2
votes
2answers
128 views

$X\!\supseteq\!K\!\simeq\!0\Rightarrow X\!\simeq\!X/K$ ($\pi_1$ of a connected graph is free)

How can I prove the following: If $X\supseteq K$ is contractible, then the quotient $X/K$ is homotopy equivalent to $X$? Since $K$ is contractible, we have a homotopy $H:id_K\!\simeq\!c_{k_0}$ ...
2
votes
1answer
170 views

Compact group actions and automatic properness

I am currently re-reading a course on basic algebraic topology, and I am focussing on the parts that I feel I had very little understanding of. There is one exercise in the chapter devoted to groups ...
7
votes
1answer
239 views

A torsion-free quotient?

Let $G$ be a group and $T$ the set of elements of finite order in $G$. If $T$ is a subgroup of $G$, then $G/T$ is a torsion-free group. Suppose $G$ is a compact Hausdorff topological group. Is it ...
1
vote
1answer
130 views

Relationship between topology of a compact group and the topology of its profinite completion

Suppose $G$ is a compact topological group. We can construct the profinite completion of $G$; let's call this $\Gamma$. My questions are: 1) Assuming that we know nothing about the (original) ...
4
votes
5answers
592 views

Why is fundamental group insufficient to classify manifolds?

I have heard that some issues in group theory prevent classifying all manifolds upto homotopy using the fundamental group invariant. Does anyone know what are those issues? Thanks, K.