3
votes
0answers
37 views

Fundamental group of 7-gon with labelling scheme $abaaab^{-1}a^{-1}$

I want to calculate the fundamental group of of a $7$-gon with labelling scheme $abaaab^{-1}a^{-1}$. I will call the quotient space $X$ with reference point $x_0$. This is what I tried: If you have ...
0
votes
2answers
44 views

How to prove the existence of this homomorphism $G_1*G_2\to H$?

Say there is a homomorphism from a group $G_1$ to another group $H$, and there is a homomorphism from $G_2$ to $H$, then there is a homomorphism from $G_1*G_2$ to $H$ induced by the former to ...
1
vote
1answer
47 views

Are the two inverses in the free group same?

Set G is a group, and the set C is contained in G. Set C contains only 2 elements a and b such that they are the inverse of each other.Set F(C) is the free group made by C. Then in F(C), it contains ...
1
vote
0answers
32 views

Geometric definition of the stable commutator length

In his book, D.Calegari proves the equivalence of the algebraic and geometric definitions of stable commutator length (Proposition 2.10, p. 15). I actually have some difficulties in understanding the ...
0
votes
0answers
66 views

Canonical topology on standard groups?

I just wanted to know whether there is any standard topology on groups like $\mathbb{Z}/n\mathbb{Z}$ or $\mathbb{Z}$ ? - The only one that I could imagine, especially for finite groups is the discrete ...
0
votes
1answer
22 views

Isotropy groups of tetrahedron after identifying its sides

If we identify the 4 sides of a regular tetrahedron in $\mathbb{R}^3$ by letting the group of all isometries of the tetrahedron act on it, what would the resulting space look like? The resulting ...
2
votes
2answers
63 views

A comment on a proof of equivalent knots

The following theorem and its proof is from A First Course in Algebraic Topology By Czes Kosniowski pp. 219-220 ...
1
vote
1answer
71 views

Period of a particular finite group

Let $G$ be a group fitting in the following exact sequence: $0 \to \mathbb{Z}/p \to G \to \mathbb{Z}/q^r \to 0.$ Here $q$ and $p$ are primes (not necessarily distinct). It is easy to check (by the ...
6
votes
1answer
95 views

Airy differential equation and Galois group

Consider the Airy equation $y^{(2)}=ry$ where $r \in \Bbb{C}(z)$ but not constant. How do you show that $G^0=G$, where $G$ is the galois group of the picard vessiot extension of solutions over ...
-1
votes
1answer
35 views

Associative property of free product of groups

I am reading Algebraic Topology by Allen Hatcher (available online at http://www.math.cornell.edu/~hatcher/AT/AT.pdf) and at line 1 of page 42, it reads: "... because of the relation ...
1
vote
0answers
62 views

Why is this a group action?

Let $G$ be a group and let $H$ be an infinite cyclic normal subgroup of $G$ of finite index. Let $K$ be the centralizer of $H$ in G, $$K=C_G(H)$$ and suppose that the index of $K$ in $G$ is 2. Let $E$ ...
3
votes
1answer
61 views

What is the maximal torus in the Lorentz group $O(m,n)$?

I'm close to certain it's just the product of the maximal tori of $O(m)$ and $O(n)$, but I can't quite prove it. I've tried the following: ...
6
votes
1answer
61 views

Baumslag–Solitar $B(1,2)$ is not hyperbolic

I have a question which asks me to show that the Baumslag–Solitar $B(1,2)$ is not hyperbolic by considering its Cayley graph and showing that triangles can be arbitrarily fat. The Cayley graph can be ...
2
votes
1answer
48 views

Computing quotients of abelian groups

Suppose that $A \cong \oplus_{i = 1}^{n} Z_{p_{i}^{k_{i}}}$ is some finite abelian group, and $(a_1, a_2, \ldots a_n)$ generates a subgroup $N$. If $\langle (a_1, a_2, \ldots a_n) \rangle$ was a ...
2
votes
0answers
41 views

group cohomology for SO(3) and SO(3,1)

I am studying relativistic quantum mechanics and I have encountered the concept of projective representation for a group. I have read in http://groupprops.subwiki.org/wiki/Projective_representation ...
8
votes
1answer
206 views

Lie groups as manifolds

In Weinberg's Classical Solutions in Quantum Field Theory, he states Lie groups, such as $SU(2)$ or $SO(3)$, may be viewed as a manifold. My questions are, If we can interpret, e.g. $SU(2)$ as a ...
1
vote
1answer
84 views

Is there a cayley graph for the Klein bottle?

When studying algebraic topology we learned about the fundamental group of the $2$-torus $T^2$ which is isomorphic to $$\langle a, b \mid aba^{-1}b^{-1} \rangle$$ (the free abelian group on two ...
3
votes
0answers
40 views

Finite index embedding of $F_{4}$ in $F_{2}$

In this question $F_{n}$ is the free group with $n$ generators. Is there a subgroup of $F_{2}$, isomorphic to $F_{4}$, which index is finite but not in the form of $3k$(not multiple of $3$)? The ...
4
votes
2answers
108 views

Fundamental group of the Poincaré Homology Sphere

I'm working on the Poincaré Homology Sphere $P_3$ and would like to compute it's Homology $H_1$ and fundamental group. I would like to identify it's fundamental group with the binary icosahedral group ...
3
votes
1answer
30 views

Appropriateness of using an interval in $\Bbb{R}$ as the parameter to a continuous path in space $X$.

The fundamental group $\pi_1(X,x)$ is usually defined using continuous paths $p : [0,1] \to X$. But... (1) Can you use other spaces besides a closed interval of $\Bbb{R}$ in the usual topo? (2) How ...
3
votes
2answers
113 views

$G$ is Topological $\implies$ $\pi_1(G,e)$ is Abelian

Hypothesis: Let $G$ be a topological group with identity element $e$. Let $\mu$ denote the multiplication mapping in $G$. Goal: Show that $\pi_1(G,e) = \pi(G)$ is an abelian group via the hint ...
10
votes
1answer
123 views

How can I understand the three-dimensional space forms?

Here is what I know: A space form is defined as a manifold admitting a Riemannian manifold of constant sectional curvature A classical result of Cartan states that a manifold is a space form if and ...
2
votes
1answer
37 views

Mapping Class Group of Simply Connected Spaces

I was wondering the following: If we take $M$ to be some orientable, simply-connected $n$-manifold. What can be said about $\pi_0(Homeo(M))$? We know that $\pi_1(M)=0$ and I know that the group is ...
0
votes
1answer
49 views

Free action of finite direct product

Let's consider free action of finite abelian group $G = G_1 \oplus G_2$ on a manifold $X$. Is it true that $X/G$ is diffeomorphic to $(X/G_1)/G_2$?
2
votes
2answers
50 views

Basis for singular chains group

The singular chain group $S_p(X)$ is defined as the free abelian group generated by continuous functions $T \in C( \Delta_p , X)$. What I understand this means is that we define $T' \in S_p(X)$ as ...
9
votes
1answer
115 views

if we set topology on a group like that, is it important?

Let $G$ be a group and $\omega$ be set of all subgroup of $G$. Since $\omega$ is closed under intersection, it is trivial to check that $\omega$ satisfies to conditions to be a base. Thus,Let $T$ be ...
2
votes
2answers
101 views

Fundamental group of Poincaré sphere

Do the two presentations below, $$G=\langle d,v \mid dv^2d=vdv, dv^3d=v^2 \rangle$$ and $$\langle r,s,t \mid r^2=s^3=t^5=rst \rangle = \langle s,t \mid (st)^2=s^3=t^5 \rangle,$$ define the same group? ...
1
vote
0answers
65 views

Set of generators of the commutator subgroup of a surface group

Good morning, I am having a hard time trying to describe the commutator subgroup of a surface group. Namely, if $S$ is a compact orientable surface and $G$ its fundamental subgroup, let's recall that ...
0
votes
0answers
46 views

Different profinite topologies on a group?

I have some general questions around the profinite topology on a group $G$. On the page http://groupprops.subwiki.org/wiki/Profinite_topology one can read, that The profinite topology on a group is ...
0
votes
0answers
51 views

Subgroup Separability translated in Profinite Topology

The normal definition of subgroup separability is: A group $G$ is said to be subgroup separable if for every finitely generated subgroup $H\leq G$ and $g\in G\setminus H$ there exists a subgroup of ...
0
votes
1answer
80 views

Abelian group with cyclic subgroup and cyclic quotient is generated by two elements

I have a number of questions that I think are related. I'm studying Algebraic Topology by Hatcher. I have essentially the same question as here. When talking about homology groups, the book says that ...
1
vote
1answer
90 views

relation between the group O(3) and SU(2)

Base on relations between groups $O(3)$, $SO(3)$ and $SU(2)$: a) $O(3)=SO(3)\otimes \{1,-1\}$ b) $SO(3)\simeq SU(2)/Z_2$ Can I say $\{1,-1\}$, i.e. $Z_2$, also the center of the group $O(3)$? If ...
3
votes
5answers
98 views

Free product of the trivial group with another group

I'm new to the idea of a free product.. Basically I was wondering if G is an arbitrary group and 1 is the trivial group then is $1\star G \cong G$. If not.. what whould it look like?
2
votes
2answers
198 views

HNN extensions as fundamental groups

I have heard that the Seifert–van Kampen theorem allows us to view HNN extensions as fundamental groups of suitably constructed spaces. I can understand the analogous statement for amalgamated free ...
2
votes
1answer
104 views

Meaning of Fundamental group of a graph

I am a computer science student working in graph algorithms. I am unable to understand what the fundamental group of a graph means. I have some intuition regarding the fundamental group of a ...
3
votes
0answers
76 views

Find the fundamental group and the Alexander polynomial

I would like to find the Alexander polynomial of the link $L$, described below. Let $K(q,r)$ be the $(q,r)$-torus knot embedded on a torus $V$. Inside the torus $V$, consider a smaller solid torus ...
1
vote
0answers
36 views

Group of continuous automorphisms of cylindrical plane is a lie group

How do I prove this theorem? Theorem: The group of continuous automorphisms of a cylindrical plane is a lie group. In this context, cylindrical means the Laguerre plane. I found a paper it ...
3
votes
0answers
47 views

Topology of a 3D wired Mandala?

There is a so called 3D-wired Mandala, based upon $2$ large circles each flowered symmetrically on its circumference by two sets of each $8$ half-circles. The circles are interconnected together by ...
3
votes
2answers
137 views

On the quotient group $\pi_{1}(K)/N$ for the Klein bottle $K$

I know that the Klein bottle $K$ is obtained from the unit square by making identifications on the boundary with the appropriate directional arrows. Usually, what is done is that we identify the point ...
2
votes
0answers
103 views

Zero exponent sum w.r.t group words in knot group's presentation

I am reading, "Plane Curves Associated to Character Varieties of 3-Manifolds" by Cooper, Culler, Gillet, Long, and Shalen and on page 28 ( http://www.math.uic.edu/~culler/papers/PlaneCurves/curves.pdf ...
11
votes
1answer
163 views

Is there a nontrivial topological group that's isomorphic to its fundamental group?

All I know is that the topological group has to be Abelian. I have no idea how to prove or disprove this statement. Thanks in advance.
2
votes
1answer
64 views

Double cover of symplectic groups

What is the normal definition of double cover of Symplectic group? I couldn't find a simple and understandable definition
3
votes
1answer
153 views

Topics of Group Theory Required to Understand Betti Numbers

I am studying Group Theory. I made sure I have a problem at hand, as part of the motivation for my study. I have chosen the problem as being able to understand as well as compute Betti Numbers where ...
3
votes
1answer
75 views

Prove that any subgroup of $F_5$ of index 3 is isomorphic to $F_{13}$

Let $F_n$ denote the free group on $n$ elements. Prove that any subgroup of $F_5$ of index 3 is isomorphic to $F_{13}$. I noted that the wedge product of 13 copies of $S^1$ is a 3 fold covering ...
3
votes
3answers
92 views

Is $\langle abab^{-1}\rangle$ a normal subgroup of $\langle a,b|\varnothing\rangle$?

Let $G=\langle a,b | \varnothing\rangle$ and let $H\leq G$ s.t $H=\langle abab^{-1}\rangle$. Is $H\triangleleft G$? I'm asking this question in order to understand the fundamental group of the Klein ...
5
votes
1answer
114 views

Showing path connected matrices of a group $G$ is a normal subgroup

Let $G$ be a subgroup of $GL_n(\Bbb{R})$. Define $$H = \biggl\{ A \in G \ \biggl| \ \exists \ \varphi:[0,1] \to G \ \text{continuous such that} \ \varphi(0)=A , \ \varphi(1)=I\biggr\}$$ Show that $H$ ...
0
votes
1answer
59 views

Consequence of injectivity of projections from covering spaces

We have the theorem which says that the induced homomorphism $p_* : \pi_1(\tilde X,\tilde x_0)\rightarrow \pi_1(X,x_0)$ is injective (hence a monomorphism). Here $\tilde X$ is a covering space of $X$. ...
4
votes
3answers
204 views

Homomorphism/map in both direction implies isomorphism/homeomorphism or not?

I was working on a homework, and my first attempt get me to a deadend, but I was eventually able to solve it using a different method. But the fail attempt make me curious, and I wonder if it could ...
3
votes
1answer
82 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 ...
0
votes
1answer
76 views

E measurable with m(E) < $\infty$?

Suppose that $E$ is measurable with $m(E)$ $<$ $\infty$. ii) Show that $\displaystyle \ \ \int_E 2f\,\,\,$ $=$ $2$$\displaystyle \ \ \int_E f\,\,\,$ if $f$ is bounded and measurable. I told my ...