Tagged Questions
0
votes
1answer
26 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 $X$.
...
3
votes
3answers
91 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
64 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
57 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 ...
3
votes
1answer
53 views
Sufficient condition for a direct limit of abelian groups to be infinitely generated
I have the following setup. The CW-complexes $\Gamma_n$ are equipped with maps $\gamma_n\colon\Gamma_{n+1}\rightarrow\Gamma_{n}$ and it is known that the rank of their first cohomology groups is ...
2
votes
2answers
50 views
free groups and bouquet of circles
For any free group $F$ generated by the set $S$, one can construct a graph specifically a bouquet of circles $X$ s.t $\pi_1(X)=F$. My question is: Does this mean free groups are isomorphic to a free ...
4
votes
1answer
112 views
Groups acting on polytopes
I am currently reading the paper "Polytopal Resolutions for Finite Groups" [1] by Graham Ellis, James Harris and Emil Skoeldberg and have a question regarding an early remark of theirs.
Their basic ...
2
votes
2answers
61 views
Given a topological space $X$, does $H_1(X)=\mathbb{Z}$ imply $\pi(X)=\mathbb{Z}$?
Let $X$ be any topological space with the first homology group $H_1(X)=\mathbb{Z}$ . I claim $\pi_1(X)=\mathbb{Z}$.
By Hurewicz, we know that $H_1(X)$ is the abelianization of $\pi_1(X)$. ...
2
votes
1answer
48 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 ...
9
votes
2answers
120 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
0answers
79 views
Groups not arising from certain centralizers
There's a lot of fuss in certain subfields of algebraic topology about giving fancy interpretations to the rings coming from the cohomology of groups, where "cohomology" is allowed to be taken to be ...
12
votes
2answers
204 views
Does $gHg^{-1}\subseteq H$ imply $gHg^{-1}= H$?
Let $G$ be a group, $H<G$ a subgroup and $g$ an element of $G$. Let $\lambda_g$ denote the inner automorphism which maps $x$ to $gxg^{-1}$. I wonder if $H$ can be mapped to a proper subgroup of ...
2
votes
1answer
135 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 ...
4
votes
2answers
274 views
Prerequisites for Algebraic Topology
I'd like to self-study Munkres's Topology. I'm already comfortable with point-set topology, so the first part of the book will serve as a nice review with some new theorems every now and then. My main ...
10
votes
2answers
130 views
Torsion on $\pi_1(X)$, $X$ connected and open in $\mathbb{R}^n$
Can the fundamental group of an open connected subset $X$ of $\mathbb{R}^n$ have a torsion element?
7
votes
1answer
134 views
Topological Meaning of semi-direct product
I know that the amalgamated free product of two groups $G\star_K H$ has a certain topological meaning. What about a semi-direct product $H \rtimes G$ ?
31
votes
0answers
1k views
Does a four-variable analog of the Hall-Witt identity exist?
Lately I have been thinking about commutator formulas, sparked by rereading the following paragraph in Isaacs (p.125):
An amazing commutator formula is the Hall-Witt identity: ...
3
votes
1answer
164 views
Why isn't this free product of groups abelian?
I'm trying to prove that the free group $A=A_1*A_2$, where $A_1, A_2\neq 1$ is not abelian. Following the hints below:
Let $x,y\in A_1*A_2$, where $x\neq y$.
Suppose now $A_1=F(S)$ and $A_2=F(T)$, ...
4
votes
1answer
82 views
Seek results in group theory obtained by applping algebraic topology tools
After studying the first section, especially the section 1.3 (covering spaces) in Hatcher's book, it is obvious that many classical topics in group theory, especially in infinite group theory, have ...
5
votes
2answers
229 views
Is the center of the fundamental group of the double torus trivial?
I know that the fundamental group of the double torus is $\pi_1(M)=\langle a,b,c,d;a^{-1}b^{-1}abc^{-1}d^{-1}cd\rangle$.
How can I calculate its center subgroup $C$? Is $C$ trivial?
Let $p$ be the ...
0
votes
1answer
86 views
Free Groups and their Generators - Specific Question
1) Let $F$ be the free group on the three generators $x,y,z$. For non-zero integers $r,s,t$
then CLAIM: the subgroup of $F$ generated by $x^r , y^s , z^t$ is freely generated by these elements.
2) ...
2
votes
1answer
74 views
Could a surface bundle over a circle have free fundamental group?
Specifically, I was wondering if the surface was non-compact with infinitely generated free fundamental group, could the surface bundle itself have infinitely generated free fundamental group. In this ...
7
votes
2answers
390 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 ...
3
votes
0answers
84 views
Multiple Dehn twists and minimal position
I have a question about a proof that I am reading in "A primer on Mapping Class Groups" by Farb and Margalit.
Let $a$ be a simple closed curve in a compact surface $S$ (possibly with marked points ...
3
votes
0answers
152 views
Double Coverings of the Double Torus
I'm trying to count all the double coverings of the double torus. I know that the fundamental group of the double torus is
$$\pi_1(X)=\langle a,b,c,d;[a,b][c,d]\rangle $$
where ...
9
votes
1answer
245 views
Can the n-string sphere braid group embed in to the (n+1)-string sphere braid group?
This question has been cross posted on MathOverflow with some very interesting answers and discussion.
I'm currently writing a project on the braid groups and their analogues on closed surfaces. ...
5
votes
3answers
100 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 ...
3
votes
1answer
110 views
Why is the group of covering transformations relative to the quotient map isomorphic to a subgroup of the Fundamental Group?
I'm trying to prove the classification theorem for covering spaces. I've got to the stage where I need to show the following:
If $H$ a subgroup of $\Pi_1(X,x_0)$ then $\exists Y$ covering space of ...
10
votes
5answers
284 views
$\varphi$ in $\operatorname{Hom}{(S^1, S^1)}$ are of the form $z^n$
I'd like to see a proof why $\varphi \in \operatorname{Hom}{(S^1, S^1)}$ looks like $z^n$ for an integer $n$.
At first I thought I could argue that if I have a homomorphism that maps $e^{ix}$ to some ...
2
votes
0answers
45 views
Dehn Twist in the sense of graphs
Does anyone knows a good book or script about Dehn Twists in the sense of graphs. More precisely: I need to know how a Dehn Twist yields an automorphism of a group or subgroups. I want to know ...
0
votes
1answer
212 views
Rank of a cohomology group, Betti numbers.
How is the rank of a cohomology group computed and what does it convey? I am trying to understand the concept behind betti numbers in a simplicial homology.
Edited with details:
Given a set of ...
1
vote
1answer
70 views
A question concerning maps of $G$-coverings
I am having difficulties thinking about how an argument for the following exercise should proceed:
Let $p: Y \rightarrow X$ and $q: Z \rightarrow X$ be $G$-coverings (i.e., covering maps such that $X ...
2
votes
1answer
132 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
126 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}, ...
1
vote
1answer
155 views
A question on Group actions
I was wondering if anyone visiting would be up for solving the following interesting little exercise out of Fulton's Algebraic Topology: A First Course.
Let $G$ a group act on a set $Y$. Say that $G$ ...
11
votes
2answers
401 views
Nails and strings and paintings
This question is based on the "Picture proof" challenges from Rankk.org...
IDEA:
You want to hold up a painting using nails on a wall and string. The string is attached to the left and right sides of ...
4
votes
0answers
85 views
free group represented by a 4-manifold
I want to show that any free group $G$ with finitely many ($n$) generators can be represented by a 4-manifold having fundamental group $G$.
I thought about the connected sum of n copies of $S^1 ...
1
vote
2answers
207 views
Are these group presentations trivial?
I just got these presentations of groups:
$\langle a,b\mid aba^{-1}b^{-1}\rangle$
$\langle a,b\mid aba^{-1}b^{-2},bab^{-1}a^{-2}\rangle$
$\langle a,b\mid abab^{-1}\rangle $
Are any of them ...
7
votes
3answers
186 views
The Hopfian property for groups
Let $G$ be a group, which for my purposes would be abelian. To say that $G$ has the Hopf property is to say that every epimorphism of $G$ is an automorphism. Does anyone happen to recall the context ...
3
votes
2answers
128 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 ...
1
vote
1answer
119 views
Question about the sum of chain groups
What's the difference between $C_n(A + B)$ (that is, $C_n(A)+C_n(B)$ in $C_n(X)$) and $C_n(A) \oplus C_n(B)$ where $A$ and $B$ are subspaces of a topological space $X$? They're the same sets, right?
...
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
132 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 ...
0
votes
2answers
247 views
Groups, quotient and direct sum question
If $A,B,C$ are groups and I have $A / B = C$:
Is it then obvious that $A = C \oplus B$? I'm asking because I think this is used in the explanation of reduced homology on page 110 in Hatcher.
Thanks ...
1
vote
1answer
199 views
Equivalent identification to get the projective plane?
I think $$ \langle a,b | abab= 1 \rangle = \langle a,b | abba = 1 \rangle $$ are 2 equivalent presentations of the fundamental group of the projective plane. To show this, I have tried to transform ...
3
votes
1answer
151 views
A question about a group presentation
I have calculated the fundamental group of the annulus and got the following group presentation:
$$ \langle a, b | ab = ba = 1 \rangle$$
This is the set of strings of the form: $1, a, a^2, a^3, ...
2
votes
1answer
97 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 ...
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 ...
2
votes
2answers
130 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 $H:id_K\!\simeq\!c_{k_0}$ ...
1
vote
1answer
72 views
Is there a finite generating set for the Torelli group $T_2$?
D.Johnson showed in 1983 that for g>2 , the Torelli group $Tg$ has a finite
set of generators. I have not been able to find out what the case is for g=1,2;
does anyone know of any result for ...
