All Questions
Tagged with free-groups algebraic-topology
96
questions
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1
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90
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Lee Mosher book definition of a tree.
I was just reading the definition of a tree in Lee Mosher book, and he said if graph is simply connected then it is contractible.
I am wondering how is this true, can someone explain this to me please?...
user965463
4
votes
0
answers
182
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A Conjecture in Low-Dimensional Topology.
Context
I looked through a book called "Problems in Low-Dimensional Topology," where Rob Kirby lists a set of problems. He provides a list of problems, states their conjectures, and ...
0
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0
answers
44
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Covering space of compact surface with free fundamental group
Let $S$ be a compact connected surface. Does $S$ have a covering space $\tilde{S}$ such that the fundamental group of $\tilde{S}$ is the free group on $n$ generators with $n>1$ ?
I know that if we ...
1
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0
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66
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How do we generate the loop $ba$ from the loops $a^2,b^2$ and $ab\ $?
In the second diagram a $2$-sheeted connected covering of the figure eight has been described. The image of the fundamental group of the covering space has the generators $a^2, b^2$ and $ab$ as ...
- 545
7
votes
2
answers
299
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The fundamental group of closed orientable surface of genus 2 contains a free group on two generators
Let $S$ be the closed orientable surface of genus $2$. It is well known that its fundamental group is given by $$ \pi_1(S)=\langle a,b,c,d:[a,b][c,d]=1\rangle.$$
How can we show that this group has a ...
- 1,630
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1
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77
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Three questions about Wikipedia's definition of Van Kampen's theorem for fundamental groups
I'm studying Algebraic Topology off of Hatcher and (unfortunately as usual) I find his definition and explanation of Van Kampen's theorem to be carelessly written and hard to follow. I happen to know ...
- 894
2
votes
1
answer
87
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Free product of finite groups that is outside graph theory
The free product of finite groups $ A * B $ naturally acts on a biregular graph see Free Product of two finite groups. This seems like one of the only places that free products of finite groups appear ...
- 8,542
4
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63
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Recovering an element of a free group from its projections
Assume you have an unknown word on an alphabet with at least three letters, and you know all the words obtained by erasing each copy of some letter. Then, you can find the first letter of the original ...
- 645
0
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1
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42
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Minimal generating set of $p_*(\pi_1(E,e))$
Consider the following degree $4$ non-normal covering space of $S_1\lor S^1$ I drew:
Here, $a$ and $b$ denote the edges which map onto the first and second circle in $S^1\lor S^1$ respectively. I ...
- 598
5
votes
1
answer
135
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Given a subgroup of a free group, find the associated covering space.
Let $R_2$ the rose with $2$ petals, that is the wedge of $S^1$ with itself. We know its fundamental group is the free group with two elements, $\pi_1(R_2)=F_2=\langle a,b\rangle$. Now given some ...
1
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0
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55
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Construction of Free abelian groups on Massey Book
I am reading the book of Massey of Algebraic topology, and I am having trouble to understand this construction.
Let $ S = \left\{ x_i : i\in I \right\}$. For each index $i$, let $S_i$ denote the ...
- 592
6
votes
0
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106
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Question on the standard algebraic topology proof that a subgroup of a free group is free [duplicate]
I have a question on the standard algebraic topology proof that a subgroup of a free group is free. My understanding of that proof (mostly from Hatcher's topology) is as follows:
We define a topology ...
- 2,068
3
votes
1
answer
443
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For level-wise free chain-complexes a quasi-isomorphism is always induced by homotopy equivalence
Let $C_{\cdot}, D_{\cdot}$ be level-wise free chain-complexes, i.e. such that each $C_n$ and $D_n$ is a free abelian group.
Let $f:C_{\cdot} \to D_{\cdot}$ be a chain-map and a quasi-isomorphism. Thus ...
- 569
0
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1
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45
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A question related to an induced homomorphism between two groups
Suppose $X$ is obtained by gluing two tori at a single point and let $r:\sum_2\to X$ be the retraction given by collapsing a circle around the middle of $\sum_2$ (surface of genus $2$) to a single ...
- 520
0
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3
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281
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Abelianization of free groups
I'm reading Hatcher's Algebraic Topology and I have some questions about an argument on Page 42:
The abelianization of a free group is a free abelian group with basis the same set of generators, so ...
- 929