All Questions
Tagged with fundamental-group at.algebraic-topology
16 questions
62
votes
9
answers
9k
views
Fundamental groups of noncompact surfaces
I got fantastic answers to my previous question (about modern references for the fact that surfaces can be triangulated), so I thought I'd ask a related question. A basic fact about surface topology ...
- 44.8k
29
votes
4
answers
3k
views
Geometric interpretation of the lower central series for the fundamental group?
For any group $G$ we can form the lower central series of normal subgroups by taking $G_0 = G$, $G_1 = [G,G]$, $G_{i+1} = [G,G_i]$. We can check this gives a normal chain
$$G_0 \ge G_1 \ge ... \ge G_i ...
- 573
51
votes
5
answers
9k
views
Fundamental group as topological group
Background
Let $(X,x)$ be a pointed topological space. Then the fundamental group $\pi_1(X,x)$ becomes a topological space: Endow the set of maps $S^1 \to X$ with the compact-open topology, endow the ...
- 63.1k
13
votes
4
answers
5k
views
Fundamental group of Lie groups
Let $T$ be a torus $V/\Gamma$, $\gamma$ a loop on $T$ based at the origin. Then it is easy to see that $$2 \gamma = \gamma \ast \gamma \in \pi_1(T).$$
Here $2 \gamma$ is obtained by rescaling $\gamma$...
- 14.7k
119
votes
6
answers
10k
views
What properties make $[0,1]$ a good candidate for defining fundamental groups?
The title essentially says it all. Consider the category $\mathfrak{Top}_2$ of triples $(J,e_0,e_1)$ where $J$ is a topological space, and $e_i \in J$. There is an obvious generalization of the ...
- 5,759
66
votes
4
answers
6k
views
Is $\mathbb{R}^3 \setminus \mathbb{Q}^3$ simply connected?
Similarly is the complement of any countable set in $\mathbb R^3$ simply connected?
Reading around I found plenty of articles discussing the path connectedness $\mathbb R^2 \setminus \mathbb Q^2$ and ...
- 1,187
17
votes
3
answers
4k
views
What is π_1(BG) for an arbitrary topological group $G$?
The classifying space $BG=|Nerve(G)|$ of an arbitrary topological group $G$ does not necessarily have the homotopy type of a CW-complex but the fundamental group should still be accessible. What is $\...
- 7,203
27
votes
3
answers
7k
views
Why are we interested in the Fundamental Groupoid of a Space?
The classical version of the van Kampen theorem is concerned about the fundamental group of a based space. In fact, it says that the functor $\pi_1$ preserves certain types of pushouts in $Top_*$.
...
- 609
27
votes
2
answers
3k
views
Teaching the fundamental group via everyday examples
This question is a "prequel" to a similar question about homology. Both questions were inspired by seeing a talk, by Tadashi Tokieda, about the interesting physics that appears in toys.
What ...
23
votes
5
answers
2k
views
Does anyone know a basepoint-free construction of universal covers?
Let $X$ be a real manifold (for simplicity). The standard construction of the universal cover $\varphi: \widetilde{X} \longrightarrow X$ involves fixing a basepoint $p \in X$ and considering homotopy ...
- 4,164
20
votes
0
answers
617
views
On a homological finiteness condition
Assumption: $X$ is a connected CW complex, and $H_{\ast}(X;\mathbb{Z})=\bigoplus_{n \geq 0} H_n (X; \mathbb{Z})$ is finitely generated.
Question: does there exist a finite CW complex $Y$ and a map $f:...
- 20.9k
16
votes
2
answers
3k
views
The fundamental group of a closed surface without classification of surfaces?
The fundamental group of a closed oriented surface of genus $g$ has the well-known presentation
$$
\langle x_1,\ldots, x_g,y_1,\ldots ,y_g\vert \prod_{i=1}^{g} [x_i,y_i]\rangle.
$$
The proof I know ...
- 20.9k
9
votes
1
answer
2k
views
Under what conditions is the induced map of etale fundamental groups surjective?
Let $f:X \to Y$ be a morphism of schemes. I am interested in sufficient conditions on $f$ which would ensure that the induced map $\pi_1^{et}(X) \to \pi_1^{et}(Y)$ of etale fundamental groups is ...
- 2,974
9
votes
2
answers
2k
views
Functoriality of fundamental group via deck transformations
Problem
I'm trying to understand this with a view towards the etale fundamental group where we can't talk about loops. What I'm missing is how the fundamental group functor should work on morphisms, ...
- 5,929
7
votes
2
answers
2k
views
The fundamental group of a $3$-manifold with a boundary of genus $>0$
Let $M$ be an orientable $3$-manifold with connected boundary $\Sigma_g$, a surface of genus $g>0$.
I would like to find a reference to the following two statements.
1) $\pi_1(M)\ne 0$.
2) $\...
- 14.3k
1
vote
1
answer
379
views
Finding the 2nd homotopy group $\pi_2(G^\mathbb{C}/P)$
Let $G$ be a compact connected and simply connected Lie group and $G^\mathbb{C}$ be the complexification of Lie group (with is diffeomorphic with $G^\mathbb{C}\cong T^*G$) then I am looking for ...
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