All Questions
Tagged with fundamental-group at.algebraic-topology
106 questions
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
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
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
38
votes
2
answers
2k
views
What is the fewest number of points you must delete from $\mathbb{R}^3$ to make it not simply connected?
This question concerns a set-theoretic aspect that I found interesting in the recent question asked by user Nick R., namely, Is
$\mathbb{R}^3\setminus\mathbb{Q}^3$ simply connected? He had asked ...
- 236k
36
votes
2
answers
5k
views
Is the fundamental group functor a left-adjoint?
Theorem 1B.9 in Hatcher's Algebraic Topology says that for a (pointed) connected CW complex $X$ and group $G$, there is a bijection $\text{Hom}(\pi_1(X), G) \cong [X,K(G,1)]$, where $\pi_1(X)$ is the ...
- 1,446
34
votes
1
answer
2k
views
Is every ''group-completion'' map an acyclic map?
I start with a longer discussion which will result in a precise version of the question. I am puzzled about an issue with the
Quillen plus construction. I have seen outstanding experts being confused ...
- 20.9k
31
votes
1
answer
3k
views
Can the fundamental group of any manifold be realized as the fund grp of a finite space?
Recently, I was asked to calculate the fundamental group of the space $X= \{a,b,c,d\}$ with open sets generated by $\{ a, c, abc, acd \}$.
Turns out, $\pi_1(X)\cong \mathbb Z$ and in fact, $X$ is the ...
- 899
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
28
votes
2
answers
6k
views
What group is $\langle a,b \,| \, a^2=b^2 \rangle$?
In teaching my algebraic topology class, this group showed up as part of an easy fundamental group computation: $\langle a,b\mid a^2=b^2\rangle$. My first instinct was that this must be $\mathbb{Z}*\...
- 5,544
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
19
votes
2
answers
1k
views
Can one compute the fundamental group of a complex variety? Other topological invariants? [duplicate]
Given a system of polynomial equations with rational coefficients, is there an algorithm to compute the geometric fundamental group of the variety defined by these equations? I'm interested in both ...
- 15.4k
18
votes
2
answers
1k
views
Fundamental group of punctured simply connected subset of $\mathbb{R}^2$
(This question is originally from Math.SE where it was suggested that I ask the question here)
Let $S$ be a simply connected subset of $\mathbb{R}^2$ and let $x$ be an interior point of $S$, meaning ...
- 2,889
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
17
votes
2
answers
4k
views
Is there a relationship between a quotient group of the fundamental group of X and the fundamental group of a quotient topology of X?
Let ($X$, $x_0$) be a topological space with a base point, and denote the fundamental group of $X$ as $\pi_1(X)$. Let $N$ be a normal subgroup of $\pi_1(X)$.
Does there necessarily exist an ...
- 271
17
votes
1
answer
574
views
Simply connected slices
Assume $\Omega$ is an open set in $\mathbb R^3$
such that the intersection of $\Omega$ with any horizontal plane is simply connected.
Can you prove that $\Omega$ is simply connected?
(Note that ...
- 45k
16
votes
6
answers
6k
views
Fundamental group of the line with the double origin.
In the simplest cases, the fundamental group serves as a measure of the number of 2-dimensional "holes" in a space. It is interesting to know whether they capture the following type of "hole".
This ...
- 3,699
16
votes
6
answers
3k
views
Fundamental groups of surfaces
What are some properties that hold for the fundamental group of a surface and do not necessarily hold for the fundamental groups of manifolds of higher dimensions?
- 311
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.
$$
The proof I know ...
- 20.9k
16
votes
0
answers
784
views
What would be the simplest analog of Langlands in algebraic topology?
It is oversimplified, I know, but just as a superficial analogy, one may think of the fact that abelianization of the fundamental group is the first homology group, as some remote relative of class ...
- 18.4k
16
votes
0
answers
645
views
Codimension Two Embeddings in Goodwillie-Weiss Manifold Calculus, and the Difficulty of Fundamental Groups
In manifold calculus, there are various analyticity estimates which run into trouble for codimension two embeddings. For instance, the functor $\operatorname{Emb}(M,N)$ is analytic in $M$ if $\dim M \...
- 3,351
15
votes
2
answers
2k
views
Topological vs pro fundamental groups
Consider the following two structure-adding refinements of the fundamental group of a topological space:
the set $\pi_1(X)$ inherits a quotient topology from the compact-open topology of $X^{S^1}$, ...
- 66.8k
14
votes
2
answers
1k
views
Are acyclic subcomplexes of finite contractible 2-complexes contractible?
Let $Y$ be a contractible finite simplicial 2-complex.
Let $X$ be an acyclic subcomplex of $Y$ (i.e. $X$ connected, $H_1(X)=0$, $H_2(X)=0$).
Is $X$ contractible? (Equivalently, is $\pi_1(X)$ trivial?)...
- 1,481
13
votes
4
answers
5k
views
Fundamental group of Lie groups
Let $T$ be a torus $V/\Gamma$, $\gamma$ a loop on $T$ based at the origin. Then it is easy to see that $$2 \gamma = \gamma \ast \gamma \in \pi_1(T).$$
Here $2 \gamma$ is obtained by rescaling $\gamma$...
- 14.7k
13
votes
1
answer
1k
views
Manifolds with prescribed fundamental group and finitely many trivial homotopy groups
Fix $G$, a finitely generated presented group.
It is known that for every $k > 3$ there is a closed $k$-manifold whose fundamental group is $G$. Similarly, there is a topological space with ...
- 3,165
13
votes
0
answers
863
views
About maps inducing bijections on homotopy classes
Let us assume that $f:X \to Y$ is a map of connected CW complexes, having the following property: if $K$ is a finite CW complex, then the induced map $f_{\ast}:[K,X] \to [K,Y]$ on \emph{free} homotopy ...
- 20.9k
12
votes
1
answer
832
views
Space with semi-locally simply connected open subsets
A topological space $X$ is semi-locally simply connected if, for any $x\in X$, there exists an open neighbourhood $U$ of $x$ such that any loop in $U$ is homotopically equivalent to a constant one in $...
- 303
11
votes
2
answers
287
views
Fundamental group under Gelfand duality
Gelfand duality states that the functor of continuous functions $C(-)$ from compact Hausdorff topological to commutative $C^*$-algebras is an equivalence of categories. In other words, all topological ...
- 21.6k
10
votes
1
answer
761
views
fundamental groups of smooth projective variety.
Is there a discrete group G which is the fundamental group of a compact Kahler
manifold but which is not the fundamental group of any smooth projective complex algebraic variety?
It is known that ...
- 3,867
9
votes
3
answers
2k
views
Computing `$\pi_1 S^1$` using groupoids
I believe it is possible to compute $\pi_1 S^1$ by applying the groupoid version of the Seifert-Van Kampen Theorem (in the version presented in May's Concise Course) to a covering of the circle by ...
- 8,803
9
votes
2
answers
710
views
For which spaces $S^n$ ($n\geq 2$) is a universal covering space?
I know that $S^n$ $(n\geq 2)$ is a universal covering space for itself and $\mathbb{RP}^n$. But my question is, for which spaces (up to homotopy equivalence) is $S^n$ ($n\geq 2$) a universal covering ...
- 1,182
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
9
votes
1
answer
657
views
Must an inverse limit of simply connected groups be simply connected?
While the fundamental group $\pi_1$ preserves products, it is not true in general that an inverse limit of simply connected topological spaces is simply connected. I would like to know if similar ...
- 7,203
9
votes
1
answer
235
views
Links and non-orientable surfaces
Let $\Sigma \subset \mathbb{R}^3$ be a compact embedded surface with boundary $\partial \Sigma$ and $i:\Sigma\setminus \partial\Sigma \to \mathbb{R}^3 \setminus \partial\Sigma$ the inclusion.
Is the ...
- 443
9
votes
1
answer
266
views
Branch cuts of $GL_n^+(\mathbb{R})$
Branch cuts
Let $GL_n^+(\mathbb{R})$ denote the group of $n\times n$ real matrices with positive determinant. Topologically, $GL_n^+(\mathbb{R})$ is connected, and
$$ \pi_1(GL_2^+(\mathbb{R})) = \...
- 13k
8
votes
6
answers
4k
views
connected compact semisimple lie group finite fundamental group
I was told that the fundamental group of a connected, compact, semisimple Lie group is finite, with the outline of a possible way to prove this fact. Is there any source however that fleshes this out ...
- 993
8
votes
2
answers
1k
views
Does the isomorphic of the fundamental groups imply the existence of a mapping inducing an isomorphism?
A pair of continuous mappings $f \colon X \to Y$ and $g \colon Y \to X$ is called $\pi_1$-equivalence if they induce mutually inverse isomorphisms of fundamental groups. Spaces are called $\pi_1$-...
- 6,962
8
votes
3
answers
943
views
Smooth projective varieties with infinite abelian fundamental group and finite $\pi_2$
Let $X$ be a smooth projective complex algebraic variety of general type. Suppose that the (topological) fundamental group of $X$ is an infinite abelian group and that $\pi_2(X^{an})$ is finite.
What ...
- 81
8
votes
1
answer
5k
views
Fundamental group of R^2-Q^2
After learning about the fundamental group, and proving that $\mathbb{R}^n$ minus any countable set is path-connected, I started wondering if the fundamental group of $\mathbb{R}^2-\mathbb{Q}^2$ is ...
- 3,079
8
votes
2
answers
5k
views
Homology of Covering Spaces
Let $A$ be a subgroup of a group $G$. Then since $A$ is a subgroup of the fundamental group $\pi_1(K(G,1))=G$, there is a covering space $p\colon Y\to K(G,1)$ with $p_*(\pi_1(Y))=A$. So the homology ...
- 1,108
8
votes
2
answers
721
views
Galois categories for topological spaces?
Can the theory of Galois categories (as developed in SGA1) be modified to produce the usual fundamental group of a topological space (maybe assumed to be path connected and locally path connected)?
...
- 3,284
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
7
votes
1
answer
2k
views
Fundamental group of a compact manifold
Why is the fundamental group of a compact manifold finitely presented?
- 217
7
votes
2
answers
566
views
Fundamental group of the space of smooth embeddings of $S^1$ into $\mathbb R^3$
Has the fundamental group of the space of smooth embeddings of $S^1$ into $\mathbb R^3$ been completely computed? Say the basepoint is an unknot. Maybe something is known for other components?
If yes,...
- 71
7
votes
1
answer
490
views
Categorical Significance of Fibrations
It is well known that the category $\text{Set}$ classifies covering spaces among $1$-categories. That is, for each topological space $X$, there is an equivalence of categories $[ \Pi (X) , \text{Set}]...
user30211
6
votes
3
answers
1k
views
Motivation of the fundamental theorem of covering spaces
The fundamental theorem of covering spaces states that for a nice topological space $X$, there is an equivalence of categories between covering spaces over $X$ and left $\pi_1(X)$-sets. "...
- 459