All Questions
Tagged with at.algebraic-topology fundamental-group
106 questions
3
votes
1
answer
169
views
Express fundamental group of $\mathcal H/\Gamma$ by $\Gamma$
Suppose $\mathcal H$ is the upper half plane, and $\Gamma$ is an arithmetic subgroup of $\operatorname{PSL}_2(\mathbb Z)$, I want to ask can we interpret the fundamental group of $\mathcal H/\Gamma$ ...
- 785
4
votes
1
answer
297
views
Fundamental group of the smooth locus of a normal algebraic surface is a quotient of that of a Zariski open subset
Let $X$ be a normal algebraic surface (over $\mathbb{C}$) and $Y$ its smooth locus, i.e., the complement of the singularities of $X$. Suppose $Z\subset Y$ is a Zariski open subset of $X$. Then is it ...
- 335
3
votes
0
answers
94
views
References for variations of Seifert–van Kampen's theorem: HNN extensions and "sensible" intersections
A basic consequence of the Seifert–van Kampen theorem is the following.
Theorem: Consider a union of topological spaces $X$, $Y$ whose intersection $X\cap Y = Z$ is open connected and $\pi_1$-...
- 1,033
2
votes
0
answers
55
views
Fundamental group of cyclic branched cover of affine plane
Let $f\in \mathbb{C}[x,y]$ be an irreducible polynomial. Let $n>0$ be an integer such that the hypersurface $S:=\{ (x,y,z)\in \mathbb{C}^3|z^n=f(x,y) \}$ is a connected complex submanifold of $\...
- 615
3
votes
2
answers
425
views
A question on the manifold $ \{n\otimes n-m\otimes m:n,m\in S^2,(n,m)=0\} $
Consider a manifold $ N $ defined as follows
$$
N=\{n\otimes n-m\otimes m:n,m\in S^2,\quad(n,m)=0\}\subset M^{3\times 3},
$$
where $ S^2 $ denotes the two dimensional sphere, $ (\cdot,\cdot) $ ...
- 1,219
2
votes
1
answer
287
views
How does hyperelliptic involution act on the standard generators of the fundamental group of surfaces of genus g with n punctures?
Let $S_{g,n}$ be the surface of genus $g$ with $n$ punctures. We know that $\pi_1(S_{g,n})$ admits a presentation:
$$\left\langle~ \alpha_1,\beta_1,\dots, \alpha_{g},\beta_{g},\gamma_{1},\dots,\gamma_{...
- 79
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
1
vote
0
answers
182
views
Does this sequence stop?
Let $\{ X_i\}$ ($i=1,2,\ldots $) be a family finite CW-complexes such that $X_{i+1}$ is homotopy domintaed by $X_i$, i.e. there exists contionuos maps $g_i:X_i \to X_{i+1}$ and $f_i :X_{i+1} \to X_i$ ...
- 1,182
1
vote
1
answer
279
views
Ways to prove that $n$-component Brunnian link is nontrivial
The attached image shows a way to construct an $n$-component Brunnian link for any $n\geq 3$. That is, this link is not trivial, but deleting any of its components makes the new link trivial. The ...
- 214
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
0
votes
1
answer
207
views
Is $\pi_2 (X_i)$ a free $\mathbb{Z}\pi_1 (X_i)$-module for $i=1,2$?
Let $X_1$ be the suspension of $\mathbb{R}P^2$ and $X_2=\bigvee_{1\leq i\leq n} (\vee_{r_i} \mathbb{S}^i)$.
Is $\pi_2 (X_i)$ a projective (or a free) $\mathbb{Z}\pi_1 (X_i)$-module for $i=1,...
- 1,182
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
0
votes
0
answers
339
views
Can someone explain this proof on aspherical manifolds?
I am trying to understand this proof that the fundamental group of an aspherical manifold is torsion free. The proof is lemma 4.1 from Aspherical manifolds at the Manifold Atlas Project. The proof is:
...
- 29
5
votes
0
answers
87
views
Fundamental groups and cellular walks
Suppose $M$ is a smooth manifold (compact if desired) with a cell structure or other nice stratification.
Call a path $\gamma : [0,1] \to M$ transverse to the stratification if there is a finite ...
- 720
6
votes
1
answer
289
views
Relationship between the holonomy pseudogroup and holonomy homomorphism (foliation)
This question is surely a duplication of https://math.stackexchange.com/questions/4343635/relationship-between-the-holonomy-pseudogroup-and-holonomy-homomorphism-foliati , however, I got no replies. ...
- 295
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
3
votes
2
answers
509
views
Can the loops in the definition of the fundamental group be considered injective?
Let $\mathrm{С}$ be some class of topological spaces that includes at least all subspaces of $\mathbb{R}^n $. Further we are in the category $\mathrm{С}_{*}$ (the category of point spaces; all ...
- 6,962
4
votes
0
answers
131
views
Fundamental group of hyperbolic 2-orbifold
Suppose $\Gamma$ is a cocompact lattice of $PSL_2(\mathbb{R})$. Then $\mathbb{H}^2/\Gamma$ has a natural structure of orbifold. My questions are:
What is $\pi_1(\mathbb{H}^2/\Gamma)$?
What is $\pi_1^{...
- 563
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
3
votes
0
answers
58
views
What's the Milnor's link group for the trivial knot in a lens space?
For a link $L$ in a 3-manifold $Y$, Milnor's paper "Link Groups" https://link.springer.com/content/pdf/10.1007/BF01393902.pdf defined the link group as some quotient of $\pi_1(Y-L)$. If $L$ ...
- 673
4
votes
1
answer
387
views
Surface bundles associated to a short exact sequence of groups
Suppose $S$ is a closed, connected, oriented surface of genus at least two and $G$ is any group. Suppose further that $\Gamma$ is any group that fits into the following short exact sequence:
$$ 1 \to \...
- 421
3
votes
1
answer
311
views
Fundamental group of twisted loop space
I'm interested in computing the fundamental group of the twisted loop space $$\Omega_f(M)=\{ \gamma \in C^{\infty}(\Bbb R,M) \mid \gamma(s+1)=f\gamma(s)\}$$
where $f \in \text{Aut}(M,x_0)$, for ...
- 503
2
votes
1
answer
282
views
Lifting of a proper map in the cover is a proper map
Let $M$ be an orientable surface without boundary$($I am not assuming $M$ is compact, it can be non-compact$)$. Let $\Phi: M\to M$ be a proper homotopy-equivalnce$($A proper homotopy-equivalence can ...
- 265
3
votes
0
answers
226
views
Is the category of covering spaces always a topos?
It is well knows that for a nice (locally path connected, semi-locally simply connected) topological spaces, the category of covering spaces over $X$ is equivalent to the functor category $\left[\Pi_1\...
- 438
5
votes
1
answer
416
views
triviality of homology with local coefficients
Let $X$ be a manifold or a CW-complex.
Let
$\pi: \tilde X\longrightarrow X$
be a covering map.
Let $\pi_1(X)$ be the fundamental group of $X$ and let $\rho: \pi_1(X)\longrightarrow O(n)$ be an ...
- 1,990
2
votes
1
answer
275
views
can the actions of fundamental groups annihilate homology?
Let $X$ be a path-connected manifold (or a CW complex).
Let $\pi_1(X)$ be the fundamental group of $X$.
Let $\pi: \tilde X\longrightarrow X$ be a covering map.
For each $m\geq 0$, let $C_m(\tilde X)$ ...
- 1,990
1
vote
0
answers
60
views
Restricted wreath product as fundamental group of a space with coinciding Reidemeister and Nielsen numbers
I am studying a group $\mathbb{Z}_n \wr \mathbb{Z}^k$, where $\wr$ denotes the restricted wreath product:
$$
\mathbb{Z}_n \wr \mathbb{Z}^k = \bigoplus_{x\in\mathbb{Z}^k}(\mathbb{Z_n})_x\rtimes\mathbb{...
- 137
6
votes
2
answers
2k
views
Action of fundamental group on homotopy fiber
For a Serre fibration of pointed topological spaces $f:X \to B$, there is an action of $\pi_1\left(B,b_0\right)$ on the fiber $F$. The construction of this action I'm familiar with uses a lift $F\...
- 834
2
votes
1
answer
275
views
Čech cocycles and monodromy
It is well known that over a topological space $X$ (and choosing an open cover $\mathfrak{U}$) every locally constant Cech cocycle $g$ on $\mathfrak{U}$ with coefficients in a group $G$ yields a $G$-...
- 553
4
votes
0
answers
397
views
Contractibility and orientation double cover
Question. Let $M$ be a triangulated non-orientable 3-manifold with non-orientable boundary. (It is possible to assume that the boundary is the Klein bottle.) Let $\ell$ be a non-orientable loop on the ...
- 976
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
5
votes
2
answers
581
views
Can we define fundamental groups functorially for non-pointed path connected topological spaces?
Let $\text{ppTop}$ denote the category of pointed and path connected topological spaces with morphisms base-preserve continuous maps. The fundamental group gives a functor $FG: \text{ppTop}\to \text{...
- 9,019
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
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
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
5
votes
3
answers
400
views
Generalize $H^1_{dR}(X)=\mathrm{Hom} (\pi_1(X),\mathbb R)$ to fundamental Groupoid
Let $X$ be a path-connected smooth manifold, it is known that: $$H^1(X):=H^1_{dR}(X)=\mathrm{Hom} (\pi_1(X),\mathbb R).$$ Explicitly, a closed one-form $\alpha$ gives a function on $\pi_1(X)$ by $[\...
- 2,789
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
1
vote
0
answers
127
views
Fundamental groups of open algebraic varieties [closed]
Let X be an algebraic variety over $\mathbb C$.
1. Is it possible to compute its fundamental group?
2. If X is two dimensional, what is its fundamental group?
3. Let $X\to \bar X$ be the inclusion to ...
- 169
3
votes
1
answer
1k
views
The (topological) fundamental group of (quasi)-projective algebraic varieties
I would like to know:
What does the fundamental group of a quasi-projective algebraic variety look like?
I remember that I have seen somewhere that for a connected, finite-type CW-complex $X$, ...
- 169
5
votes
2
answers
457
views
Finite etale covers of products of curves
Probably this question can be phrased in a much greater generality, but I will just state it in the generality I require. I work over $\mathbb{C}$.
Let $C_1, C_2 \subset \mathbb{P}^1$ be non-empty ...
- 21.4k
6
votes
2
answers
1k
views
Fundamental group of a topological group
It is well known that the fundamental group of a path-connected topological group is abelian. Suppose that $G$ is a connected topological group and let $Ab(G)$ the abelianization of the topological ...
- 451
3
votes
1
answer
84
views
Concerning the Spanier group relative to an open cover
Let $\mathcal{U} = \{ U_i \; |\; i\in I \}$ be an open covering of $X$. Spanier defined $\pi (\mathcal{U}, x)$ to be the subgroup of $\pi_1 (X, x)$ which contains all homotopy classes having ...
- 1,182
3
votes
1
answer
173
views
Approximation of homotopy avoiding a point in $\mathbb{R}^3$
For a proof that $\mathbb{R}^3\setminus \mathbb{Q}^3$ is simply connected using Baire category theorem I need to approximate an homotopy $H : [0,1]\times \mathbb{S}^1 \to \mathbb{R}^3$ from a loop $\...
- 179
4
votes
1
answer
270
views
Invariant lifts of a closed curve on a surface of genus > 1
I am learning some things about surfaces of genus greater than $1$, and I am trying to answer this question :
Let $S$ be a compact and orientable surface of genus $g \geq 2$, and $c$ a closed curve ...
- 41
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
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
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
3
votes
1
answer
429
views
A projective (or free) $\mathbb{Z}\pi_1$-module
Suppose that $Z$ is a finite wedge of spheres containing circles and there exist maps $f:Y\to Z$ and $g:Z\to Y$ so that $g\circ f\simeq 1_Y$. Assume that there exists a map $h:X\to Y$ which induces ...
- 139
6
votes
1
answer
237
views
Example similar to the Griffiths twin cone but with fundamental group that allows surjection onto $\mathbb Z$
The Griffiths twin cone is an example of a wedge sum of two contractible spaces being non-contractible. Namely, it is the wedge sum $\mathbb G=C\mathbb H\vee_p C\mathbb H$ of two coni over the ...
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