All Questions
Tagged with at.algebraic-topology fundamental-group
106 questions
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
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
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
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
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 ...
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
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
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
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 $\gamma$...
- 14.8k
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
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
7
votes
1
answer
2k
views
Fundamental group of a compact manifold
Why is the fundamental group of a compact manifold finitely presented?
- 217
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
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 ...
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
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
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
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
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:
$$
\mathbb{Z}_n \wr \mathbb{Z}^k = \bigoplus_{x\in\mathbb{Z}^k}(\mathbb{Z_n})_x\rtimes\mathbb{...
- 137
4
votes
1
answer
270
views
Invariant lifts of a closed curve on a surface of genus > 1
I am learning some things about surfaces of genus greater than $1$, and I am trying to answer this question :
Let $S$ be a compact and orientable surface of genus $g \geq 2$, and $c$ a closed curve ...
- 41
3
votes
1
answer
173
views
Approximation of homotopy avoiding a point in $\mathbb{R}^3$
For a proof that $\mathbb{R}^3\setminus \mathbb{Q}^3$ is simply connected using Baire category theorem I need to approximate an homotopy $H : [0,1]\times \mathbb{S}^1 \to \mathbb{R}^3$ from a loop $\...
- 179
1
vote
0
answers
127
views
Fundamental groups of open algebraic varieties [closed]
Let X be an algebraic variety over $\mathbb C$.
1. Is it possible to compute its fundamental group?
2. If X is two dimensional, what is its fundamental group?
3. Let $X\to \bar X$ be the inclusion to ...
- 169
8
votes
3
answers
943
views
Smooth projective varieties with infinite abelian fundamental group and finite $\pi_2$
Let $X$ be a smooth projective complex algebraic variety of general type. Suppose that the (topological) fundamental group of $X$ is an infinite abelian group and that $\pi_2(X^{an})$ is finite.
What ...
- 81
16
votes
2
answers
3k
views
The fundamental group of a closed surface without classification of surfaces?
The fundamental group of a closed oriented surface of genus $g$ has the well-known presentation
$$
\langle x_1,\ldots, x_g,y_1,\ldots ,y_g\vert \prod_{i=1}^{g} [x_i,y_i]\rangle.
$$
The proof I know ...
- 20.9k
4
votes
2
answers
1k
views
Fundamental groups of closed hyperbolic 3-manifolds are freely indecomposable
I believe the following statement is true, and I've even seen it referenced here. Could someone point me to a proof?
The fundamental group of a closed hyperbolic 3-manifold is not a free product.
- 726
3
votes
1
answer
84
views
Concerning the Spanier group relative to an open cover
Let $\mathcal{U} = \{ U_i \; |\; i\in I \}$ be an open covering of $X$. Spanier defined $\pi (\mathcal{U}, x)$ to be the subgroup of $\pi_1 (X, x)$ which contains all homotopy classes having ...
- 1,182
7
votes
2
answers
2k
views
The fundamental group of a $3$-manifold with a boundary of genus $>0$
Let $M$ be an orientable $3$-manifold with connected boundary $\Sigma_g$, a surface of genus $g>0$.
I would like to find a reference to the following two statements.
1) $\pi_1(M)\ne 0$.
2) $\...
- 14.3k
5
votes
1
answer
1k
views
What is the algebraic fundamental groups of $SO(n)$ and $Sp(2n)$?
Let $k$ be an algebraically closed field of characteristic zero. and let $$\sigma: SL_n(k)\rightarrow SL_n(k)$$
be an involution.
My questions are:
How could one calculate the fundamental group of $...
- 307
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
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
5
votes
1
answer
353
views
Fundamental groups of open sets in $R^n$ with $n=3,4$
It is well known that every finitely presented group may be realised as fundamental group of some closed $4$-manifold.
What groups can be obtained as fundamental groups of open subsets of $R^4$? I'...
- 9,340
6
votes
2
answers
595
views
Fundamental group of a manifold with an $S^1$-action
Let $M$ be a compact connected manifold with an $S^1$-action. Suppose that $S^1$ has a fixed point in $M$. Is it true that $\pi_1(M)=\pi_1(M/S^1)$?
I is there some reference or a short proof of this ...
- 14.3k
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
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
2
votes
3
answers
651
views
question about the induced homomorphism of etale fundamental groups
Background/Setup
For any connected scheme $S$, let $\text{FEt}_S$ denote the category of finite etale $S$-schemes. Let $f : X\rightarrow Y$ be a morphism of connected schemes, then for any finite ...
- 10.7k
5
votes
1
answer
1k
views
On the fundamental group of closed 3-manifolds
I know that every finitely presented group can be realized as the fundamental group of a compact, connected, smooth manifold of dimension 4 (or higher). In dimension 2 there are strong restriction on ...
- 683
4
votes
1
answer
375
views
What is kernel $\phi:G\rightarrow \pi_1(X/G,p(x_0))$?
Let $G$ be a discontinuous group (this means that it acts discontinuously with finite stabilizers) of homeomorphisms of a simply connected, locally compact metric space $X$. Let $p:X\rightarrow X/G$ ...
- 399
1
vote
1
answer
1k
views
Computing the fundamental group of a flag variety
Let $G$ be a compact and connected and simply connected Lie group and $\mathfrak{g}$ be its Lie algebra and $x\in\mathfrak{g}^*$. How can we compute the fundamental group of $G/G_x$ where $G_x$ is ...
user21574
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
4
votes
1
answer
573
views
A lower-dimensional algebraic topology problem between homology group and fundamental group
Let
\begin{equation}
A\stackrel{\alpha}{\longrightarrow}B\stackrel{\beta}{\longrightarrow}C\quad\quad (1)
\end{equation}
be a short sequence of (abelian or nonabelian) groups and homomorphisms. We say ...
- 1,881
1
vote
1
answer
438
views
When is the class of functions between sets a set?
I'm reading the paper 'The big fundamental group, big Hawaiian earrings and the big free groups'. The authors state that the class of homotopy equivalences of loops in the space he dubs as the big ...
- 189
6
votes
3
answers
1k
views
$\pi_1$ Sequence of Topological Groups
Consider a connected topological group $G$ (not necessarily Lie). You have some maps $G\times G\to G$, such as projection to either summand, or multiplication $(g,h)\mapsto gh$. Now let's look at a ...
- 17.5k
5
votes
2
answers
399
views
Conjugation of homogeneous spaces
Let $X$ be a smooth irreducible algebraic variety
over the field of complex numbers ${\mathbb{C}}$.
Let $x\in X({\mathbb{C}})$.
Let $\tau$ be an automorphism of ${\mathbb{C}}$ (not necessarily ...
- 14.2k
4
votes
1
answer
1k
views
Computing fundamental groups of the complement of plane curves
This paper of Zariski contains this statement: If $C$ is a curve in $\mathbb{CP}^2$, and $L$ is a generic line, then the injection $L\setminus C \hookrightarrow \mathbb{CP}^2\setminus C$ induces an ...
- 5,359
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
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
2
votes
0
answers
317
views
A homomorphism in the long exact sequence of a fibration for a homogeneous space of a Lie group
Let $G$ be a connected Lie group, and let $H\subset G$ be a (closed) Lie subgroup, not necessarily connected. Set $X=G/H$.
The fibration $j\colon G\to X$ with fiber $H$ induces an exact sequence
$$
\...
- 14.2k
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
1
vote
1
answer
307
views
The fundamental group of an $S^1$-quotient
Let $M$ be a compact manifold with an $\mathbb S^1$-action that fixes a point on $M$.
Is it correct that $\pi_1(M/S^1)=\pi_1(M)$?
I believe this is correct and is a corollary of some well-known ...
- 14.3k
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