Questions tagged [fundamental-group]

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6
votes
2answers
351 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\...
6
votes
1answer
199 views

An extension of symplectomorphism group

$\DeclareMathOperator\GL{GL}\DeclareMathOperator\Sp{Sp}$Let $\omega=\sum dx_i\wedge dy_i$ be the standard symplectic structure of $\mathbb{R}^{2n}=\mathbb{R}^{n}\times \mathbb{R}^n$. We consider the ...
2
votes
1answer
125 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 $G$-...
7
votes
1answer
232 views

Understanding fundamental group of Poincare homology sphere

I'm currently reading Knots, Links, Braids, and 3-Manifolds by V. V. Prasolov and A. B. Sossinsky. I have trouble understanding the following picture. The dashed line denotes a trefoil whose tubular ...
4
votes
0answers
159 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 ...
1
vote
2answers
267 views

Non-self-intersecting paths on $\mathbb{C}\setminus\{0,1\}$ [closed]

Let us make two small holes around points $0$ and $1$ on the complex plane and consider non-self-intersecting paths that start on the boundary of one hole and finish at the boundary of the another. It ...
1
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0answers
55 views

Glueing local systems over union of compact Riemann surfaces

Let $X,Y$ be two connected, non-singular compact Riemann surfaces such that $X$ intersects $Y$ transversely at two distinct points. Let $L$ be a $\mathbb{C}$-local system on $X$. Let $L'$ be the ...
1
vote
0answers
102 views

Canonical étale path between a point and its ''nearby'' point

Consider the punctored line $X=\Bbb{A}^1_k\setminus \{s_1,\ldots,s_n\}$ over some field $k$. A(n étale) path in $X$ between two geometric points $x$ and $y$ is, by definition, an isomorphism between ...
8
votes
1answer
273 views

The direct product of the geometric fundamental group and the absolute Galois group

Given a geometrically connected variety $X$ over $\mathbb{Q}$ we have a short exact sequence $$ 1\to \pi_1(X_{\overline{\mathbb{Q}}})\to \pi_1(X)\to Gal(\overline{\mathbb{Q}}/\mathbb{Q})\to 1. $$ A ...
18
votes
2answers
525 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 ...
5
votes
2answers
351 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{...
10
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1answer
156 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 ...
0
votes
1answer
149 views

Fundamental group of the complement of cell subcomplexes

Given a regular CW complex stucture on a manifold $C$ of dimension $n$ and a subcomplex $D$ of dimension $n-2$, I want to compute the fundamental group of the complement $\pi_1(C\setminus D)$. A ...
22
votes
5answers
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 ...
3
votes
0answers
208 views

Is there a reasonable notion of universal cover for schemes over arbitrary fields?

Let $X$ be smooth projective variety, such as an elliptic curve. We know that $X$ does not have a universal cover in the category of schemes. However, if $k = \mathbf{C}$, then $\tilde{X}$ exists as ...
2
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0answers
76 views

How to compute fundamental groups of slice disk complements?

To compute the fundamental group of the complement $S^3 \setminus K$ of a knot, one usually uses the Wirtinger algorithm. Is there a similarly well-established procedure for computing the fundamental ...
5
votes
1answer
231 views

Étale fundamental group of multiplicative group over an algebraically/separably closed field

This is a repost of my question here. Do we know the structure of the étale fundamental group $\pi^\text{et}_1(\mathbb{G}_{m,K^\text{sep}})$ of the multiplicative group, for a given field $K$? For ...
2
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0answers
70 views

Realizing an amalgamated product of groups by splitting a closed manifold along a codimension 1 submanifold

In the paper "A splitting theorem for manifolds" by S.E. Cappell, https://www.maths.ed.ac.uk/~v1ranick/papers/capsplit.pdf the following "inverse" of the Seifert-van Kampen theorem for closed ...
0
votes
0answers
152 views

Galois cover corresponding to finite quotient of the étale fundamental group

Let $X$ be a connected scheme,$\pi_1(X,\bar{x})$ its étale fundamental group for some geometric point $\bar{x} : Spec(K) \rightarrow X$ and $E = \pi_1(X,\bar{x})/N$ a finite quotient of $\pi_1(X,\bar{...
11
votes
2answers
349 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 $...
3
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0answers
198 views

First thoughts about fundamental group of a topological (Lie) groupoid

I am reading the paper Chern-Weil map for principal bundles over groupoids. In page number $13$, authors say let us recall the definition of fundamental group of a topological groupoid. But, they ...
5
votes
3answers
335 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 $[\...
7
votes
1answer
424 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}]...
1
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0answers
113 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 ...
1
vote
1answer
264 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 $X$, ...
5
votes
2answers
281 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 ...
6
votes
2answers
514 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 ...
3
votes
0answers
135 views

Recover an etale fundamental group from local fundamental groups?

Suppose we have a scheme $S$ and an etale covering $\{ U_i \to S \}$. How much information about $\pi_1^{et} (S)$ can we recover from all the $\pi_1^{et} (U_i)$ ? Are there special conditions we can ...
3
votes
1answer
63 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 ...
3
votes
1answer
145 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 $\...
4
votes
1answer
163 views

Geodesic representatives in the orbifold fundamental group

Does every element in the orbifold fundamental group $\pi_1^{orb}(X,x)$ of a closed hyperbolic 2-orbifold $X$ admit a unique geodesic arc representing it? Does every free homotopy class in $X$ admit ...
4
votes
1answer
161 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 ...
8
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0answers
263 views

Relationships among constructions of fundamental group for schemes

There seem to be several constructions of fundamental group for schemes and stacks: by Grothendieck, Deligne, Nori, Noohi, Esnault-Hai, Vakil-Wickelgren, perhaps others as well. I am trying to ...
5
votes
2answers
374 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)? ...
15
votes
0answers
511 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 ...
3
votes
1answer
162 views

Multivariate Alexander polynomial vs single variable (Conway) Alexander polynomial

I consider the multivariate Alexander polynomial $\Delta(t_1,\ldots,t_n)$ for a $n$-component link (defined using e.g. the Fox derivative). If we wish to construct a 1-variable polynomial $A(t)$, we ...
19
votes
2answers
718 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 ...
3
votes
1answer
393 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 ...
6
votes
1answer
150 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 ...
8
votes
0answers
180 views

Fundamental group of moduli space of K3's

According to Rizov (https://arxiv.org/abs/math/0506120), the moduli stack of primitively polarized K3 surfaces of degree 2d $\mathcal{M}_{d}$ is a Deligne-Mumford stack over $\mathbb{Z}$. I'm looking ...
4
votes
0answers
171 views

Geometric fundamental group and algebraically closed residue field

my questions relates to the following talk of Tsuji: https://www.youtube.com/watch?v=2brDj26phP0 At around 10:30 of the video, Tsuji is interrupted by a man stating that his construction does not ...
6
votes
1answer
198 views

Finite étale covers of concentrated schemes and extension of base field

Let $k'/k$ be an extension of algebraically closed fields of characteristic $0$, and $X$ a concentrated (i.e. quasi-compact and quasi-separated) scheme over $k$. Question: is the pullback functor ...
17
votes
2answers
2k 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 ...
6
votes
0answers
188 views

Is the stack of smooth canonically polarized surfaces uniformisable (or a good orbifold)

This is a question about the fundamental group of the moduli of canonically polarized surfaces. Let $\mathcal{M}$ be the (locally finite type separated Deligne-Mumford) algebraic stack of smooth ...
8
votes
1answer
871 views

Galois theory, topos vs fundamental groups

Classical Galois theory states that the etale topos X of a field k is equivalent to the classifying topos of the absolute Galois group of k. (Marc Hoyois, Higher Galois theory, $\S$3, arXiv:1506....
6
votes
1answer
327 views

Fundamental groups of non-orientable closed four-manifolds

The fundamental group of a closed orientable manifold is finitely presented, and every finitely presented group arises as the fundamental group of a closed orientable four-manifold; see this question. ...
9
votes
1answer
386 views

Mapping class group and representation of fundamental group of Riemann surfaces

Let $S$ be a Riemann surface with genus $g>0$. Let $M$ be the mapping class group of $S$. $Hom(\pi_1(S),Gl(n, \mathbb{C}))$ is the representation space of fundamental group of $S$ Question: Is ...
1
vote
1answer
242 views

What does the group of automorphisms corresponding to $\mathfrak{g}$

I am reading a book titled "Lectures on An Introduction to Grothendieck's Theory of the Fundamental Group" by J.P. Murre. I am in the chapter 4 titled "Fundamental groups". Here he fixes a base ...
3
votes
0answers
132 views

How does the fundamental group of $\mathbb G_{m,S}$ depend on the base scheme

Let $S$ be an integral noetherian regular scheme, and let $X =\mathbb{G}_{m,S}$. How to compute $\pi_1^{et}(X)$? Note. I am only interested in the part not coming trivially from the finite etale ...
8
votes
1answer
422 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 ...