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Questions tagged [fundamental-groups]

For questions about or involving the fundamental group.

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Every endomorphism $\psi$ of $\pi_1(S^1\times S^1, (1,1))$ can be expressed as $\psi = f_\ast$?

I have to prove the following: Using the canonical isomorphism $$\pi_1(S^1\times S^1,(1,1)) \cong \pi(S^1,1)\times \pi_1(S^1,1),$$ show that every endomorphism of the group $\pi_1(S^1\times S^1,(1,...
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Etale fundamental group of $Spec \mathbb Z_p[x,y]/(xy-p)$

One can compute etale fundamental group of complex algebraic varieties using topological fundamental group, and we know etale fundamental group of $\mathbb Z_p$ is the same as $\mathbb F_p$. How ...
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Writing every homomorphism $\Upsilon : \pi_{1}(S^1,1) \to \pi_{1}(S^1,1)$ as $\Upsilon = f_\ast.$

I'm proposed the following problem: Show that every group homomorphism $\Upsilon : \pi_1(S^1,1) \to \pi_1(S^1,1)$ can be written as $\Upsilon = f_\ast$ where $f : S^1 \to S^1$ is a continuous map. ...
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$\pi_{1}(X,x_{0})$ is abelian $\iff$ $ \hat{\alpha}=\hat{\beta}$ for all paths $\alpha,\beta$ in a path connected space. [duplicate]

Let $x_{0},x_{1}\in X$, where $X$ is path connected. Let $\pi_{1}(X,x_{0})$ the fundamental group of $X$ based at $x_{0}$ If $\alpha$ is a path in $X$ from $x_{0}$ to $x_{1}$, we define $\hat{\alpha}...
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1answer
61 views

Fundamental group of the complement of $k$ points in $\mathbb{R}^2$

Let $S = \{p_1, \ldots, p_k\}$ be a set of $k$ points in $\mathbb{R}^2$ ($1 \leq k < \infty$). My goal is to calculate the fundamental group of $\mathbb{R}^2 \setminus S$ using van Kampen's theorem....
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If $ E \rightarrow X $ is a covering, then $ |\pi_1 (X)|$ divides to $ \chi (E) $

I want to prove that no finite group acts free in $ \mathbb{R}^n $ in the process I found the following doubt: How can you prove that: If $ E \rightarrow X $ is a covering, then $ |\pi_1 (X)|$ ...
2
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1answer
53 views

Bundle over circle; Monodromy action on cohomology of fiber

I want to understand the following sentence: Let $\pi: M \to S^1$ be a fiber bundle with path connected fiber $F$. Then its monodromy action on $H^k(F; \mathbb C)$ satisfies... How is the ...
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0answers
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Fundamental group of punctured simply connected subset of $\mathbb{R}^2$

Let $S$ be a simply connected subset of $\mathbb{R}^2$ and let $x$ be an interior point of $S$, meaning that $B_r(x)\subseteq S$ for some $r>0$. Is it necessarily the case that $\pi_1(S\setminus\{x\...
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1answer
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The fundamental group of the gluing of two genus $g$ $3$-dimensional handlebodies

I was given the problem above, and I'd appreciate some help. I think I have a general direction, but I'm not entirely sure if what I'm doing is true, so it'd be great if someone could tell me if what ...
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3answers
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Fundamental group of the real projective plane and its universal cover

I have some questions regarding the real projective plane $\mathbb{R}\mathbb{P}^{2}$. If we choose to represent it using the following identification on the square $[0, 1]^{2}$ : how can we see that ...
3
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1answer
56 views

Calculate the quotient space of three real projective planes

Consider the following quotient space: Take $a,b \in RP^2$, $Y = RP^2 \times \{1, 2, 3\}$, and $X$ = the quotient space of $Y$ quotient by $(b,1) = (a,2)$, $(b,2) = (a,3)$, $(b,3) = (a,1)$. How do ...
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1answer
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The notation for fundamental group!

I am new in algebraic topology and I am wondering why the notation for the fundamental group is $\pi_1$? I mean what is this "1" for? I searched on the Web and did not find anything! Any idea? Also ...
2
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1answer
101 views

Local system associated to monodromy representation

How can I associate a local system to a representation $\rho: \pi_1(X) \to \mathbb C^*$? I have seen some construction, but it doesn't click for me. I know that the idea is to use a diagonal action ...
2
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1answer
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What is the fundamental group of $RP^{2}$ # $\cdots$ # $RP^{2}$ [duplicate]

I want to know about fundamental group of $RP^{2}$ # $\cdots$ # $RP^{2}$ by Seifert-Van Kampen theorem. In my guessing, that is $\langle a_1, a_2 ,... a_n | a_1^{2}a_2^{2}\cdots a_n^{2}=1\rangle$. ...
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1answer
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Classification of compact connected manifolds by fundamental group [duplicate]

Every compact connected 2-manifold (I define this as a surface) is homeomorphic to a 2-sphere, a connected sum of tori or a connected sum of projective planes. Since the fundamental groups of the ...
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1answer
63 views

Automorphism covering space and universal cover

I have again a new question about the automorphism of covering space and the universal cover of a topological space $B$. Actually, let $p : X \rightarrow B$ the universal cover of $B$. I take a normal ...
5
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2answers
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How can I visualise the fundamental group of the projective plane?

The real projective plane $\mathbb{RP}^2$ has fundamental group $C_2$. We can understand this via the universal covering mapping $S^2 \to \mathbb{RP}^2$ which identifies antipodal points: the ...
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1answer
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Let $f : B^2 \rightarrow \mathbb{R}^2$ a continuous map, is $B^2 \subset \operatorname{im}(f)$?

Consider a continuous map $f : B^2 \rightarrow \mathbb{R}^2$ such that $f(S^1) \subset S^1$ and $deg(f_{|S^1}) \ne 0.$ Prove that $B^2 \subset \operatorname{im}(f).$ [Note: Here, $B^2$ is the ...
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1answer
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Problem with set of cycles covering a graph

My question is the following: is there always, for any non-directed graph $G$, a choice of generators of the fundamental group or, more in general, a set of cycles, covering the graph and with the ...
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1answer
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Fundamental group of the complement of $3$ disjoint hypersurfaces in $\mathbb{C}P^2$

Let $X$ be the union of $3$ hypersurfaces in $\mathbb{C}P^2$, then how to compute the $\pi_1(\mathbb{C}P^2\setminus X)$? What I know is the complement of a hypersurface in $\mathbb{C}P^2$ is $\mathbb{...
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1answer
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Fundamental group of group of homeomorphism of a compact surface

I'm reading "A Primer on Mapping Class Group", and there is something I don't understand in the proof of Theorem 4.6. Define $\mathrm{Homeo}^+(S)$ to be the group of orientation-preserving ...
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1answer
46 views

Using Van Kampen's Theorem to determine fundamental group

I'm trying to calculate the fundamental group of a surface using (i) deformation retracts and (ii) Van Kampen's Theorem. I'm really struggling understanding the group theory behind it and the ...
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1answer
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Equivalence between Category of Covers and $\pi_1(X)$ Sets

I have a question about an argument used in Szamuely's "Galois Groups and Fundamental Groups" in the excerpt below (or look up at page 38): In order to show the category equivalence claimed in Thm 2....
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1answer
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Notions of Fundamental Groups for semisimple algebraic groups

Let $G$ be a connected, semisimple linear algebraic group over a field $k$. In Springer's Linear Algebraic group, the Fundamental Group of $G$ is defined by a certain quotient of groups $P/Q$, which ...
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How to see correspondence between G-covers and homomorphisms

I found this paper on van Kampen's theorem (https://www3.nd.edu/~andyp/notes/SeifertVanKampen.pdf), and I was wondering how to prove Lemma 1: Let Z be a reasonable nonempty path-connected space, let ...
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Algebraic Fundamental Group $π_1(U)$

I have a $1 \frac{1}{2}$questions about two examples introduced in Szamuely's "Galois Groups and Fundamental Groups" in the excerpt below (see page 122): Let $X$ be an integral proper normal $k$-...
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1answer
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Homotopy equivalence between quotients by free actions

Let $X,Y$ two contractible spaces. Assume there is a free action of a group $G$ on both spaces. $X$ and $Y$ are obviously homotopy equivalent. In particular, we can consider the homotopy equivalence ...
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1answer
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Real plane with two holes and circumference are not homotopy equivalent

I´d like to obtain an argument to prove that the real plane with two holes, for example $\mathbb{R} \setminus \{p,q\}$ is not homotopy equivalent to the circumference $S^1$. I know they have ...
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3answers
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Can I use Seifert-van Kampen theorem infinite times

I know the definition of Seifert-van Kampen theorem for a topological space "made" with 2 parts. Is not difficult to see that if I use the theorem a finite number of times to calculate a the ...
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1answer
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fundamental group of torus minus a point

I want to calculate fundamental group of torus minus a point using Van-Kampen Theorem.I know that result is $\Bbb Z* \Bbb Z$. I proved to consider like open $U$ the torus minus a disk $S$ that ...
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1answer
26 views

Simply connected and direct products

A topological space $X$ is simply connected if and only if $X$ is path-connected and the fundamental group of $X$ at each point is trivial, that is, $\pi(X,x_0) = 0$ for any $x_0 \in X$. Now, we know ...
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Constructing a Universal Cover--Proving Injectivity

Here is a quote from Hatcher's Algebraic Topology: Given a set $U \in \mathcal{U}$ and a path $\gamma$ in $X$ from $x_0$ to a point in $U$, let $$U_{[\gamma]} = \{[ \gamma \cdot \eta ] \mid \eta \...
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Pencil and fundamental group

Let $f_i(x,y,z)$ be a homogeneous polynomial of degree $i$. Then for pencil $C_{a,b}=a(f_2)^3+b(f_3)^2$, we have a map $\phi:\mathbb{C}P^2\setminus B\to \mathbb{C}P^1$, where $B$ is the base locus of ...
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Homomorphism Induced by Inclusion is Trivial

On page 64 of Hatcher's book on Algebraic Topology, he writes the following: ...if the map $\pi_1(U) \to \pi_1 (X)$ is trivial for one choice of basepoint in $U$, it is trivial for all choices of ...
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Is this proof that $\widehat{G_p}$ is pro-$p$ free correct?

Let $G$ be an abstract group with the following presentation: $$G \simeq \langle x,y \mid x^2y^2 = 1 \rangle $$ Let $p \neq 2$ be an odd prime. I want to show that $\widehat{G_p} \simeq \mathbb{Z}_p$...
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Why is the fundamental group of a closed hyperbolic $n$-manifold is a (uniform) lattice in $SO(n,1)$?

Why is the fundamental group of a closed hyperbolic $n$-manifold is a (uniform) lattice in $SO(n,1)$? Why is $SO(n,1)$ the (orientation-preserving) isometry group of real hyperbolic $n$-space? Is ...
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1answer
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Fundamental group of $\mathbb{R}P^2$ in 2 models

I know that $\pi_1(\mathbb{R}P^2)\cong\mathbb{Z}_2$, but in the square model, I get that $\pi_1(\mathbb{R}P^2)=\langle a,b\colon abab\rangle$. These groups must be isomorphic, but I can't find the ...
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1answer
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Fundamental group of $\mathbb{C^*}/G$

Calculate the fundamental group of $\mathbb{C^*/G}$ where $G=\{\phi^n:n\in \mathbb{Z}\}$ is the group of the homeomorphisms s.t. $\phi (z)=2z$ and $\mathbb{C^*}=\mathbb{C}-\{0\}$ Any hints on how to ...
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Deck Transformations Question

Using the Hatcher's notation on proposition 1.39. Let $G(\tilde{X}):=\{ \varphi: \tilde{X} \to \tilde{X} \; \mathrm{Isomorphism} \}$ be the Deck transformation group for the covering space $p: \tilde{...
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1answer
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Extension of surface group by cyclic is residually finite

Let $G$ be the fundamental group of a surface, and consider an extension $1 \to \mathbb{Z}/p\mathbb{Z} \to E \to G \to 1$. Is $E$ residually finite? I'm interested in proving the injectivity of the ...
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1answer
28 views

Same cell structure have isomorphic fundamental groups

Let $X$ and $Y$ be two topological spaces both with the following cell structure 2 0-cells 4 1-cells 3 2-cells 1 3-cell Can I conclude anything about the fundamental groups of $X$ and $Y$? Are ...
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1answer
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Fundamental group of tetrahedron

Consider the following double tetrahedron We glue $DCE$ to $CBA$, $CBE$ to $BDA$ and $BDE$ to $DCA$. We call the resulting space $L$. I want to find a cell-structure on $L$ with only two $0$-cells ...
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Modify the Alexander horned sphere for an embedding $S^2 \hookrightarrow ֓\mathbb{R}^3$ s.t. neither component of $\mathbb{R}^3 − S^2$ is 1-connected.

Question 2.B.6 in Allen Hatcher's Algebraic Topology page 176: Modify the construction of the Alexander horned sphere to produce an embedding $S^2 \hookrightarrow ֓\mathbb{R}^3$ for which neither ...
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Computing fundamental group of a circle

I'm very new to Topology but we just covered in class an algorithm for calculating the Fundamental Group of a surface given a triangulation. The algorithm goes as follows: 1) Find a spanning tree in ...
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1answer
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What is the fundamental group of the $3$-manifold bounded by a genus-$2$ torus?

As in the question, let $X$ be the $3$-manifold bounded by $\partial X$ which is a torus of genus $2$.What is $\pi_1(X)$? I noted that $\pi_1(\mathbb{R} ^3\backslash X) \cong \langle a, b | aba^{-1} ...
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0answers
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Monodromies of complex differential equation

Let $A:\mathbb{C}^*\to\mathrm{M}_n(\mathbb{C})$ be a holomorphic map. Consider the system of first order differential equations \begin{equation} \begin{cases} \frac{dY}{dz} = A Y\\ Y(1)= I, \end{cases}...
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1answer
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Constructing the flat vector bundle associated to a given linear representation of the fundamental group

I'm reading this notes, and I have some questions about the contruction on page 18. Let $M$ be a connected manifold and $E\rightarrow M$ a flat vector bundle over $M$. Consider $\{(U_\alpha, \phi_{...
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1answer
84 views

Fundamental group of torus knot without thickening

The calculation of the fundamental group of a $(m, n)$ torus knot $K$ is usually done using Seifert-Van Kampen theorem, splitting $\mathbb{R}^3\backslash K$ into a open solid torus (with fundamental ...
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2answers
195 views

Compute fundamental group _visually_ by the polygonal representation of the space

Some time ago, before I learnt about covering spaces and Seifert-Van Kampen theorem, I tried to compute visually the fundamental group of some spaces. For example I figure out by myself that the ...
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1answer
26 views

References for injectivity/surjectivity $\pi_i(U(n-k)) \rightarrow \pi_i(U(n))$

I have read (Theorem 29.3, line 7, pg 144) the following statement: $\pi_i(U(n-k)) \rightarrow \pi_i(U(n))$ is surjective for $i \le 2(n-k)+1$ and injective for $i \le 2(n-k)$. and ...