For questions about or involving the fundamental group.

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Explicitly building Top. space with trivial homology group and non trivial fundamental group, w/o CW-complexes

I know this is a pretty famous question here, but I was asked to show explicitly such space, during a bachelor lecture, without using any CW-complex result. I started working using some covering ...
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1answer
45 views

Fundamental group of a genus-2 surface

I want to calculate the fundamental group of a genus-2 surface, i.e. a double torus. Using Van-Kampen I obtain ( with the notation generators- relations) $$\Pi_1(X,p) = < \alpha, \beta, \alpha_1, ...
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1answer
34 views

Finding the fundamental group of the torus

I'm trying to prove that the fundamental group of the torus is $\Bbb Z \times \Bbb Z$, and I've been checking different questions on this site, however in many of the questions people use the ...
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1answer
47 views

Fundamental group of $GL(n, \mathbb{C}) $

I want to prove that the fundamental group of $GL(n, \mathbb{C}) $ is infinite. I don't know how to proceed, any hint ?
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1answer
32 views

$\mathbb{R}P^2$ and its fundamental group by identification of edges of unity square

Suppose we identify edges of the the unity square $[0, 1] \times [0, 1]$, as in the picture: http://de.wikipedia.org/wiki/Datei:ProjectivePlaneAsSquare.svg Now to compute the fundamental group, for ...
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1answer
58 views

Showing that Möbius band can't retract onto the boundary circle

Trying to prove that Möbius band can't retract onto the boundary circle, I got an idea that I must show the below thing. If $\alpha\in\pi_1 (\partial M)$ is a generator, its image $i^*(\alpha) \in ...
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1answer
36 views

Given a group $G$, the existence of a space such that $\pi_1(X)\simeq G$.

I'm having trouble understanding Corollary 1.28 of Hatcher which proves that for every group $G$ there is a space $X_G$ such that $\pi_1(X_G)=G$. Since $G$ is the quotient of a free group $F$, we ...
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1answer
58 views

Every Group is a Fundamental Group

I am studying elementary Algebraic Topology recently. I have seen that a topological space is identified with a group. We are telling the group as Fundamental Group. So every topological space $X$ and ...
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1answer
40 views

Map from $\mathbb{R}\mathbb{P}^{2}$ to Klein bottle homotopic to constant map

I'm currently struggling with the following exercise: Show that any continuous map $$f: \mathbb{R}\mathbb{P}^{2} \to K$$ where $K$ is the Klein bottle, is homotopic to a constant map. I know that ...
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1answer
72 views

Fundamental group of the complement of an eight shape

What is $\pi_1(\mathbb{R}^3 \setminus (S^1 \vee S^1))$? I would say that the space deformation retracts to a sphere $S^2$ surrounding the missing eight with two sticks stuck in the two loops of the ...
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2answers
38 views

How do we prove that the fundamental group is a group?

My understanding of the fundamental group is that it's the set of all loops starting and ending at a point $x_0$ in a space $X$, along with the operation of composition. For it to be a group, ...
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lifting loops on surface with abelian fundamental group for decrease their self-intersection number

I found a proof of following statement (which is available here ), but I'm not sure if we must assume that fundamental group of surface $M$ is non-abelian: Lemma: Let $M$ be a compact orientable ...
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1answer
37 views

Classical presentation of fundamental group of surface with boundary

It is well known fact about fundamental group of orientable compact surface: Letting $g$ be the genus and $b$ the number of boundary components of surface $M$. There is a generating set ...
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3answers
66 views

Hatcher, p.35, $\mathbb{R^{n}} - \{x\}$ is homeomorphic to $S^{n-1} \times \mathbb{R}$

Could some help me understand how $\mathbb{R^{n}} - \{x\}$ is homeomorphic to $S^{n-1} \times \mathbb{R}$. Also, if the one point set is replaced by any finite set, how does the argument work. This ...
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1answer
71 views

The nature of isomorphism between fundamental groups with different base points

New to algebraic topology. Munkres (Topology, 2 ed.) in the last paragraph on page 332 says that "If $X$ is path-connected, all the groups $\pi_1(X,x)$ are isomorphic, so it is tempting to try to ...
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1answer
39 views

Does the Wirtinger presentation extend to compliments of graphs and links?

In a previous question I asked about a specific fundamental group problem, which was resolved via SVK but I was also interested in whether or not the Wirtinger presentation was valid in some way. In ...
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3answers
112 views

Fundamental group of complement of circle with a line passing through it.

Let $X= S^1 \cup y$-axis in the $xy$-plane. Compute the fundamental group of $\mathbb{R}^3 - X$ Edit: To clarify, $S^1 =\{(x,y)|x^2 + y^2 = 1\}$ is the unit circle in the $xy$-plane, so that the ...
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2answers
62 views

what is the fundamental group of a torus with $k$ points removed

I was about to use the Seifert-van Kampen to compute the fundamental group of a torus of genus 2. In the process, I need to know the fundamental group of a torus (of genus one) with a hole removed. is ...
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0answers
43 views

Is there fundamental group “with coefficients”

Constructing fundamental group in usual way, one uses $I=[0,1] \in \mathbb R$, so uses real numbers, and gets a group classifying coverings for nice spaces: locally connectible and so on, in fact ...
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1answer
39 views

Fundamental group of the quotient of a cylinder by a rotation at either end

The following is a past qual problem: let $X = S^1 \times [0,1]$ be the cylinder, and define an equivalence relation on $X$ by $(z,1) \sim (iz,1)$ and $(w,0) \sim (e^{i\pi /7} w, 0)$. Compute ...
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1answer
107 views

An algebraic topology proof of a result from analysis

A colleague of mine recently brought up the following result from real analysis: Theorem: If $f:\mathbb{R}^2\to\mathbb{R}^2$ is continuous and $|f(x)-f(y)|\geq |x-y|$ for all $x,y$, then $f$ is onto. ...
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1answer
79 views

Fundamental group of quotient of $S^1 \times [0,1]$

I have a past qual question here: Let $X = S^1 \times [0,1] /{\sim}$, where $(z,0) \sim (z^4,1)$ for $z \in S^1 = \{ z \in \mathbb{C} \colon \| z \| = 1 \}$. Compute $\pi_1(X)$. I've been trying to ...
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38 views

How to describe a homomorphism from a fundamental group to a finite group?

Let $S$ be a connected locally noetherian scheme with $s$ a geometric point of $S$. I read something like this: to give a surjective continuous homomorphism from $\pi_1(S, \bar{s})$ to a finite group ...
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1answer
31 views

Fixed point equivalence [duplicate]

Let $\Bbb D^n$ be n-dimensional ball and $S^{n-1}$ the $n-1$ dimensional sphere realized as boundary of $\Bbb D^n$. Prove that following are equivalent. There is no retraction $\Bbb D^n\to S^{n-1}$ ...
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1answer
65 views

For compact surface $M$ and loop $f$ there exists a surjection $\phi: H_1(M) \to \mathbb{Z}/8\mathbb{Z}$ such that $f \notin \ker(\phi)$

Why is this sentence true? For every not nullhomologous loop $f$ without selfintersections on orientable compact surface $M$ there exists a surjection $\phi: H_1(M) \to \mathbb{Z}/8\mathbb{Z}$ such ...
4
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1answer
89 views

Fundamental Group of a Hexagon with Edge Identifications

What's the easiest way to compute this thing's fundamental group? I've been playing with it for a little while, and I'm getting $\mathbb{Z}+\mathbb{Z}$. After making the ID's I think the 1-cells ...
4
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3answers
145 views

Do freely homotopic maps induce the same homomorphism on fundamental groups?

Let $f,g\colon X\to Y$ be two continuous maps that are freely homotopic, such that there is some $x_0\in X$ with $f(x_0)=g(x_0)$. Is it true that the induced homomorphisms $f_*,g_*\colon ...
4
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2answers
84 views

Funky Fundamental Group Question

Let $D$ be a closed disk (w/ boundary $C$) and let $D_a$, $D_b$ be two disjoint closed disks in the interior of $D$ (w/ boundaries $C_a$ and $C_b$, resp.) . Now remove the interiors of $D_a$ and ...
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1answer
57 views

the fundamental group of punctured surface

Let $S_{g,m}$ be a surface of genus $g$ with $m$ punctured, we know the fundamental group of $S_{g,0}$ is $$ \pi_1(S_{g,0}) = \left\langle a_1, b_1, \dots, a_g, b_g {~\large\mid~} [a_1, b_1] \dots ...
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1answer
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4
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1answer
93 views

Homology and Fundamental group of $\mathbb{R}^4\setminus S^1$

I was attempting to help someone with this problem and realized I could not solve it myself. Let $S^1=\{(x,y,0,0):x^2+y^2=1\}$ be the unit circle in $\mathbb{R}^4$ and consider $M=\mathbb{R}^4 ...
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1answer
118 views

Is the homotopy type of an aspherical space determined by its fundamental group?

Question: Let $X$ and $Y$ be path-connected spaces that admit a contractible universal cover, with $\pi_1(X) \cong \pi_1(Y)$. Is $X$ homotopy equivalent to $Y$? Comments: $X$ and $Y$ are both ...
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1answer
78 views

Fundamental group of a connected graph

Is this a legit way to prove that a fundamental group of a connected graph $\Gamma$ is a free group? Without using quotient and homotopy extension property from Hatcher's "Algebraic Topology": Take ...
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0answers
21 views

Dual of etale fundamental group, field functoriality

Suppose $k$ is an arbitrary field and $X/k$ is a finite-type and integral scheme (hence connected). Let $K$ be the fraction field of $X$. Via the canonical morphism $$\phi: \rm{Spec}(K)\rightarrow ...
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1answer
41 views

Determing if the fundamental group of the following is isomorphic to either the trivial, infinite cyclic, figure eight fundamental groups

Hello there i am having trouble to determine isomorphisms of the following fundamental groups: 1) the torus $T$ with a removed point. 2) $\mathbb{R}^3$ with nonnegative axes 3) $S^1 \cup ...
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6answers
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Hanging a picture on the wall using two nails in such a way that removing any nail makes the picture fall down

A friend of mine told me that it's possible to hang a picture on the wall from a string using two nails in such a way that removing either of the two nails will make both the string and picture fall ...
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2answers
73 views

Showing that $\pi(G/H, 1) = H$ under a condition

Problem: Let $G$ be a simply connected (i.e., $\pi(G)=1$) topological group, and let $H$ be a discrete normal subgroup. Prove that $\pi(G/H,1) = H$. I know that since $H$ is a discrete subgroup ...
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0answers
31 views

Isomorphism of the etale fundamental group

Given a birational proper morphism $f\colon X \rightarrow Y$ of complex algebraic varieties. It is always true that $f^* \colon \pi^{et}_1 (X)\rightarrow \pi^{et}_1(Y)$ is an isomorphism? I think ...
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2answers
46 views

Higher Homotopy Group of a Product of Spaces

I want to show that for two toplogical spaces $ X_1,X_2$ and for $x_1\in X_1 , x_2 \in X_2$ we have an isomorphism between $\pi_n (X_1 \times X_2 , (x_1,x_2)) $ and $ \pi_n (X_1, x_1) \times \pi_n ...
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52 views

Isomorphism of the fundamental groups

Given two complex algebraic varieties $X$ and $Y$. If there exists a birational proper morphism $f\colon X\rightarrow Y$ then a Theorem of Grothendieck (SGA.X) say that $\pi_1^{et}(X)\cong ...
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1answer
53 views

Van Kampen theorem on n-manifold remove one point

I'm trying to prove for a ($n \geq 3$)-dimensional path connected manifold M, if remove a point in M, then use Van Kampen theorem can somehow show the fundamental group of original $M$ and $M-{x}$, ...
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1answer
125 views

Homotopy quantum field theories as functors

A homotopy quantum field theory is a symmetric monoidal functor $\tau:\mathrm{HCobord}(n,X)\to\mathrm{Vect}_{\mathbb{K}}$, with $X$ a path connected space with basepoint $\ast$. There is the following ...
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1answer
106 views

How to compute the fundamental group from first homology group?

I have been reading about the fundamental group and its connection to the first homology group. In fact, there is an isomorphism $$\pi_1^{ab}(X,x_0) \to H_1(X)$$ for every path-connected topological ...
2
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1answer
88 views

The fundamental group of $U(n)/O(n)$

Since $O(n)$ is a subgroup of $U(n)$, we can consider the quotient space $L_n=U(n)/O(n)$. The quotient space is homotopic to the ($n$-th) Lagrangian Grassmannian, and it is known that its fundamental ...
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2answers
172 views

What is the suspension used in the Freudenthal suspension theorem?

The theorem states: The suspension map $\pi_{i}( S^{n})\rightarrow \pi_{i}(S^{n+1})$ is an isomorphism when $i<2n-1$ and a surjection when $i=2n-1$. In the case where $X$ is an ...
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0answers
23 views

How to prove homomorphisms with 'lifting'

for topology i just started a chapter called lifting and i'm having trouble using this concept to prove a homomorphism of a lifting correspondance. This is my following question: let $m \in ...
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1answer
118 views

Isomorphism Finite Topological Space

Does there exist a finite topological space with fundamental group isomorphic to $\mathbb{Z_2}$?
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Gromov-Hausdorff convergence to a circle

I am working on the book A course in metric geometry written by D. Burago, Y. Burago and S. Ivanov, and more precisely on exercice 7.5.9: Exercice: Let $\{X_n\}$ be a sequence of compact length ...
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1answer
31 views

Appropriateness of using an interval in $\Bbb{R}$ as the parameter to a continuous path in space $X$.

The fundamental group $\pi_1(X,x)$ is usually defined using continuous paths $p : [0,1] \to X$. But... (1) Can you use other spaces besides a closed interval of $\Bbb{R}$ in the usual topo? (2) How ...
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2answers
134 views

Spaces with fundamental group $\mathbb{Z}$

Let $X=A\cup B$ be an open cover of $X$ where $A,B$ are simply connected and $A\cap B$ consists of 2 simply connected components $C_1, C_2$. Show that $\pi_1(X)=\mathbb{Z}$. I tried different ...