0
votes
0answers
35 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 ...
2
votes
1answer
25 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 $\Bbb D^n\to S^{n-1}$ ...
3
votes
1answer
51 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
votes
1answer
70 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
votes
3answers
121 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
votes
2answers
76 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 ...
1
vote
1answer
45 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 ...
0
votes
1answer
48 views
4
votes
1answer
76 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 ...
9
votes
1answer
104 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 ...
2
votes
1answer
61 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 ...
1
vote
1answer
30 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 ...
40
votes
6answers
7k views

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 ...
3
votes
1answer
58 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 ...
4
votes
0answers
30 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 ...
2
votes
2answers
43 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 ...
2
votes
1answer
78 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
votes
1answer
81 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 ...
3
votes
2answers
162 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 ...
0
votes
0answers
21 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 ...
7
votes
1answer
115 views

Isomorphism Finite Topological Space

Does there exist a finite topological space with fundamental group isomorphic to $\mathbb{Z_2}$?
3
votes
1answer
30 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 ...
3
votes
2answers
124 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 ...
2
votes
4answers
156 views

A simple fundamental group

I think I have this one, but I want to make sure: Using Van Kampen's theorem, Find the fundamental group of two disjoint spheres with each north pole identified, and each south pole identified. The ...
0
votes
1answer
57 views

Show $\otimes$ and $*$ are the same operation on $\pi_1(G, x_0)$

Show $\otimes$ and $*$ are the same operation on $\pi_1(G, x_0)$ where $(f\otimes g)(s) = f(s) \cdot g(s)$ where $\cdot$ is the group operation on the topological group $G. $ This is a question from ...
0
votes
2answers
71 views

Prove Simply Connected

If $X = U \cup V$ with $U,V$ open and simply connected and $U \cap V$ is path connected, why is $X$ simply connected?
3
votes
1answer
62 views

Is Orbit Criterion an abstract nonsense? Different induced fundamental groups.

(Orbit Criterion) Let $p:\tilde X \to X$ be a covering map. If $\tilde q, \tilde q' \in \tilde X$ are two points in the same fiber $p^{-1}(q)$, there exists a covering transformation taking $\tilde ...
2
votes
1answer
67 views

Fundamental group: Attaching a disc to a path connected triangulable space

What's the effect on the fundamental group of a path connected triangulable space X if you attach a disc? So far I've figured out that a disc has trivial fundamental group. So the free product of ...
0
votes
0answers
62 views

exercise 17 of hatcher page 80 chapter 1.3

Given a group $G$ and a normal subgroup $ N$, show that there exists a normal covering space $\widetilde{X} \rightarrow X $ with $\pi_{1}(X)\approx G ,\pi_{1}(\widetilde{X})\approx N $, and deck ...
2
votes
1answer
101 views

nice space with wild fundamental group

I would like to know an example of nice space with very strange fundamental group. With simplices and similar things I only get finitely presented groups. Edit. I know from comments that Hawaiian ...
0
votes
0answers
27 views

is there a specific way to find deck transformation and its related group?

is there a specific way to find deck transformation and its related group? this question came to my mind at the first time I studied deck transformation and related topics of covering space. it ...
3
votes
2answers
112 views

$G$ is Topological $\implies$ $\pi_1(G,e)$ is Abelian

Hypothesis: Let $G$ be a topological group with identity element $e$. Let $\mu$ denote the multiplication mapping in $G$. Goal: Show that $\pi_1(G,e) = \pi(G)$ is an abelian group via the hint ...
0
votes
0answers
18 views

why $i_{1_{*}}(r)=aba^{-1}b^{-1}$?

my question is about this example,I just didn't understand why $i_{1_{*}}(r)=aba^{-1}b^{-1}$? I want to visualize how the generator $r$ changed to $aba^{-1}b^{-1}$ in $S^{1} \vee S^{1}$ but I ...
1
vote
0answers
39 views

Fundamental groups of configuration spaces

In a previous answer see here by Samuel Reid, I read the following: "The configuration space of $n$ points in a topological space $X$ is usually defined to be, $$C_{\hat{n}}(X) = \{(z_{1},...,z_{n}) ...
1
vote
3answers
47 views

Why is $\pi_1(\Bbb{R}^n,x_0)$ the trivial group in $\Bbb{R}^n$?

My Algebraic Topology book says Let $\Bbb{R}^n$ denote Euclidean n-space. Then $\pi_1(\Bbb{R}^n,x_0)$ is the trivial subgroup (the group consisting of the identity alone). I wonder why that is. ...
2
votes
0answers
30 views

problem to find open subset that contains sphere for wedge of two spheres.

I want to use van kampen theorem to show that the fundamental group of wedge union of two spheres is zero,I know that I must choose $U$ and $V$ which they are open and its union is the whole space.I ...
2
votes
2answers
76 views

Computation of fundamental group of pseudo circle

A common example of a weak homotopy equivalence which isn't symmetric is the pseudo circle $\mathbb{S}$. Wikipedia gives the following map $f\colon S^{1}\rightarrow\mathbb{S}$ ...
3
votes
3answers
153 views

how should I show that it is wedge of infinite circles?

I know that the shape that we see it below is homotopy equivalent of wedge of infinite circles,so the fundamental group of it is $\prod _{1}(\vee _{\alpha \in A}S^{1})=\ast _{\alpha \in A ...
0
votes
0answers
70 views

Fundametal Group and Étale Fundamental Group

For what kind of schemes the fundamental group and the étale fundamental group coincide? Is there a relation between this groups on a variety? I'm interested on toric varieties (more generally, ...
2
votes
2answers
103 views

The fundamental group of a topological group is abelian

I want to show the fundamental group of a topological group is abelian. In fact, the question says the topological group is path connected. I do not know where I should use path-connectedness. I ...
1
vote
1answer
75 views

Prove fundamental group is the direct product

Suppose that $A$ is a retract of $X$ with retraction $r : X \rightarrow A$. Also suppose that $i_*(\pi(A,a))$ is a normal subgroup of $\pi(X,a)$. Prove that $\pi(X,a)$ is the direct product of the ...
4
votes
3answers
267 views

Fundamental group of Hawaiian earring

I am trying to understand how the fundamental group of the infinite shrinking wedge of circles is $G=\prod_{i=1}^\infty\mathbb{Z}$. I understand that it is something more than ...
2
votes
2answers
100 views

Fundamental group of Poincaré sphere

Do the two presentations below, $$G=\langle d,v \mid dv^2d=vdv, dv^3d=v^2 \rangle$$ and $$\langle r,s,t \mid r^2=s^3=t^5=rst \rangle = \langle s,t \mid (st)^2=s^3=t^5 \rangle,$$ define the same group? ...
4
votes
2answers
97 views

Independence of the path implies that the fundamental group is abelian

a) Prove that two paths $f,g$ from $x$ to $y$ give rise to the same isomorphism from $\pi(X,x)$ to $\pi(X,y)$ (i.e. $u_f=u_g)$ if and only if $[g*\bar{f}] \in Z(\pi(X,x))$. b) Let $u_f: \pi(X,x) ...
0
votes
1answer
118 views

Non-homotopy equivalent spaces with isomorphic fundamental groups

I know that if two spaces $X,Y$ have the same fundamental group : $\pi_1(X,a) \cong \pi_1(Y,b) $ does not imply that they are homotopy equivalent. But I can't find an example. I was thinking about ...
2
votes
0answers
147 views

Fundamental group of CW-Complex only depends on 2-Skeleton

I was just about to write down my answer to an exercise in algebraic topology and I wanted to use the fact that $\pi_1 (X)$ only depends on the $2$-Skeleton of $X$ for any $CW$-Complex $X$. I am very ...
0
votes
1answer
105 views

cohomology groups of K(Zp x Zp, 1)

I have a question regarding the cohomology groups of the Eilenberg-MacLane space $K(\mathbb{Z}_p \times \mathbb{Z}_p,1)$. For $n$ > $2$, is there a way to show that $H^n(K(\mathbb{Z}_p \times ...
3
votes
2answers
379 views

Fundamental group of projective plane is $C_{2}$???

I just recently know that there are topology with finite nontrivial fundamental group (homotopy curve). I just can't wrap my mind around it at all. If you have a curve, and somehow cannot shrunk it ...
2
votes
2answers
258 views

Dunce Cap triangle homotopy equivalent to $S^1$

Why is Dunce Cap triangle homotopy equivalent to $S^1$. Yes, maybe it is not the correct expression, but i mean that the dunce cap can be represented as a quotient space of a triangle by ...
1
vote
1answer
119 views

Any finitely presented group is the fundamental group of some topological space?

I have come across a problem in topology as described in the title. Here is my intuitive construction with analogy to the fundamental group of closed surfaces. Let $G=(a_1,a_2,\dots,a_n \mid ...