What is fundamental group: Definition and 44 Discussions
In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of the topological space. The fundamental group is the first and simplest homotopy group. The fundamental group is a homotopy invariant—topological spaces that are homotopy equivalent (or the stronger case of homeomorphic) have isomorphic fundamental groups. The fundamental group of a topological space
What is the way to compute ##\pi_1(PGL_2(R))##?
Is it related to defining an action of ##PGL_2(R)## on ##S^3##?
it would be helpful if you can provide me with relevant information regarding this
This space is homotopy equivalent to the complement of the three coordinate axes in ##R^3##.
This is in the chapter about the Seifert-Van Kampen Theorem, so I'm expecting to invoke that theorem.
The thing is, how should we choose our open sets such that the intersection is path connected and...
Well, for starters, ##\pi(T)##, the fundamental group of the torus, is ##\pi(S^1)x\pi(S^1)=## which is in turn isomorphic to the direct product of two infinite cyclic groups. Before I tackle the case of n connect tori with one point removed, I'm trying to just understand a torus with a point...
edit: fixed typo's andrewkirk pointed out, oops
I can cover the projective plane with 2 open sets U,V where each of these neighborhood contains the point that is missing, and the intersection of these two neighborhoods will be simply connected.
I was then hoping to invoke the Seifert-Van-Kampen...
They seem the same to me. So I can have many paths between a and b that are continuously deformable into each other while keeping the endpoints fixed. We say these function form a equivalence class [f]. This should be regardless if the endpoints are the same or not.
The fundamental group seems...
Homework Statement
Let p: E-->B be a covering map, let p(e_0)=b_0 and let E be path connected. Show that there is a bijection between the collection of right cosets of p*F(E,e_0) in F(B,b_0) (where p* is the homomorphism of fundamental groups induced by p and F(E,e_0),F(B,b_0) are the...
i: B to Y is an inclusion, p: X to Y is a covering map. Define $D=p^{-1}(B)$, we assume here B and Y are locally path-connected and semi-locally simply connected. The question 1: if B,Y, X are path-connected in what case D is path-connected (dependent on the fundamental groups)? 2 What's the...
In Chapter 7 of John M. Lee's book on topological manifolds, we find the following text on composable paths and the multiplication of path classes, [f] ... ...
Lee, writes the following:In the above text, Lee defines composable paths and then defines path multiplication of path classes (not...
Suppose we have a group with presentation G = <A|R> i.e G is the quotient of the free group F(A) on A by the normal closure <<A>> of some subset A of F(A). Is it true that that fundamental group of the Cayley graph of G (with respect to the generating set A) will be isomorphic to the subgroup...
Hello,
I'm reading a textbook and in the textbook we are discussing the fundamental group of the unit circle and having some difficulty making out what a degree of a map is and why when there is a homotopy between two continuous maps f,g from S^{1} to S^{1} why the deg(f)=deg(g)
We have...
Hi,
I am reading J.P. May's book on "A Concise Course in Algebraic Topology" and have approached the calculation where \pi_{1}(S^{1})\congZ
He defines a loop f_{n} by e^{2\pi ins}
I want to show that [f_{n}][f_{m}]=[f_{m+n}]
I understand this as trying to find a homotopy between...
So I'm revamping the question I had posted here, after a bit of work.
I'm concerned with the homomorphism induced by the inclusion of the Figure 8 into the Torus, and why it is surjective. There seem to be a lot of semi-explanations, but I just wanted to see if the one I thought of makes...
On page 333 in Section 52: The Fundamental Group (Topology by Munkres) Munkres writes: (see attachement giving Munkres pages 333-334)
"Suppose that h: X \rightarrow Y is a continuous map that carries the point x_0 of X to the point y_0 of Y.
We denote this fact by writing:
h: ( X...
Homework Statement
χ is the Riemann Surface defined by P(w, z) = 0, where P is a complex polynomial of two variables of degree 2 in w and of degree 4 in z, with no mixed products. Find the fundamental group of χ.Homework Equations
A variation of the Riemann-Hurwitz Formula states that if χ is...
I've been thinking about complex residues and how they relate to the topology of a function's Riemann's surface. My conclusion is this: it definitely tells us something, but it relates more directly to the Riemann surface of its antiderivative. Specifically:
A closed contour in the plane is...
Hi
I don't know how to attack the following question, any hints would be appreciated:
If G is a simply connected topological group and H is a discrete subgroup, then \pi_1(G/H, 1) \cong H .Thank you
James
How does fundamental group determines number of possible quantum statistics?
Why is number of possible statistics equal to number of different possible paths?
Homework Statement
Given two tori, the two-holed torus can be formed by removing the interior of a small disk from each and identifying the boundaries. Compute the fundamental group of the two torus.
Homework Equations
\pi_1(T^2) = \mathbb{Z} \times \mathbb{Z}
The van Kampen Theorem...
There is a theorem:If |K| is connected,abelianizing its fundamental group gives the first homotopy group of K.
How to abelianize a group? And how to understand this theorem more obviously?Can anyone show me an example to see it?
I myself will think this problem for more time because I...
why is S^n/S^m homotopic to S^n-m-1. the book just made this remark how do you see this geometrically.
how do you compute fundamental groups of matrices like O(3) and SO(3) or SL(2) and whatnot.
So I've read through beginning alg topology really fast and there are a lot of theorems and methods for computing fundamental groups but what are the most useful tools? When asked to compute the fundamental group what should one do? try to find a deformation retract and compute the fund group...
Hi, everyone:
Given a top space X, and a homeo. h: X--->X , we get an induced map
(by functoriality ) h_*: Pi_1(X)---> Pi_1(X) . We can also write
the map as a map g: Aut(X) --->Hom(Pi_1(X),Pi_1(X))
Is the map g always surjective.? . Almost definitely...
Compute the fundamental group of the space
X:=((S^1\times S^1) \sqcup (S^1\times S^1))/\sim
where ~ is the equivalence relation
(e^{it},e^{it}) \sim (1,e^{2it})
meaning the diagonal of the first torus is identified and wrapped around twice the second generating circles.
Call T_A the...
How do you compute the Fundamental group of the 1-skeleton of the 3-cube I^{3} = [0,1]^{3} ? What about the Fundamental group of the 1- skeleton of the 4-cube I^{4} ?
I know the Fundamental group of a space X at a point x_{0} is the set of homotopy classes of loops of X based at x_{0} . And...
The fundamental group of a torus is Z*Z,then the fundamental group of a punctured torus is Z*Z*Z.
But I've ever done a problem,it said a punctured torus can be continuously deformed into two cylinders glued to a square patch.Really?
If that is right,then the fundamental group of punctured...
In a smooth compact 3 manifold there is an embedded loop - a diffeomorph of the circle
Consider a torus that is the boundary of a tubular neighborhood of this loop.
If the loop is not null homotopic does that imply that the torus is not null homologous?
If I take a plane with n holes, would the fundamental group be that of the "bouquet of n circles"? (http://en.wikipedia.org/wiki/Rose_(topology ).) The bouquet of circles is the same as the unit line with n-1 points identified. All three spaces initially appear quite different so it would be...
I am doing some revision and trying to do fundamental groups and I was wondering if the fundamental group of the following space is {1} i.e. all loops based p are homotopic.
fundamental group of (X,p) = D^2\{(x,0) : 0<=x<=1} where p=(-1,0)
What I understand from the definition of the fundamental group is:
Pi1(X.x) is "the set of rel {0,1} homotopy classes [a] of closed paths"
Ok, when I think about one [a] it consists of all:
1.Closed paths like a and b with a(0)=a(1)=x & b(0)=b(1)=x --->since
they are closed.
2.And since...
I am reading Munkres and know exactly how to find the fundamental groups of surfaces, using pi_1 and reducing it down to simpler problems. However, I'm completely lost when looking at my final exam it says to find the fundamental groups of matrices! How do you go about doing that! There are...
I'm studying for an exam which is a couple months away and I found an old exam which asks the following:
Find the fundamental group of:
a) The closed subset in R3 given by the equation x - y^2 -z^2 = in the standard coordinates.
b) The closed subset in R3 given by the equation x - y^2 -z^2...
The Law of Biot and Savart Law tells us how to find a differential form that generates the first de Rham cohomology of S3- embedded loop. Run a steady current through the loop.
This form is just the dual of the induced magnetic field (using the Euclidean metric).
Ampere's Law tells us...
Homework Statement
Is the Fundamental group of the circle (S^1) abelian?
Not a homework question, just something I want to use.
Homework Equations
The Attempt at a Solution
Intuitively it appears to be and it is isomorphic to the additive group of integers which is abelian. I...
So I have been wondering, what is the fundamental group of a projective plane after we remove n points?
I tried doing this using Van Kampens Theorem, maybe I am applying in incorrectly, I am getting that it is the Free group on n generators.
However, when I think of RP^2 as a quotient of...
This is not important, but it's been bugging me for a while.
I'm struggling to see how the locally constant sheaves of vector spaces on X give rise to representations of the fundamental group of X.
The approach I've been thinking of is the following. Given a locally constant sheaf F on X...
How do we show that the fundamental group of the disk D^2={(x,y) in RxR: x^2 +y^2< or eq. to 1} is trivial?
I know how to show that the fundamental group of the circle is isomorphic to the group of the integers under addition, but for some reason, I don't see a way to show that the...
The question is to prove that the fundamental group of the circle S^1 is isomorphic to the group of integers under addition.
So I think I should show that the following map Phi is an isomorphism.
Phi: F(S^1, (1,0)) --> Z defined by Phi([f])= f*(1) where f* is the lifting path of f (...
Fundamental group of RP^n by recurrence!?
Homework Statement
That's it. Find the fundamental group of RP^n by recurrence.
The Attempt at a Solution
It's just obvious to me that it's Z/2 no matter n but what is this recurrence argument that I'm supposed to use?
I have a little hard time understanding the definition of a simply connected space in terms of a fundamental group. A space is simply connected if its fundamental group is trivial, has only one element?
It's been some time since I played around with homotopy. My understanding is that a set...