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


{\displaystyle X}
is denoted by




{\displaystyle \pi _{1}(X)}

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  1. A

    A About calculating a fundamental group

    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
  2. PsychonautQQ

    A Fundamental group of a sphere with 6 points removed

    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...
  3. PsychonautQQ

    A Fundamental group of n connect tori with one point removed

    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...
  4. PsychonautQQ

    A Fundamental group of Project Plane with 2 points missing

    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...
  5. FallenApple

    I Homotopy Class vs Fundamental Group.

    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...
  6. PsychonautQQ

    Fundamental Group Coset to preimage bijection

    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...
  7. 1

    A The fundamental group of preimage of covering map

    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...
  8. D

    Proving the Fundamental Group of SO(2) is Z: How Can it Be Done Explicitly?

    Good morning. I was wondering how do you prove explicitly that the fundamental group of SO(2) is Z?
  9. Math Amateur

    Multiplication of Path Classes and the Fundamental Group

    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...
  10. M

    Fundamental Group of a Cayley Graph

    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...
  11. T

    A question on degrees of maps of the fundamental group of the unit circle

    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...
  12. T

    Showing the Fundamental Group of S^1 is isomorphic to the integers

    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...
  13. S

    Fundamental Group of the Torus-Figure 8

    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...
  14. Math Amateur

    Algebraic Topology - Fundamental Group and the Homomorphism induced by h

    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...
  15. P

    Find the fundamental group of a Riemann Surface

    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...
  16. A

    Residues and the fundamental group

    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...
  17. J

    Fundamental Group of Quotient Space

    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
  18. V

    How is the fundamental group related to quantum statistics?

    How does fundamental group determines number of possible quantum statistics? Why is number of possible statistics equal to number of different possible paths?
  19. J

    Fundamental Group of Genus 2 Surface

    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...
  20. S

    Hi,i want to understand how fundamental group of a closed oriented

    Hi, i want to understand how fundamental group of a closed oriented 3-mfd determines all its homology and cohomology gorups. Please can you help me.
  21. S

    Fundamental Group of lens space

    Hi i want to see why the fundamental group of lens space L(p,q) is Z_p. Can you help me?
  22. K

    How to abelianizing the fundamental group?

    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...
  23. M

    How can you geometrically see the homotopy between S^n/S^m and S^n-m-1?

    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.
  24. M

    Fast Computation of Fundamental Groups: Practical Methods and Tricks

    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...
  25. B

    Ontoness and Induced Maps on Fundamental 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...
  26. quasar987

    What is the fundamental group of X?

    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...
  27. M

    Algebraic Topology: Fundamental group of a cube

    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...
  28. K

    What's the fundamental group of a punctured torus?

    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...
  29. L

    Fundamental group to second homology group

    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?
  30. M

    Fundamental group with n holes

    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...
  31. M

    Fundamental Group of (X,p): D^2\{(x,0) : 0<=x<=1}

    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)
  32. Z

    Understanding the Fundamental Group: Exploring Pi1(X.x) and Its Definition

    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...
  33. C

    How Do You Determine the Fundamental Groups of Specific Matrix Spaces?

    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...
  34. C

    Finding Fundamental Group: Step-by-Step Guide

    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...
  35. W

    Current Loops and fundamental group

    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...
  36. F

    Is the Fundamental group of the circle abelian?

    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...
  37. T

    Fundamental Group

    Can someone please explain this to me. Let X = fundamental group of a genus 2 surface Let Y = fundamental group of a genus 1 surface Can X embed Y? Ty
  38. G

    Fundamental Group of the projective plane after we remove n points?

    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...
  39. O

    Representations of the Fundamental Group

    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...
  40. M

    The fundamental group of the disk is trivial, why?

    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...
  41. M

    Fundamental group of the circle S^1

    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 (...
  42. quasar987

    Fundamental group of RP^n by recurrence?

    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?
  43. quasar987

    Is the Fundamental Group of a Pointed Space Dependent on the Base Point?

    Simple question: is the fundamental group of a pointed space independant of the base point?
  44. W

    Simply Connected and Fundamental Group

    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...