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

In summary, The fundamental group of a space can be defined as all loops based at a point that are homotopic. In the case of the space X = D^2\{(x,0) : 0<=x<=1} with base point p=(-1,0), the fundamental group is {1} as all loops are homotopic to the constant map c(t)=p. For nice subsets of \mathbb{R}^2, the fundamental group is trivial if the space has no holes. To show that the fundamental group of a product of topological spaces is isomorphic to the product of fundamental groups, an isomorphism can be constructed by projecting paths onto each space and showing that these are hom
  • #1
Mikemaths
23
0
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)
 
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  • #2
Yes. Just consider a loop [itex]f : [0,1] \to X[/itex]. This is homotopic to the constant map c(t)=p by the usual straight-line homotopy [itex](t,i) \mapsto (1-i)f(t) + ip[/itex]. You can verify yourself that this works.

The intuitive idea when dealing with fairly nice subsets of [tex]\mathbb{R}^2[/tex] is that the fundamental group is trivial if and only if the space has no holes (since then you can wrap a loop around that hole).
 
  • #3
Ok thanks I get that as I thought.

I am trying to show that the fundamental group of a product of Topological Spaces is isomorphic to the product of fundamental groups:

pi1(X x Y , (p,q)) -> pi1(X,p) x pi1(Y,q)

I can understand this but want to construct the isomorphism and am struggling?
 
  • #4
Just construct it in the obvious way. Given two loops [itex]f_1 : I \to X[/itex] and [itex]f_2 : I \to Y[/itex] you can construct a path [itex]I \to X \times Y[/itex] where [itex]t \mapsto (f_1(t),f_2(t))[/itex]. Conversely given a path [itex]f : I \to X \times Y[/itex] you can construct paths [itex]I \to X[/itex] and [itex]I \to Y[/itex] by projecting onto X and Y. You can show that both of these give homomorphisms of the fundamental groups and that they are inverses.
 
  • #5


The fundamental group of (X,p) is indeed {1}, meaning that all loops based at p are homotopic. This is because the space X is homotopy equivalent to a point, as it can be continuously deformed to a single point without leaving the space. Therefore, any loop in X based at p can be continuously deformed to a constant loop, which is homotopic to the trivial loop.
 

1. What is the Fundamental Group of (X,p)?

The Fundamental Group of (X,p) is a mathematical concept that describes the topological structure of a space X around a point p. This group is denoted by π1(X,p) and is defined as the set of all possible loops that can be formed around the point p in X, up to homotopy equivalence.

2. How is the Fundamental Group of (X,p) calculated?

The Fundamental Group of (X,p) is calculated using the fundamental group theorem, which states that the group is isomorphic to the quotient group of the fundamental group of X and the normal subgroup of all loops that start and end at the point p. This can be further simplified using Van Kampen's theorem, which states that the fundamental group can be calculated by taking the free product of the fundamental groups of the individual open sets of X.

3. What is the significance of the Fundamental Group of (X,p) in topology?

The Fundamental Group of (X,p) is a fundamental tool in topology that helps to classify and distinguish topological spaces. It is an important invariant that remains unchanged even when a space is deformed or distorted, making it useful in understanding the topological properties of a space.

4. Can the Fundamental Group of (X,p) be used to determine the shape of a space?

Yes, the Fundamental Group of (X,p) can be used to determine the shape of a space. For example, if the group is trivial (i.e. contains only the identity element), then the space is simply connected and is topologically equivalent to a point. On the other hand, if the group is non-trivial, then the space is not simply connected and has a more complex shape.

5. Are there any applications of the Fundamental Group of (X,p) in other fields of science?

Yes, the Fundamental Group of (X,p) has applications in various fields such as physics, computer science, and biology. In physics, it is used to study the topology of spacetime and in computer science, it is used in the analysis of algorithms and data structures. In biology, it is used to study the structure and function of biological molecules.

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