Calculate Fundamental Groups of X_1-X_5

• latentcorpse
In summary, the conversation discusses the calculation of fundamental groups for different spaces, including X_1, X_2, X_3, X_4, and X_5. The definition of a fundamental group is mentioned, and the speaker expresses uncertainty about understanding it. They also mention seeking help from a coursemate and provide their own answers for the first three spaces. The speaker also asks for confirmation of their answers for the fourth and fifth spaces, expressing their understanding that the fundamental group is trivial for X_4 and isomorphic to \mathbb{Z} for X_5.
latentcorpse
I need to calculate the fundamental group of the following spaces:

$X_1 = \{ (x,y,z) \in \mathbb{R}^3 | x>0 \}$
$X_2 = \{ (x,y,z) \in \mathbb{R}^3 | x \neq 0 \}$
$X_3 = \{ (x,y,z) \in \mathbb{R}^3 | (x,y,z) \neq (0,0,0) \}$
$X_4 = \mathbb{R}^3 \backslash \{ (x,y,z) \in \mathbb{R}^3 | x=0,y=0, 0 \leqslant z \leqslant 1 \}$
$X_5 = \mathbb{R}^3 \backslash \{ (x,y,z) \in \mathbb{R}^3 | x=0, 0 \leqslant y \leqslant 1 \}$

I fundamentally do not understand what a fundamental group is of how to calculate it. I have read the notes on this but they are so so abstract.

What is the definition of a fundamental group?

ok. i think i have answers for the first 3 that I am happy with (ive discussed this with a coursemate)

the 4th one i believe the fundamental group is trivial as any path can be contracted to a point. is this correct?

and the 5th one is R£ with a "sheet" removed is was going to say that we can pull the space in around the sheet making a rectangle taht can then be deformed into a circle. this means the fundamental group of X5 is isomoprhic to that of S1 i.e. it is $\mathbb{Z}$. is this correct?

thanks.

e(ho0n3 said:
What is the definition of a fundamental group?

thanks. could you take a look at my above post please?

1. What is the purpose of calculating the fundamental groups of X_1-X_5?

The fundamental group is a mathematical concept used to study the connectivity and topological properties of a space. Calculating the fundamental groups of X_1-X_5 allows us to understand the fundamental structures and properties of these spaces, which can then be applied to various fields such as physics, engineering, and computer science.

2. How is the fundamental group of X_1-X_5 calculated?

The fundamental group of a space is calculated by determining all possible loops in the space and then classifying them based on their homotopy equivalence. This involves identifying all continuous deformations of a loop that can be continuously transformed into each other without breaking or gluing any points along the way.

3. What types of spaces can be used to calculate the fundamental group?

The fundamental group can be calculated for any topological space, including X_1-X_5. However, it is most commonly used for spaces that are path-connected and locally path-connected, meaning that any two points in the space can be connected by a path and that the space has a neighborhood for every point that is path-connected.

4. How can the fundamental groups of X_1-X_5 be applied in real-world situations?

The fundamental groups of X_1-X_5 can have various applications in real-world situations. For example, in physics, they can be used to understand the topology of spacetime, while in engineering, they can be used to analyze the deformation and stability of structures. In computer science, they can be used to study the connectivity and communication networks.

5. Are there any limitations or assumptions when calculating the fundamental groups of X_1-X_5?

Yes, there are some limitations and assumptions when calculating the fundamental groups. One limitation is that the space must be able to be described by a set of points and their connections, which may not always be possible for more complex spaces. Additionally, calculations may become more complicated for non-simply connected spaces, where there are multiple fundamental groups to consider. Assumptions may also be made about the space, such as it being locally path-connected or simply connected, which may not always hold true in reality.

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