Fundamental Group of Quotient Space

In summary, the fundamental group of quotient space is a mathematical concept used in topology to study spaces obtained by "gluing" smaller spaces together. It is calculated by taking the direct product of the fundamental groups of these smaller spaces. The significance of this group lies in its ability to classify and distinguish spaces based on their topological properties and study their connectivity and shape. It can also be used to prove properties of a space and has various applications in mathematics, physics, and engineering.
  • #1
James4
14
0
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 [itex]\pi_1(G/H, 1) \cong H [/itex].Thank you

James
 
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  • #2
Do you know how to find the fundamental group of [itex]S^1[/itex] (using covering spaces). Can you adjust the proof a little bit such that it holds in this case??
 

FAQ: Fundamental Group of Quotient Space

1. What is the fundamental group of quotient space?

The fundamental group of quotient space, also known as the fundamental group of a quotient manifold, is a mathematical concept used in topology to study the properties of spaces that have been obtained by "gluing together" smaller spaces. It is a group, which means it follows the rules of group theory, and is used to classify and distinguish spaces from each other.

2. How is the fundamental group of quotient space calculated?

The fundamental group of quotient space is calculated using the fundamental group of the smaller spaces that were "glued" together to create the quotient space. This is done by taking the direct product of the fundamental groups of the smaller spaces.

3. What is the significance of the fundamental group of quotient space?

The fundamental group of quotient space is significant because it helps classify and distinguish spaces based on their topological properties. It also provides a way to study the connectivity and shape of a space, which can have implications in fields such as physics and engineering.

4. Can the fundamental group of quotient space be used to prove properties of a space?

Yes, the fundamental group of quotient space can be used to prove properties of a space, such as whether it is simply connected or not. It can also be used to determine if a space is homeomorphic to another space.

5. Are there any applications of the fundamental group of quotient space?

Yes, the fundamental group of quotient space has various applications in mathematics, physics, and engineering. It is used in the study of knot theory, homotopy theory, and algebraic topology. It also has applications in robotics, computer graphics, and computer vision.

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