Hatcher 1.3.25: Finding Π 1 \pi_1 Π 1 ​ Of The Quotient Of The Punctured Place By A Hyperbolic Z \mathbb{Z} Z -action

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Introduction

In this article, we will delve into the world of algebraic topology and explore the concept of the fundamental group of a quotient space. Specifically, we will be working with the quotient of the punctured plane under a hyperbolic Z-action, as described in Problem 1.3.25 of Hatcher's "Algebraic Topology". This problem requires us to find the fundamental group of the quotient space, denoted as Y, which is obtained by applying the Z-action to the punctured plane X = ℝ² \ {0}. Our goal is to understand the properties of this quotient space and determine its fundamental group.

The Punctured Plane and the Hyperbolic Z-Action

The punctured plane X = ℝ² \ {0} is a topological space that consists of the Euclidean plane with a single point removed. This space is often used as a model for the hyperbolic plane, which is a non-Euclidean geometry. The hyperbolic Z-action on X is generated by the map (x, y) ↦ (2x, y/2), which stretches the x-coordinate by a factor of 2 and shrinks the y-coordinate by a factor of 1/2. This action is a type of hyperbolic motion, which preserves the hyperbolic metric on X.

The Quotient Space Y

The quotient space Y is obtained by applying the Z-action to the punctured plane X. This means that we identify points in X that are related by the Z-action, i.e., points (x, y) and (2x, y/2) are considered equivalent. The resulting space Y is a topological space that has a similar structure to the punctured plane, but with a different topology.

Finding the Fundamental Group of Y

The fundamental group of a topological space is a fundamental concept in algebraic topology that encodes information about the connectedness and holes of the space. In this case, we want to find the fundamental group of the quotient space Y. To do this, we need to understand the properties of the Z-action and how it affects the topology of X.

The Covering Space X

The punctured plane X is a covering space of the quotient space Y, meaning that there is a continuous map from X to Y that is a local homeomorphism. This map is often referred to as the covering map. The covering map is a key tool in understanding the topology of Y, as it allows us to transfer information from X to Y.

The Fundamental Group of X

The fundamental group of the punctured plane X is a well-known result in algebraic topology. It is isomorphic to the free group on two generators, denoted as F₂. This means that the fundamental group of X is generated by two elements, often denoted as a and b, subject to the relation ab = 1.

The Fundamental Group of Y

Using the covering space X and the fundamental group of X, we can determine the fundamental group of Y. Since X is a covering space of Y, the fundamental group of Y is a quotient group the fundamental group of X. Specifically, the fundamental group of Y is isomorphic to the quotient group F₂ / N, where N is a normal subgroup of F₂.

Determining the Normal Subgroup N

To determine the normal subgroup N, we need to understand the properties of the Z-action and how it affects the fundamental group of X. The Z-action is a type of hyperbolic motion, which preserves the hyperbolic metric on X. This means that the Z-action induces a homomorphism from the fundamental group of X to the fundamental group of the circle, denoted as ℤ.

The Fundamental Group of the Circle

The fundamental group of the circle is a well-known result in algebraic topology. It is isomorphic to the integers, denoted as ℤ. This means that the fundamental group of the circle is generated by a single element, often denoted as c, subject to the relation c^n = 1.

Determining the Normal Subgroup N

Using the homomorphism from the fundamental group of X to the fundamental group of the circle, we can determine the normal subgroup N. The normal subgroup N is the kernel of this homomorphism, meaning that it consists of all elements of the fundamental group of X that are mapped to the identity element in the fundamental group of the circle.

Conclusion

In this article, we have explored the concept of the fundamental group of the quotient of the punctured plane by a hyperbolic Z-action. We have determined the fundamental group of the quotient space Y, which is isomorphic to the quotient group F₂ / N. We have also determined the normal subgroup N, which is the kernel of the homomorphism from the fundamental group of X to the fundamental group of the circle. This result provides a deeper understanding of the topology of the quotient space Y and its fundamental group.

References

  • Hatcher, A. (2002). Algebraic Topology. Cambridge University Press.
  • Massey, W. S. (1991). A Basic Course in Algebraic Topology. Springer-Verlag.
  • Munkres, J. R. (2000). Topology. Prentice Hall.

Further Reading

For further reading on algebraic topology and the fundamental group, we recommend the following resources:

  • Hatcher, A. (2002). Algebraic Topology. Cambridge University Press.
  • Massey, W. S. (1991). A Basic Course in Algebraic Topology. Springer-Verlag.
  • Munkres, J. R. (2000). Topology. Prentice Hall.

Introduction

In our previous article, we explored the concept of the fundamental group of the quotient of the punctured plane by a hyperbolic Z-action. We determined the fundamental group of the quotient space Y, which is isomorphic to the quotient group F₂ / N. In this article, we will answer some common questions related to this topic.

Q: What is the punctured plane X?

A: The punctured plane X is a topological space that consists of the Euclidean plane with a single point removed. This space is often used as a model for the hyperbolic plane, which is a non-Euclidean geometry.

Q: What is the hyperbolic Z-action on X?

A: The hyperbolic Z-action on X is generated by the map (x, y) ↦ (2x, y/2), which stretches the x-coordinate by a factor of 2 and shrinks the y-coordinate by a factor of 1/2. This action is a type of hyperbolic motion, which preserves the hyperbolic metric on X.

Q: What is the quotient space Y?

A: The quotient space Y is obtained by applying the Z-action to the punctured plane X. This means that we identify points in X that are related by the Z-action, i.e., points (x, y) and (2x, y/2) are considered equivalent.

Q: What is the fundamental group of X?

A: The fundamental group of the punctured plane X is a well-known result in algebraic topology. It is isomorphic to the free group on two generators, denoted as F₂. This means that the fundamental group of X is generated by two elements, often denoted as a and b, subject to the relation ab = 1.

Q: What is the fundamental group of Y?

A: The fundamental group of Y is a quotient group of the fundamental group of X. Specifically, it is isomorphic to the quotient group F₂ / N, where N is a normal subgroup of F₂.

Q: What is the normal subgroup N?

A: The normal subgroup N is the kernel of the homomorphism from the fundamental group of X to the fundamental group of the circle. This means that it consists of all elements of the fundamental group of X that are mapped to the identity element in the fundamental group of the circle.

Q: How do I determine the normal subgroup N?

A: To determine the normal subgroup N, you need to understand the properties of the Z-action and how it affects the fundamental group of X. The Z-action induces a homomorphism from the fundamental group of X to the fundamental group of the circle, denoted as ℤ. The normal subgroup N is the kernel of this homomorphism.

Q: What is the fundamental group of the circle?

A: The fundamental group of the circle is a well-known result in algebraic topology. It is isomorphic to the integers, denoted as ℤ. This means that the fundamental group of the circle is generated by a single element, often denoted as c, subject to the relation c^n = 1.

Q: How do I apply the results to other problems?

A: The results we obtained in this article can be applied to other problems in algebraic topology. For example, you can use the fundamental group of the quotient space Y to study the topology of other spaces that are related to the punctured plane X.

Conclusion

In this article, we have answered some common questions related to the fundamental group of the quotient of the punctured plane by a hyperbolic Z-action. We hope this Q&A article has provided a useful resource for students and researchers in algebraic topology.

References

  • Hatcher, A. (2002). Algebraic Topology. Cambridge University Press.
  • Massey, W. S. (1991). A Basic Course in Algebraic Topology. Springer-Verlag.
  • Munkres, J. R. (2000). Topology. Prentice Hall.

Further Reading

For further reading on algebraic topology and the fundamental group, we recommend the following resources:

  • Hatcher, A. (2002). Algebraic Topology. Cambridge University Press.
  • Massey, W. S. (1991). A Basic Course in Algebraic Topology. Springer-Verlag.
  • Munkres, J. R. (2000). Topology. Prentice Hall.

We hope this Q&A article has provided a useful resource for students and researchers in algebraic topology.