Homology Relative To A Point

Introduction

In algebraic topology, the concept of homology relative to a point is a fundamental idea that helps us understand the topological properties of a space. It is a powerful tool that allows us to study the connectivity and holes of a space by considering it as a pair of spaces, one of which is a point. In this article, we will explore the concept of homology relative to a point and prove the isomorphism between the reduced homology of a space and the homology of the pair of the space and a point.

What is Homology Relative to a Point?

Homology relative to a point is a way of studying the homology of a space by considering it as a pair of spaces, one of which is a point. The idea is to take a space X and a point * (often referred to as the "base point") and consider the pair (X, *) . The homology of this pair is denoted by H_n(X, *) and is called the homology of X relative to the point *.

Reduced Homology

Reduced homology is a variant of homology that is used to study the homology of a space in a way that is more suitable for certain applications. It is defined as the homology of the pair (X, *) with the point * removed. The reduced homology of a space X is denoted by \widetilde{H_n}(X) and is defined as:

$$\widetilde{H_n}(X) = H_n(X, \ast)$$

The Long Exact Sequence for Pairs

One of the most powerful tools in algebraic topology is the long exact sequence for pairs. This sequence is a way of relating the homology of a space to the homology of the pair of the space and a point. The long exact sequence for pairs is given by:

$$\cdots \to H_n(X) \to H_n(X, \ast) \to H_{n-1}(\ast) \to H_{n-1}(X) \to \cdots$$

Proving the Isomorphism

We want to prove that the reduced homology of a space X is isomorphic to the homology of the pair of the space and a point. In other words, we want to prove that:

$$\widetilde{H_n}(X) \cong H_n(X, \ast)$$

For n > 0, the long exact sequence for pairs gives us:

$$H_n(X) = \widetilde{H_n}(X) \cong H_n(X, \ast)$$

This shows that the reduced homology of a space X is isomorphic to the homology of the pair of the space and a point for n > 0.

The Case n = 0

For n = 0, the long exact sequence for pairs gives us:

$$\widetilde{H_0}(X) \cong H_0(X, \ast) \cong \ker(H_0(X) \to H_0(\ast))$$

Since H_0(\ast) is isomorphic to the integers, we have:

$$\widetilde{H_0}(X) \cong \ker(H_0(X) \to \mathbb{Z})$$

This shows that the reduced homology of a space X is isomorphic to the kernel of the map from H_0(X) to the integers.

Conclusion

In this article, we have explored the concept of homology relative to a point and proved the isomorphism between the reduced homology of a space and the homology of the pair of the space and a point. We have shown that the reduced homology of a space X is isomorphic to the homology of the pair of the space and a point for n > 0, and that the reduced homology of a space X is isomorphic to the kernel of the map from H_0(X) to the integers for n = 0.

References

• [1] Hatcher, A. (2002). Algebraic Topology. Cambridge University Press.
• [2] Spanier, E. (1966). Algebraic Topology. McGraw-Hill.
• [3] Hilton, P. J., & Roitman, J. (1997). Localization of Modules. Cambridge University Press.

Further Reading

• [1] Algebraic Topology by Allen Hatcher
• [2] Topology and Geometry by Glen E. Bredon
• [3] Homology Theory by Edwin H. Spanier
  Homology Relative to a Point: Q&A =====================================

Q: What is the main difference between homology and homology relative to a point?

A: The main difference between homology and homology relative to a point is that homology relative to a point considers the space as a pair of spaces, one of which is a point, whereas homology considers the space as a single space.

Q: Why do we need to consider the space as a pair of spaces?

A: We need to consider the space as a pair of spaces because it allows us to study the connectivity and holes of the space in a more nuanced way. By considering the space as a pair of spaces, we can capture the information about the space that is lost when we consider it as a single space.

*Q: What is the significance of the point * in the pair (X, )?

A: The point * in the pair (X, *) is a reference point that allows us to define the homology of the pair. It is often referred to as the "base point".

Q: How does the long exact sequence for pairs relate to homology relative to a point?

A: The long exact sequence for pairs is a way of relating the homology of a space to the homology of the pair of the space and a point. It shows that the homology of the pair is isomorphic to the reduced homology of the space.

Q: What is the reduced homology of a space?

A: The reduced homology of a space is a variant of homology that is used to study the homology of a space in a way that is more suitable for certain applications. It is defined as the homology of the pair (X, *) with the point * removed.

Q: How does the reduced homology of a space relate to the homology of the pair of the space and a point?

A: The reduced homology of a space is isomorphic to the homology of the pair of the space and a point. This is shown by the long exact sequence for pairs.

Q: What are some of the applications of homology relative to a point?

A: Homology relative to a point has many applications in algebraic topology, including the study of the connectivity and holes of spaces, the classification of spaces, and the study of the properties of spaces.

Q: What are some of the challenges of working with homology relative to a point?

A: One of the challenges of working with homology relative to a point is that it can be difficult to compute the homology of a space relative to a point. However, the long exact sequence for pairs provides a powerful tool for relating the homology of a space to the homology of the pair of the space and a point.

Q: What are some of the future directions for research in homology relative to a point?

A: Some of the future directions for research in homology relative to a point include the development of new tools and techniques for computing the homology of spaces relative to a point, the study of the properties of spaces using homology relative to a point, and the application of homology relative to a point to other areas of mathematics.

References

• [1] Hatcher, A. (2002). Algebraic Topology. Cambridge University Press.
• [2] Spanier, E. (1966). Algebraic Topology. McGraw-Hill.
• [3] Hilton, P. J., & Roitman, J. (1997). Localization of Modules. Cambridge University Press.

Further Reading

• [1] Algebraic Topology by Allen Hatcher
• [2] Topology and Geometry by Glen E. Bredon
• [3] Homology Theory by Edwin H. Spanier