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, *) of the space and the point. The homology of this pair is denoted by H_n(X, *) and is called the homology of X relative to the point *.

Reduced Homology

The reduced homology of a space X, denoted by \widetilde{H_n}(X), is a way of studying the homology of X by ignoring the trivial homology group H_0(X). The reduced homology is defined as the homology of the pair (X, x_0) where x_0 is a fixed point in X. In other words, \widetilde{H_n}(X) = H_n(X, x_0).

The Long Exact Sequence for Pairs

The long exact sequence for pairs is a fundamental tool in algebraic topology that helps us study the homology of a pair of spaces. It is a sequence of homology groups that relates the homology of the pair (X, A) to the homology of X and A. The long exact sequence for pairs is given by:

... → H_n(A) → H_n(X) → H_n(X, A) → H_{n-1}(A) → ...

Proving the Isomorphism

We want to prove the isomorphism between the reduced homology of a space X and the homology of the pair (X, *):

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

We can use the long exact sequence for pairs to prove this isomorphism. For n > 0, we have:

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

This is because the long exact sequence for pairs gives us the isomorphism H_n(X) = H_n(X, *) for n > 0.

The Case n = 0

For n = 0, we have:

$$\widetilde{H_0}(X) \cong H_0(X, *)$$

However, we need to be careful here. The reduced homology \widetilde{H_0}(X) is defined as the kernel of the map H_0(X) → H_0(*), which is the map that sends a 0-cycle in X to the 0-cycle in * that is the constant function. The homology H_0(X, ) is the kernel of the map H_0(X) → H_0(), which is the map that sends a 0-cycle in X to the 0-cycle in * that is the constant function.

Conclusion

In conclusion, we have proved the isomorphism between the reduced homology of a space X and the homology of the pair (X, *):

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

This isomorphism is a fundamental result in algebraic topology that helps us study the topological properties of a space by considering it as a pair of spaces, one of which is a point.

References

• [1] Hatcher, A. (2002). Algebraic Topology. Cambridge University Press.
• [2] Spanier, E. (1966). Algebraic Topology. Springer-Verlag.

Further Reading

• [1] Bredon, G. E. (1993). Topology and Geometry. Springer-Verlag.
• [2] Munkres, J. R. (2000). Topology. Prentice Hall.

Glossary

• Homology: A way of studying the connectivity and holes of a space by considering it as a chain complex.
• Reduced Homology: A way of studying the homology of a space by ignoring the trivial homology group H_0(X).
• Long Exact Sequence for Pairs: A sequence of homology groups that relates the homology of the pair (X, A) to the homology of X and A.
• Isomorphism: A bijective homomorphism between two groups.
  Homology Relative to a Point: Q&A =====================================

Q: What is the main idea behind homology relative to a point?

A: The main idea behind homology relative to a point is to study the homology of a space by considering it as a pair of spaces, one of which is a point. This allows us to understand the topological properties of a space by considering it as a pair of spaces, one of which is a point.

Q: What is the difference between homology and reduced homology?

A: Homology is a way of studying the connectivity and holes of a space by considering it as a chain complex. Reduced homology, on the other hand, is a way of studying the homology of a space by ignoring the trivial homology group H_0(X).

Q: What is the long exact sequence for pairs?

A: The long exact sequence for pairs is a sequence of homology groups that relates the homology of the pair (X, A) to the homology of X and A. It is a fundamental tool in algebraic topology that helps us study the homology of a pair of spaces.

Q: How do we prove the isomorphism between reduced homology and homology relative to a point?

A: We can use the long exact sequence for pairs to prove the isomorphism between reduced homology and homology relative to a point. For n > 0, we have:

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

This is because the long exact sequence for pairs gives us the isomorphism H_n(X) = H_n(X, *) for n > 0.

Q: What is the case for n = 0?

A: For n = 0, we have:

$$\widetilde{H_0}(X) \cong H_0(X, *)$$

However, we need to be careful here. The reduced homology \widetilde{H_0}(X) is defined as the kernel of the map H_0(X) → H_0(*), which is the map that sends a 0-cycle in X to the 0-cycle in * that is the constant function. The homology H_0(X, ) is the kernel of the map H_0(X) → H_0(), which is the map that sends a 0-cycle in X to the 0-cycle in * that is the constant function.

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

A: Homology relative to a point has many applications in algebraic topology, including:

• Studying the connectivity and holes of a space
• Understanding the topological properties of a space
• Relating the homology of a space to the homology of its subspaces
• Studying the homotopy groups of a space

Q: What are some common mistakes to avoid when working with homology relative to a point?

A: Some common mistakes to avoid when working with homology relative to a point include:

• Confusing the reduced homology with the homology of the pair (X, *)
• Not being careful with the definition of the reduced homology
• Not using the long exact sequence for pairs correctly

Q: What are some resources for learning more about homology relative to a point?

A: Some resources for learning more about homology relative to a point include:

• [1] Hatcher, A. (2002). Algebraic Topology. Cambridge University Press.
• [2] Spanier, E. (1966). Algebraic Topology. Springer-Verlag.
• [3] Bredon, G. E. (1993). Topology and Geometry. Springer-Verlag.
• [4] Munkres, J. R. (2000). Topology. Prentice Hall.

Glossary

• Homology: A way of studying the connectivity and holes of a space by considering it as a chain complex.
• Reduced Homology: A way of studying the homology of a space by ignoring the trivial homology group H_0(X).
• Long Exact Sequence for Pairs: A sequence of homology groups that relates the homology of the pair (X, A) to the homology of X and A.
• Isomorphism: A bijective homomorphism between two groups.