Problem 5 - Algebraic Topology

Question

Let \(S\) be an embedded \(k\) -sphere in \(S^{n}\) for which there exists a disk \(D^{n} \subset S^{n}\) intersecting \(S\) in the disk \(D^{k} \subset D^{n}\) defined by the first \(k\) coordinates of \(D^{n} .\) Let \(D^{n-k} \subset D^{n}\) be the disk defined by the last \(n-k\) coordinates, with boundary sphere \(S^{n-k-1} .\) Show that the inclusion \(S^{n-k-1} \hookrightarrow S^{n}-S\) induces an isomorphism on homology groups.

Step-by-Step Solution

Answer

The inclusion \(S^{n-k-1} \hookrightarrow S^n - S\) induces an isomorphism on homology groups.

Step 1: Understand the Setup

We are given an embedded \(k\)-sphere \(S\) in \(S^n\) that intersects with a \(n\)-disk \(D^n \subset S^n\) in a \(k\)-disk \(D^k\). We also have a \((n-k)\)-disk \(D^{n-k}\) defined by the last \(n-k\) coordinates of \(D^n\), with a boundary sphere \(S^{n-k-1}\). The task is to show that the inclusion \(S^{n-k-1} \hookrightarrow S^n - S\) induces an isomorphism on homology groups.

Step 2: Analyze the Homology Groups

Before proceeding with calculations, note that the homology of a sphere \(S^n\) is well-known: - \(H_0(S^n) = \mathbb{Z}\), - \(H_n(S^n) = \mathbb{Z}\), - all other homology groups are zero.Since \(S\) is a \(k\)-sphere, it affects the homology of \(S^n\), specifically \(H_k(S^n)\) will change due to the presence of \(S\). However, since we're examining \(S^{n-k-1} \hookrightarrow S^n - S\), we will focus on the impact of removing \(S\) on the homology of \(S^n\).

Step 3: Apply the Excision Theorem

The Excision Theorem tells us that removing a subset of a space that intersects a certain open set in a particularly 'nice' way gives a space homotopy equivalent to the original. Here, removing \(S\) leaves us with \(S^n - S\) having a homotopy type of \(S^{n-k-1}\). This is because the portion of \(D^n\) that remains after removing \(D^k\) is homotopy equivalent to \(S^{n-k-1}\). Thus, \(H_*(S^n - S) \cong H_*(S^{n-k-1})\).

Step 4: Identify the Resulting Homology

The homology groups of \(S^{n-k-1}\) are: - \(H_0(S^{n-k-1}) = \mathbb{Z}\),- \(H_{n-k-1}(S^{n-k-1}) = \mathbb{Z}\), - all other homology groups are zero.Because the inclusion \(S^{n-k-1} \hookrightarrow S^n - S\) is essentially the identity from a homological perspective, the induced map on homology groups is an isomorphism.

Step 5: Conclusion

Therefore, the inclusion of the boundary sphere \(S^{n-k-1} \hookrightarrow S^n - S\) indeed induces an isomorphism on all homology groups, as \(S^{n-k-1} \) and \(S^n - S\) share the same homology.

Key Concepts

Homology Groups

In algebraic topology, homology groups provide a way to analyze and classify topological spaces based on their "holes" or "cycles." Each homology group represents cycles of different dimensions: 0-dimensional points, 1-dimensional loops, 2-dimensional voids, and so on. These groups are useful because they give a robust way to capture the topology of a space.
For a sphere like the one in our exercise, the homology groups can be straightforward:

• The 0-dimensional homology group, denoted as \(H_0(S^n)\), is \(\mathbb{Z}\), representing that any sphere is connected.
• The nth-dimensional group \(H_n(S^n)\), is also \(\mathbb{Z}\) because any sphere encloses a void of that dimension.
• All other homology groups are zero because there are no other "holes".

If you remove a \(k\)-sphere embedded in \(S^n\), it modifies the homology groups of \(S^n\). Our exercise investigates how this affects the inclusion map \(S^{n-k-1} \hookrightarrow S^n - S\) in terms of homology. Knowing homology groups helps us understand the topological structure without looking at it physically.

Excision Theorem

The Excision Theorem is a powerful tool in algebraic topology. It allows simplification by helping us "excise," or remove, parts of a space without affecting the homology in most cases.
The theorem essentially says that, if you have a subspace that intersects nicely (usually as an open or contractible set) with part of your space, you can remove this part without changing the homology of the larger space. This gives a homological equivalence, meaning the remaining space after removal still "looks the same" in a homological sense.

• In the context of our exercise, the Excision Theorem is applied when removing the \(k\)-sphere \(S\) from \(S^n\).
• The theorem tells us that the homology of \(S^n - S\) is essentially the same as that of a lower-dimensional sphere, \(S^{n-k-1}\), because of the way the removal is set up.

This concept simplifies complex topological expressions to more manageable terms, which is extremely useful in applications involving complex spaces.

Sphere Homotopy

Homotopy is a central concept in topology, referring to the continuous transformation between spaces. When it comes to spheres, sphere homotopy frequently concerns understanding how various transformations can be "deformed" into each other, or when two spheres can be considered equivalent under such transformations.
In our example, after the \(k\)-sphere is removed from \(S^n\), what's left is homotopy equivalent to a lower-dimensional sphere, \(S^{n-k-1}\), due to the way the transformation can occur.

• Sphere homotopy types are often used to compare spaces by considering them equivalent if they can be continuously transformed into one another.
• In this case, it directly aids in simplifying the topology we're working with: \(S^n\) minus a \(k\)-sphere is simplified homotopically to \(S^{n-k-1}\).

Homotopy equivalence is important because it preserves fundamental properties like homology, which is exactly why in the exercise's solution, the inclusion induces an isomorphism on homology groups.