Problem 27 - Algebraic Topology

Question

For a universal cover \(p: \tilde{X} \rightarrow X\) we have two actions of \(\pi_{1}\left(X, x_{0}\right)\) on the fiber \(p^{-1}\left(x_{0}\right),\) namely the action given by lifting loops at \(x_{0}\) and the action given by restricting deck transformations to the fiber. Are these two actions the same when \(X=S^{1} \vee S^{1}\) or \(X=S^{1} \times S^{1} ?\) Do the actions always agree when \(\pi_{1}\left(X, x_{0}\right)\) is abelian?

Step-by-Step Solution

Answer

These actions differ when \( X = S^1 \vee S^1 \) but agree for \( X = S^1 \times S^1 \) and any abelian fundamental group.

Step 1: Understanding the Setup

We are given a universal cover \( p: \tilde{X} \rightarrow X \) with two actions of \( \pi_{1}(X, x_{0}) \) on the fiber \( p^{-1}(x_{0}) \). These actions are from lifting loops at \( x_{0} \) and restricting deck transformations to the fiber. Our task is to check if these actions are the same when \( X = S^{1} \vee S^{1} \), \( X = S^{1} \times S^{1} \), and when \( \pi_{1}(X, x_{0}) \) is abelian.

Step 2: Analyzing the Space X = S^1 ∨ S^1

For \( X = S^{1} \vee S^{1} \), the fundamental group \( \pi_{1}(X, x_{0}) \) is the free group on two generators, say \( a \) and \( b \). Non-abelian groups often lead to different actions, as the way loops lift in the universal cover might not commute with the deck transformations, which are generally non-trivial. Thus, in general, these two actions would differ.

Step 3: Analyzing the Space X = S^1 × S^1

For \( X = S^{1} \times S^{1} \), the fundamental group \( \pi_{1}(X, x_{0}) \) is \( \mathbb{Z} \times \mathbb{Z} \), which is abelian. Since the group is abelian, any sequence of applying group elements will be commutative, causing the two actions to coincide as the sequence of performing loops and deck transformations commutes perfectly.

Step 4: Testing with Abelian Fundamental Groups

If \( \pi_{1}(X, x_{0}) \) is abelian, transformations based on group actions commute due to the abelian property of the group. Subsequently, lifting loops and deck transformation actions would be the same, as there would be no variance introduced by the order of operations.

Key Concepts

Fundamental Group

The fundamental group, denoted as \(\pi_{1}(X, x_{0})\), is a key concept in algebraic topology.
It measures the basic shape, or 'holes,' of a topological space \(X\) around a base point \(x_{0}\).
In simpler terms, it captures how loops in the space can be transformed into each other.

Some important properties of the fundamental group include:

• The fundamental group may contain different generators depending on the space. For instance, the space \(S^{1} \vee S^{1}\) has a fundamental group that's a free group with two generators.
• In contrast, the space \(S^{1} \times S^{1}\) has a fundamental group \(\mathbb{Z} \times \mathbb{Z}\), which is abelian.

These properties are crucial for understanding how actions like lifting loops and deck transformations interact with different spaces.

Deck Transformations

Deck transformations are self-homeomorphisms of a covering space that map fibers to themselves while preserving the entire structure of the cover.
In the context of universal covers, these transformations help in understanding how different layers of the cover relate.
Deck transformations have these key characteristics:

• They form a group, often known as the deck transformation group.
• The way these transformations apply to specific points within fibers \(p^{-1}(x_{0})\) plays a critical role in how different actions unfold in the universal covering space.

When analyzing spaces like \(S^{1} \vee S^{1}\) versus \(S^{1} \times S^{1}\), we see that deck transformations and their interaction with loops can differ based on whether the fundamental group is abelian or not.

Abelian Group

An abelian group has the property that the group operation is commutative. This means for any elements \(a\) and \(b\) in the group, \(a \cdot b = b \cdot a\).
This commutative property simplifies many group interactions.

In terms of the universal cover exercise, the importance of the abelian group lies in:

• When dealing with the abelian fundamental group \(\mathbb{Z} \times \mathbb{Z}\) for spaces like \(S^{1} \times S^{1}\), actions like lifting loops and deck transformations align—these actions don't depend on the order of operations.
• This contrasts with non-abelian groups, where such actions do not necessarily align, leading to different results.

Understanding when a fundamental group is abelian helps predict how loops and transformations interact.

Lifting Loops

Lifting loops involve projecting loops from the base space \(X\) into its universal cover \(\tilde{X}\), offering a way to analyze how loops behave in more complex spaces.
These lifts are often discussed with a focus on whether such loops close up or if they behave differently when lifted to the cover.

Lifting loops is essential in:

• Identifying whether different loop actions will produce the same results in the universal cover.
• Examining how these actions interact with the deck transformations, especially in spaces like \(S^{1} \times S^{1}\), where the fundamental group plays a crucial role in ensuring actions commute when the group is abelian.

Thus, lifting loops is a valuable tool in understanding the geometry of the underlying space.