Problem 2 - Algebraic Topology

Question

Show that the change-of-basepoint homomorphism \(\beta_{h}\) depends only on the homotopy class of \(h\).

Step-by-Step Solution

Answer

The change-of-basepoint homomorphism \( \beta_{h} \) depends only on the homotopy class of the path \( h \).

Step 1: Understand the Problem Statement

We need to show that if two paths from a point \( x \) to a point \( y \) are homotopic, then the change-of-basepoint homomorphism, denoted \( \beta_h \), depends only on the homotopy class of the path, not the specific path itself.

Step 2: Recall the Definition of Homotopy

Two continuous functions \( f, g : X \to Y \) are homotopic if there exists a continuous map \( H : X \times [0,1] \to Y \) such that \( H(x, 0) = f(x) \) and \( H(x, 1) = g(x) \) for all \( x \in X \). In our case, \( h_1 \sim h_2 \) if there exists a homotopy between two paths \( h_1 \) and \( h_2 \).

Step 3: Define the Change-of-Basepoint Homomorphism

The change-of-basepoint homomorphism \( \beta_{h} \) is defined as a map \( \beta_h : \pi_1(Y, x) \to \pi_1(Y, y) \), where \( \pi_1 \) denotes the fundamental group and \( h \) is a path from \( x \) to \( y \). It takes a loop based at \( x \) and translates it to a loop based at \( y \) using the path \( h \).

Step 4: Establish Homotopy Invariance of the Change-of-Basepoint

Suppose we have two homotopic paths \( h_1 \sim h_2 \) and need to show that \( \beta_{h_1} = \beta_{h_2} \). Consider a loop \( \alpha \) in \( \pi_1(Y, x) \). Then, \( \beta_{h_1}(\alpha) = [h_1^{-1} \cdot \alpha \cdot h_1] \) and \( \beta_{h_2}(\alpha) = [h_2^{-1} \cdot \alpha \cdot h_2] \).

Step 5: Use Midpoint Homotopy

Since \( h_1 \sim h_2 \), there exists a homotopy \( H : [0, 1] \times [0, 1] \to Y \) such that \( H(s, 0) = h_1(s) \) and \( H(s, 1) = h_2(s) \). Define \( K : [0, 1] \to Y \) as \( K(t) = H(t, 0.5) \). This connects both \( h_1 \) and \( h_2 \).

Step 6: Show the Equivalence of the Paths

Using the homotopy, we have \( [h_1^{-1} \cdot \alpha \cdot h_1] \sim [h_2^{-1} \cdot \alpha \cdot h_2] \). Thus, \( \beta_{h_1}(\alpha) = \beta_{h_2}(\alpha) \) if the loops are homotopic, proving that \( \beta_h \) depends only on the homotopy class of \( h \).

Step 7: Final Step: Conclusion

We conclude that the change-of-basepoint homomorphism \( \beta_h \) does not depend on the specific path \( h \), but only on its homotopy class. The path homotopy ensures that any two homotopic paths produce the same homomorphism when changing the basepoint.

Key Concepts

Change-of-basepoint homomorphism

In algebraic topology, the concept of a change-of-basepoint homomorphism is quite interesting. When working with paths on a topological space and their associated fundamental groups, we often need to switch the starting point of our analysis from one place to another. This is where the change-of-basepoint homomorphism comes into play.

This homomorphism is a function, specifically denoted as \( \beta_h \), and it helps us transition between fundamental groups based at different points. Suppose we're dealing with a point \( x \) and we want to shift our focus to a new point \( y \). The path \( h \) serves as our bridge. It's crucial to understand that \( \beta_h \) is defined from \( \pi_1(Y, x) \) to \( \pi_1(Y, y) \). This means it maps loops starting and ending at \( x \) to equivalent loops at \( y \) by incorporating the path \( h \) leading between these points.

Here's a simplification:

• Loop starts at \( x \), travels path \( h \).
• Our original loop completes a journey, returning to \( x \).
• Path \( h^{-1} \) brings the loop to \( y \).

The powerful thing about this is that such a transformation only leans on the path's homotopy class, not its precise shape or trajectory.

Homotopy

The notion of homotopy is fundamental when discussing continuous transformations in topology. Think of it as a flexible mapping that allows us to warp one path smoothly into another within a certain space, without tearing or disconnecting it.

Formally, two paths, say \( f \) and \( g \), are homotopic if there exists a continuous map \( H : X \times [0, 1] \to Y \). This functions in such a way that for each time point—0 through 1—\( H(x, 0) = f(x) \) and \( H(x, 1) = g(x) \) for every \( x \) in \( X \). Imagine \( H \) as slowly altering \( f \) into \( g \).

This concept ensures that if two paths can be compressed into one another, they are considered equivalent under homotopy. In topology:

• These paths might look different initially.
• Yet, a seamless morphing exists between them, emphasizing their underlying symmetry.
• Such equivalence makes analyzing their properties much more simplified.

Homotopy becomes central when we explore behaviors and characteristics of spaces, underpinning many theorems and ideas in algebraic topology.

Fundamental group

The fundamental group is a key tool in studying the shape and structure of topological spaces. It's essentially a collection of loops based at a point, modulated by homotopy equivalence, offering profound insights into the topological properties of spaces.

Let's concentrate on what it means practically. Imagine a fixed base point \( x \) in a space \( Y \). The fundamental group, denoted \( \pi_1(Y, x) \), consists of all loops anchored at \( x \), with these loops understood up to homotopy. This translates to allowing loops to be "reshaped" if they are homotopically equivalent.

Here's why the fundamental group matters:

• It captures the essential 'loopiness' of a space.
• The group operations are defined by concatenating and reversing these loops.
• Each loop is treated as an equivalence class, depending only on its homotopy level.

In simpler terms, the fundamental group's elements reveal possible ways to navigate around obstacles within the space. Hence, through a fundamental group, we can explore whether certain features, like holes, exist in a topological space.