Problem 8 - Algebraic Topology

Question

We can regard \(\pi_{1}\left(X, x_{0}\right)\) as the set of basepoint- preserving homotopy classes of maps \(\left(S^{1}, s_{0}\right) \rightarrow\left(X, x_{0}\right) .\) Let \(\left[S^{1}, X\right]\) be the set of homotopy classes of maps \(S^{1} \rightarrow X\) with no conditions on basepoints. Thus there is a natural map \(\Phi: \pi_{1}\left(X, x_{0}\right) \rightarrow\left[S^{1}, X\right]\) obtained by ignoring basepoints. Show that \(\Phi\) is onto if \(X\) is path-connected, and that \(\Phi([f])=\Phi([g])\) iff \([f]\) and \([g]\) are conjugate in \(\pi_{1}\left(X, x_{0}\right) .\) Hence \(\Phi\) induces a oneto-one correspondence between \(\left[S^{1}, X\right]\) and the set of conjugacy classes in \(\pi_{1}(X)\) when \(X\) is path-connected.

Step-by-Step Solution

Answer

\(\Phi\) is onto if \(X\) is path-connected, and \(\Phi\([f]\)=\Phi\([g]\) iff \([f]\) and \([g]\) are conjugate. Thus, \(\Phi\) induces a one-to-one correspondence between \([S^1, X]\) and conjugacy classes of \(\pi_1(X)\) when \(X\) is path-connected.

Step 1: Define the Maps and Key Terms

We start by identifying the elements involved in the exercise. Given a topological space \(X\) with a basepoint \(x_0\), \(\pi_1(X, x_0)\) refers to the fundamental group of \(X\) at \(x_0\), which consists of basepoint-preserving homotopy classes of loops in \(X\). The set \([S^1, X]\) represents all homotopy classes of maps from \(S^1\) to \(X\) without requiring the maps to preserve the basepoint. The map \(\Phi: \pi_1(X, x_0) \rightarrow [S^1, X]\) sends basepoint-preserving homotopy classes to non-basepoint-preserving homotopy classes.

Step 2: Show \(\Phi\) is Onto if \(X\) is Path-Connected

To prove \(\Phi\) is surjective, take an arbitrary homotopy class \([f] \in [S^1, X]\). Since \(X\) is path-connected, for any point \(f(s_0)\) (the image of some point in \(S^1\)), there exists a path \(\gamma\) in \(X\) from \(x_0\) to \(f(s_0)\). Construct a new map \(g: S^1 \rightarrow X\) as \(g(t) = \gamma(t)\cdot f(t)\cdot \bar{\gamma}(t)\), where \(\bar{\gamma}\) is the inverse path of \(\gamma\). \(g\) is basepoint-preserving and its class in \(\pi_1(X, x_0)\) maps to \([f]\) under \(\Phi\), making \(\Phi\) onto.

Step 3: Establish \(\Phi([f])=\Phi([g])\) If and Only If \([f]\) and \([g]\) are Conjugate

For \(\Phi([f]) = \Phi([g])\), \(f\) and \(g\) must map to the same homotopy class in \([S^1, X]\). Homotopically, this means there exists a path \(\delta\) in \(X\) such that \(g\) can be transformed into \(f\) via conjugation by \(\delta\), i.e., \(g(t) = \delta(0)f(t)\delta^{-1}(1)\) for all \(t\). This is the definition of conjugation in the fundamental group \(\pi_1(X, x_0)\), showing \([f]\) and \([g]\) are conjugate if and only if they map to the same element in \([S^1, X]\) through \(\Phi\).

Step 4: Conclude One-to-One Correspondence Between \([S^1, X]\) and Conjugacy Classes in \(\pi_1(X)\)

From step 2, \(\Phi\) is onto, so every element in \([S^1, X]\) corresponds to an element in \(\pi_1(X, x_0)\). Step 3 then shows the kernel of \(\Phi\) consists of conjugate elements, meaning \(\Phi\) identifies elements based on conjugacy. Together, these prove that \(\Phi\) creates a bijective map between \([S^1, X]\) and the set of conjugacy classes of \(\pi_1(X)\) when \(X\) is path-connected.

Key Concepts

Homotopy Classes

Homotopy classes are fundamental in topology, especially when studying continuous functions between topological spaces. Essentially, two functions from one topological space to another are considered homotopically equivalent if one can be continuously deformed into the other without breaking or tearing. This relation defines homotopy classes of functions.
In the context of the fundamental group, the homotopy classes are those of loops, which are functions from the circle, denoted by the unit circle \(S^1\), into a space \(X\). We say two loops are homotopic, or belong to the same homotopy class, if one can be continuously transformed into the other while keeping the basepoint fixed.

• This leads to what is known as basepoint-preserving homotopy classes, which are central to the definition of the fundamental group \(\pi_1(X, x_0)\).
• For a map \(f: S^1 \to X\) without basepoint preservation, its homotopy class belongs to \([S^1, X]\).

Understanding homotopy classes is crucial for grasping more complex topological structures that involve these transformations and equivalences.

Path-Connected Spaces

Path-connectedness is a crucial aspect of analyzing topological spaces. A space \(X\) is said to be path-connected if any two points in \(X\) can be joined by a continuous path. This kind of connectivity plays a major role in simplifying the study of topological spaces, especially in algebraic topology.
Why does path-connectedness matter? Mainly because it ensures the possibility of moving continuously from any point of the space to any other point.

• In the case of the fundamental group, a path-connected space implies the loosely specified maps (not preserving basepoints) can be related back to specific basepoint-based transformations.
• This property is essential for proving that the map \(\Phi\), connecting \(\pi_1(X, x_0)\) to \([S^1, X]\), is onto (or surjective), as it allows reversing arbitrary transformations back to the logoic of the basepoint-preserving scope.

In both theoretical and practical terms, path-connectedness means the space behaves cohesively, supporting the concept of simplifying analysis through homotopy and conjugacy.

Conjugacy Classes in Algebraic Topology

Conjugacy classes emerge as a significant concept when considering the structure of groups within algebraic topology. Within the fundamental group, loops can be transformed through conjugation. This is akin to shifting or twisting a loop around a key point in the space.
How do conjugacy classes tie into homotopy and the fundamental group?

• Two elements \([f]\) and \([g]\) in the fundamental group are considered to be in the same conjugacy class if one can be transformed into the other by a particular type of continuous deformation involving a third loop. This is expressed algebraically as \([g] = \delta \cdot [f] \cdot \delta^{-1}\), where \(\delta\) is another path or loop.
• The connection with the homotopy groups lies in how \(\Phi\), as a function, equates two different homotopy classes if and only if they are conjugate, effectively creating a one-to-one correspondence between these classes and the set \([S^1, X]\).

Understanding the interplay between conjugacy and homotopy offers deeper insights into transformational behaviors and symmetries within topological spaces, simplifying the algebraic characterization of these spaces and expanding the horizon of algebraic topology.