Problem 6 - Algebraic Topology

Question

Given a space \(X\) and a path-connected subspace \(A\) containing the basepoint \(x_{0}\) show that the map \(\pi_{1}\left(A, x_{0}\right) \rightarrow \pi_{1}\left(X, x_{0}\right)\) induced by the inclusion \(A \hookrightarrow X\) is surjective iff every path in \(X\) with endpoints in \(A\) is homotopic to a path in \(A\).

Step-by-Step Solution

Answer

The map is surjective if every path in \(X\) with endpoints in \(A\) is homotopic to a path in \(A\).

Step 1: Understanding Surjectivity

To show that the map \( ext{incl}_*: \pi_1(A, x_0) \to \pi_1(X, x_0)\) is surjective, we need to prove that for every loop \(\gamma: [0,1] \to X\) based at \(x_0\), there exists a loop \(\alpha: [0,1] \to A\) with the same basepoint such that \([\gamma] = [\alpha]\) in \(\pi_1(X, x_0)\).

Step 2: Implication: Surjectivity Implies Homotopy

Suppose the map \(\pi_1(A, x_0) \to \pi_1(X, x_0)\) is surjective. Take any path \(\gamma: [0,1] \to X\) with endpoints in \(A\). Since \(X\) is path-connected, we can extend \(\gamma\) to a loop at \(x_0\) by appending paths in \(A\). By surjectivity, this loop is homotopic to a loop in \(A\). Thus, \(\gamma\) is homotopic to a path entirely contained in \(A\).

Step 3: Implication: Homotopy Implies Surjectivity

Assume every path in \(X\) with endpoints in \(A\) is homotopic to a path in \(A\). For any loop \(\gamma: [0,1] \to X\) based at \(x_0\), consider it tentatively as a path from \(x_0\) to \(x_0\). By assumption, \(\gamma\) is homotopic to a path \(\alpha\) in \(A\) with endpoints \(x_0\). This implies \(\gamma\) is homotopic to a loop in \(A\), making the inclusion map surjective.

Step 4: Wrapping Up

Combining both directions, we conclude: \(\pi_1(A, x_0) \to \pi_1(X, x_0)\) is surjective if and only if every path in \(X\) with endpoints in \(A\) is homotopic to a path in \(A\). Thus, the conditions on paths guarantee the surjectivity of the fundamental group homomorphism induced by inclusion.

Key Concepts

Surjectivity in Algebraic Topology

In algebraic topology, the concept of surjectivity deals with the idea of having every element in one set being mapped to from another set. When we discuss the map \[ \pi_1(A, x_0) \to \pi_1(X, x_0) \] induced by the inclusion of a subspace, we want this map to be surjective. This means every possible path class loop (or homotopy class of loops) in the space \(X\) must correspond to some loop in the subspace \(A\).

Surjectivity ensures that:

• All loops in the larger space \(X\) can be reflected as loops within the subspace \(A\).
• Loosely, elements of \( \pi_1(X, x_0) \) actually "come" from elements of \( \pi_1(A, x_0) \), showing a comprehensive image mapping.

Understanding surjectivity in this context is crucial for showing a deeper connection between the larger space and its subspace, emphasizing the role of path-connectedness in topology.

Path Homotopy

Path homotopy refers to a concept where two paths with the same starting and ending points can be continuously deformed into each other. In simple terms, you can "pull" and "stretch" one path to look like the other, without cutting or breaking it.

When we say that a path \( \gamma \) in the larger space \(X\), starting and ending at points in \(A\), is homotopic to a path in \(A\), it means:

• You can transform \( \gamma \) into another path that lies completely within \(A\).
• This transformation respects the endpoints, keeping them fixed and continuous.

In topological terms, path homotopy helps show how two loops (or paths) belong to the same equivalence class in terms of their fundamental group classification. Thus, proving that every path in \(X\) is homotopic to a path in \(A\) supports the idea of mapping path classes from one space onto another.

Subspace Inclusion

Subspace inclusion involves embedding one topological space into another as a subset. Here, we consider including the subspace \(A\) into the space \(X\) with natural mapping. This inclusion is noted as \(A \hookrightarrow X\).

For our case, where the map induces a homomorphism between fundamental groups:

• Subspace inclusion acts as a bridge connecting the path-connected subspace \(A\) to the entire space \(X\).
• Through inclusion, paths (loops) in \(A\) "fit" within \(X\), and crucially, any path in \(X\) with endpoints in \(A\) must respect its topology, often modeled by such an inclusion.

Inclusion plays a key role in understanding how properties of the smaller space \(A\) propagate to and illuminate properties of the larger space \(X\), especially when discussing the fundamental group and surjectivity of induced mappings.