Let \(G\) be a group; let \(x\) be an element of \(G\); let \(N\) be the least normal subgroup of \(G\) containing \(x\). Show that if there is a normal, path-connected space whose fundamental group is isomorphic to \(G\), then there is a normal, path-connected space whose fundamental group is isomorphic to \(G/N\).
Solution
Goal: Prove that attaching a 2-cell \(e^2\) to a normal path-connected space \(X\) along a loop representing \(x \in \pi_1(X, p) \cong G\) produces a normal path-connected space \(Y\) with fundamental group isomorphic to \(G/N\), where \(N = \langle\langle x \rangle\rangle\).
1. Construction of the space \(Y\):Proof:
1.1. Let \(X\) be a normal, path-connected topological space with \(\pi_1(X, p) \cong G\).
1.2. Choose a representative loop \(\alpha: [0, 1] \to X\) based at \(p\) such that \([\alpha] = x \in \pi_1(X, p)\).
1.3. Let \(D^2 = \{z \in \mathbb{C} \mid |z| \le 1\}\) be the standard closed 2-disk with boundary \(S^1 = \partial D^2\).
1.4. Define the attaching map \(f: S^1 \to X\) by \(f(e^{2\pi i t}) = \alpha(t)\) for \(t \in [0, 1]\).
1.5. Form the adjunction space \(Y = X \cup_f D^2 = (X \sqcup D^2) / \sim\), where \(z \sim f(z)\) for \(z \in S^1\).
2. Topological properties of \(Y\) (path-connected and normal):Proof:
2.1. Path-connectedness: \(X\) and \(D^2\) are path-connected, and the gluing identifies the non-empty connected set \(S^1 \subset D^2\) with \(f(S^1) \subset X\). Thus \(Y\) is path-connected.
2.2. Normality: Adjoining a cell along a closed subspace of a compact metric space (\(S^1 \subset D^2\)) to a normal space \(X\) preserves normality. (Any closed disjoint subsets in \(Y\) pull back to closed disjoint sets in \(X\) and \(D^2\), which are separated by continuous Urysohn functions compatible along \(S^1\).)
3. Computation of \(\pi_1(Y, p)\) via Seifert-van Kampen:Proof:
3.1. Let \(\mathbf{0} \in D^2\) be the center of the attached disk.
3.2. Define open subsets of \(Y\):\[U = Y \setminus \{\mathbf{0}\} \quad \text{and} \quad V = \operatorname{Int}(D^2) \subset Y.\]
3.3. We have \(U \cup V = Y\).
3.4. The open set \(V\) is homeomorphic to the open disk, so \(V\) is contractible: \(\pi_1(V) = \{1\}\).
3.5. The set \(U\) deformation retracts onto \(X\) along the radial lines of the punctured disk \(D^2 \setminus \{\mathbf{0}\}\). Hence the inclusion induces an isomorphism \(\pi_1(U, p) \cong \pi_1(X, p) \cong G\).
3.6. The intersection \(U \cap V = \operatorname{Int}(D^2) \setminus \{\mathbf{0}\} \cong S^1 \times (0, 1)\) is path-connected with \(\pi_1(U \cap V) \cong \mathbb{Z}\), generated by a standard loop \(\gamma\) around the origin.
3.7. Under the inclusion \(U \cap V \hookrightarrow U\), the generator \([\gamma]\) maps to \([\alpha] = x \in G\).
3.8. By the Seifert-van Kampen Theorem:\[\pi_1(Y, p) \cong \pi_1(U, p) *_{\pi_1(U \cap V, p)} \pi_1(V, p) \cong G *_{\mathbb{Z}} \{1\} \cong G / \langle\langle x \rangle\rangle = G/N.\]
4. Conclusion:The space \(Y = X \cup_f D^2\) is normal, path-connected, and has \(\pi_1(Y, p) \cong G/N\). Q.E.D.