Let \(X\) be a Hausdorff space; let \(A\) be a closed path-connected subspace. Suppose that \(h: B^n \to X\) is a continuous map that maps \(S^{n-1}\) into \(A\) and maps \(\operatorname{Int} B^n\) bijectively onto \(X - A\). Let \(a\) be a point of \(h(S^{n-1})\). If \(n > 2\), what can you say about the homomorphism of \(\pi_1(A, a)\) into \(\pi_1(X, a)\) induced by inclusion?
Solution
1. Cell attachment description:
1.1. The space \(X\) is homeomorphic to the adjunction space \(A \cup_\phi B^n\) formed by attaching an \(n\)-cell \(B^n\) to \(A\) along the attaching map \(\phi = h|_{S^{n-1}}: S^{n-1} \to A\).
definition of cell attachment quotient topology.
2. Open cover for Seifert–van Kampen Theorem:
2.1. Let \(p = h(0) \in X \setminus A\) be the center of the attached cell.Define the open subsets:\[ U = X \setminus \{p\}, \qquad V = h\big(\{x \in B^n \mid \|x\| < 3/4\}\big). \]
openness in the quotient topology.
2.2. Retraction of \(U\) onto \(A\):The punctured ball \(B^n \setminus \{0\}\) deformation retracts radially onto the boundary \(S^{n-1}\).This radial deformation extends continuously to \(X \setminus \{p\}\) by fixing \(A\) pointwise, so \(A\) is a deformation retract of \(U\).Thus the inclusion \(A \hookrightarrow U\) induces an isomorphism:\[ \pi_1(A, a) \cong \pi_1(U, a). \]
homotopy invariance of fundamental group.
2.3. Contractibility of \(V\):\(V \cong \operatorname{Int}(B^n)\) is homeomorphic to the open Euclidean ball, which is convex and contractible.Thus \(\pi_1(V) = 0\).
contractibility of open Euclidean balls.
2.4. Simply-connectedness of \(U \cap V\) for \(n > 2\):The intersection \(U \cap V = h(\{x \in B^n \mid 0 < \|x\| < 3/4\})\) is homeomorphic to the punctured Euclidean ball \(\mathbb{R}^n \setminus \{0\}\), which deformation retracts onto the sphere \(S^{n-1}\).Since \(n > 2\), the dimension \(n - 1 \ge 2\), so \(S^{n-1}\) is simply connected:\[ \pi_1(U \cap V) \cong \pi_1(S^{n-1}) = 0. \]
\(\pi_1(S^k) = 0\) for all \(k \ge 2\).
3. Seifert–van Kampen Theorem:
3.1. By the Seifert–van Kampen Theorem, since \(U \cap V\) is path-connected:\[ \pi_1(X, a) \cong \pi_1(U, a) *_{\pi_1(U \cap V)} \pi_1(V) \cong \pi_1(A, a) *_0 0 \cong \pi_1(A, a). \]
Seifert–van Kampen Theorem with trivial amalgamating subgroup.
3.2. The isomorphism \(\pi_1(A, a) \xrightarrow{\sim} \pi_1(X, a)\) is precisely the homomorphism \(i_*\) induced by the inclusion map \(i: A \hookrightarrow X\).
inclusion factorization \(A \hookrightarrow U \hookrightarrow X\).
4. Conclusion:For \(n > 2\), the inclusion-induced map \(i_*: \pi_1(A, a) \to \pi_1(X, a)\) is an isomorphism. Q.E.D.
2 and 3.