With the notation of the preceding problem, show that \(\operatorname{Aut}(\bigvee_n S^k) \approx \operatorname{GL}_n(\mathbb{Z})\) for \(k > 1\), where \(\bigvee_n S^k\) denotes the wedge sum of \(n\) copies of \(S^k\) and \(\operatorname{GL}_n(\mathbb{Z})\) is the group of \(n \times n\) matrices with entries in \(\mathbb{Z}\) having an inverse matrix of the same form.
Solution
1. Compute \(\pi_k(X)\) for \(X = \bigvee_{j=1}^n S^k\) (\(k > 1\)):
1.1. Since \(k > 1\), the space \(X\) is \((k-1)\)-connected: \(\pi_1(X) = \cdots = \pi_{k-1}(X) = 0\).
Seifert–van Kampen theorem and cellular approximation for \(k\)-dimensional CW complexes.
1.2. By the Hurewicz Theorem:\[ \pi_k(X) \cong H_k(X; \mathbb{Z}) \cong \bigoplus_{j=1}^n H_k(S^k; \mathbb{Z}) \cong \mathbb{Z}^n. \]
Hurewicz isomorphism theorem for \((k-1)\)-connected spaces and homology of wedge sums.
1.3. Let \(\iota_j: S^k \hookrightarrow X\) denote the inclusion of the \(j\)-th sphere summand; their homotopy classes \(\{[\iota_1], \dots, [\iota_n]\}\) form a standard \(\mathbb{Z}\)-basis for \(\pi_k(X)\).
generators of the homology of the wedge sum.
2. Characterize the monoid of based homotopy classes \([X, X]_*\):
2.1. By the universal property of the wedge sum, a based map \(f: X \to X\) is determined up to based homotopy by its restrictions \(f \circ \iota_j: S^k \to X\) for \(j = 1, \dots, n\).
\([A \vee B, X]_* \cong [A, X]_* \times [B, X]_*\).
2.2. Each restriction \([f \circ \iota_j] \in \pi_k(X) \cong \mathbb{Z}^n\).
definition of the \(k\)-th homotopy group.
2.3. Thus \([X, X]_* \cong \prod_{j=1}^n \pi_k(X) \cong M_n(\mathbb{Z})\), where \(f\) corresponds to the matrix \(M(f) \in M_n(\mathbb{Z})\) whose \(j\)-th column is the coordinate vector of \(f_*([\iota_j])\) in the basis \(\{[\iota_1], \dots, [\iota_n]\}\).
column-by-column representation of linear maps on \(\mathbb{Z}^n\).
2.4. The assignment \(f \mapsto M(f)\) preserves composition: \(M(g \circ f) = M(g) M(f)\).
functoriality of induced maps on homotopy groups \((g \circ f)_* = g_* \circ f_*\).
3. Characterize homotopy equivalences via Whitehead’s Theorem:
3.1. \(X\) is a finite CW complex of dimension \(k\) with cells only in dimensions \(0\) and \(k\).
cellular structure of a wedge of spheres.
3.2. By Whitehead’s Theorem, a based map \(f: X \to X\) between simply connected CW complexes of dimension \(k\) is a homotopy equivalence if and only if \(f_*: \pi_k(X) \to \pi_k(X)\) is an isomorphism of abelian groups.
Whitehead’s Theorem for CW complexes.
3.3. The group homomorphism \(f_*: \mathbb{Z}^n \to \mathbb{Z}^n\) is an isomorphism if and only if its representation matrix \(M(f)\) is invertible over \(\mathbb{Z}\), which means \(M(f) \in \operatorname{GL}_n(\mathbb{Z})\) (i.e. \(\det M(f) = \pm 1\)).
invertible \(\mathbb{Z}\)-linear endomorphisms of \(\mathbb{Z}^n\) form the group \(\operatorname{GL}_n(\mathbb{Z})\).
4. Conclusion: The map \([f] \mapsto M(f)\) induces an isomorphism of groups:\[ \operatorname{Aut}\left(\bigvee_{j=1}^n S^k\right) \cong \operatorname{GL}_n(\mathbb{Z}). \]
2 and 3.
Q.E.D.