Using the Künneth formula, show that \(\Sigma K(\mathbb{Z}_m \times \mathbb{Z}_n, 1) \simeq \Sigma K(\mathbb{Z}_m, 1) \vee \Sigma K(\mathbb{Z}_n, 1)\) if \(m\) and \(n\) are relatively prime.
Solution
1. Suspension of a Cartesian product:
1.1. For any pointed CW complexes \(X\) and \(Y\), there is a natural homotopy equivalence:\[ \Sigma(X \times Y) \simeq \Sigma X \vee \Sigma Y \vee \Sigma(X \wedge Y). \]
Hatcher Proposition 4I.1 (suspension splits product into wedges with the smash product).
1.2. Let \(X = K(\mathbb{Z}_m, 1)\) and \(Y = K(\mathbb{Z}_n, 1)\). Since \(K(G \times H, 1) \simeq K(G, 1) \times K(H, 1)\), we have:\[ \Sigma K(\mathbb{Z}_m \times \mathbb{Z}_n, 1) \simeq \Sigma K(\mathbb{Z}_m, 1) \vee \Sigma K(\mathbb{Z}_n, 1) \vee \Sigma\big(K(\mathbb{Z}_m, 1) \wedge K(\mathbb{Z}_n, 1)\big). \]
product of Eilenberg–MacLane spaces.
2. Show that \(\Sigma(X \wedge Y)\) is contractible when \(\gcd(m, n) = 1\):
2.1. The reduced homology groups \(\widetilde{H}_i(X)\) are non-zero only in odd degrees, where \(\widetilde{H}_{2k+1}(K(\mathbb{Z}_m, 1)) \cong \mathbb{Z}_m\). In particular, \(m \cdot \widetilde{H}_i(X) = 0\) for all \(i \ge 0\).
homology of lens spaces / \(K(\mathbb{Z}_m, 1)\).
2.2. Symmetrically, \(n \cdot \widetilde{H}_j(Y) = 0\) for all \(j \ge 0\).
homology of \(K(\mathbb{Z}_n, 1)\).
2.3. By Bézout’s Identity, \(\gcd(m, n) = 1\) implies there exist integers \(u, v \in \mathbb{Z}\) such that \(um + vn = 1\).
Euclidean algorithm.
2.4. For any \(m\)-torsion abelian group \(A\) and \(n\)-torsion abelian group \(B\):- \(A \otimes_\mathbb{Z} B = 0\), because \(a \otimes b = (um + vn)(a \otimes b) = u(ma \otimes b) + v(a \otimes nb) = 0\).- \(\operatorname{Tor}_1^\mathbb{Z}(A, B) = 0\), because Tor is annihilated by both \(m\) and \(n\), hence by \(\gcd(m, n) = 1\).
algebra of torsion abelian groups.
2.5. By the Künneth formula for smash products:\[ \widetilde{H}_k(X \wedge Y) \cong \bigoplus_{i+j=k} \big(\widetilde{H}_i(X) \otimes \widetilde{H}_j(Y)\big) \oplus \bigoplus_{i+j=k-1} \operatorname{Tor}_1^\mathbb{Z}\big(\widetilde{H}_i(X), \widetilde{H}_j(Y)\big) = 0 \quad \text{for all } k \ge 0. \]
Künneth formula for spaces and 4.
2.6. The suspension \(\Sigma(X \wedge Y)\) is simply connected and has all reduced homology groups zero:\[ \widetilde{H}_*(\Sigma(X \wedge Y)) \cong \widetilde{H}_{*-1}(X \wedge Y) = 0. \]
suspension isomorphism in homology.
2.7. By Whitehead’s Theorem for simply connected CW complexes with trivial homology, \(\Sigma(X \wedge Y) \simeq *\).
Whitehead’s Theorem and Hurewicz Theorem.
3. Conclusion:\[ \Sigma K(\mathbb{Z}_m \times \mathbb{Z}_n, 1) \simeq \Sigma K(\mathbb{Z}_m, 1) \vee \Sigma K(\mathbb{Z}_n, 1) \vee * \simeq \Sigma K(\mathbb{Z}_m, 1) \vee \Sigma K(\mathbb{Z}_n, 1). \]Q.E.D.
1 and 2.