Show that if \(M\) is a manifold of dimension \(2n\) for which there exists a fiber bundle \(S^1 \to S^{2n+1} \to M\), then \(M\) is simply-connected and \(H^*(M; \mathbb{Z}) \approx H^*(\mathbb{CP}^n; \mathbb{Z})\) as rings. Conversely, if \(M\) is simply-connected and \(H^*(M; \mathbb{Z}) \approx H^*(\mathbb{CP}^n; \mathbb{Z})\) as rings, show there is a bundle \(S^1 \to E \to M\) where \(E \simeq S^{2n+1}\).
Solution
Goal. Prove the two directions relating a circle bundle \(S^1 \to S^{2n+1} \to M\) to the cohomology ring of \(\CP^n\).
1. (\(\Rightarrow\)) Given \(S^1 \to S^{2n+1} \to M\), show \(M\) is simply-connected and \(H^*(M;\ZZ) \cong H^*(\CP^n;\ZZ)\).
1.1. \(M\) is simply-connected.
the long exact sequence of the fibration gives \(\pi_1(S^{2n+1}) = 0 \to \pi_1(M) \to \pi_0(S^1) = 0\), so \(\pi_1(M) = 0\) (using that \(S^{2n+1}\) is simply-connected and the fiber \(S^1\) is connected).
1.2. The Gysin sequence of the circle bundle gives \(H^*(M;\ZZ) \cong \ZZ[\alpha]/(\alpha^{n+1})\) with \(|\alpha| = 2\).
the Gysin sequence for the \(S^1\)-bundle \(S^1 \to S^{2n+1} \to M\) (with Euler class \(e \in H^2(M)\)) gives \(H^*(M) \cong \ZZ[e]/(e^{n+1})\), matching \(H^*(\CP^n;\ZZ) = \ZZ[\alpha]/(\alpha^{n+1})\).
1.3. Hence \(H^*(M;\ZZ) \cong H^*(\CP^n;\ZZ)\) as rings.
2.2.
2. (\(\Leftarrow\)) Given \(M\) simply-connected with \(H^*(M;\ZZ) \cong H^*(\CP^n;\ZZ)\), construct \(S^1 \to E \to M\) with \(E \simeq S^{2n+1}\).
2.1. \(H^2(M;\ZZ) = \ZZ\), generated by a class \(\alpha\).
\(H^2(\CP^n;\ZZ) = \ZZ\).
2.2. Let \(E\) be the total space of the principal \(S^1\)-bundle with Euler class \(\alpha\) (the generator of \(H^2(M;\ZZ)\)).
principal \(S^1\)-bundles over \(M\) are classified by \(H^2(M;\ZZ) = \ZZ\), and we take the bundle corresponding to the generator.
2.3. The Gysin sequence shows \(H^*(E;\ZZ) \cong H^*(S^{2n+1};\ZZ)\).
the Gysin sequence for this bundle, with Euler class \(\alpha\) (a generator), gives \(H^*(E) \cong H^*(S^{2n+1})\) (the cohomology of a sphere).
2.4. Since \(E\) is simply-connected (by the LES, \(\pi_1(E) = 0\)), and \(H^*(E) \cong H^*(S^{2n+1})\), Whitehead’s theorem gives \(E \simeq S^{2n+1}\).
a simply-connected space with the cohomology of \(S^{2n+1}\) is homotopy equivalent to \(S^{2n+1}\).
3. Q.E.D.
1 and 2 give both directions.