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Let $B$ be an even dimensional closed manifold. Suppose that $\pi: E \rightarrow B$ is a fibre bundle with fibre $F = S^{2}$. Suppose furthermore that there is a section $s : B \rightarrow E$ (i.e. such that $\pi \circ s = id_{B}$). Is it true that $H_{2}(E,\mathbb{R})$ is generated by $H_{2}(s(B),\mathbb{R})$ and $H_{2}(F,\mathbb{R})$? (after the appropriate inclusions).

Nick L
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  • I think De Rham's theorem might be useful here. – Camilo Arosemena-Serrato May 03 '18 at 18:10
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    This is at least true for orientable $S^2$-bundles. Use the Gysin sequence, the fact that the Euler class of an odd-rank vector bundle is zero in real cohomology, and that the existence of a section splits the short exact sequence. For the general case, try to relate to the orientable double cover (I haven't thought very hard about this). –  May 05 '18 at 11:30

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