I am trying to find if there is a orientable compact manifold $M$ of dimension 10 with the 5th cohomology group of De Rham $H^5_{DR}(M)\cong \mathbb R$.

But, I can find such an example or prove that it is not possible. Anyone have some ideas?

Thank you.

  • $\begingroup$ Do you know about Poincaré duality? $\endgroup$ – Najib Idrissi Dec 9 '14 at 18:21

It's not possible. There is a nondegenerate antisymmetric pairing $H^5\otimes H^5\to \mathbb R$. This means that $H^5$ can't be $1$ dimensional.

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    $\begingroup$ ...because in fact it must be even-dimensional. More generally if $5$ is replaced by any odd number. $\endgroup$ – Qiaochu Yuan Dec 9 '14 at 19:39
  • $\begingroup$ @QiaochuYuan Nice! But how to show it? $\endgroup$ – Chieh LIU Dec 9 '14 at 22:06
  • $\begingroup$ @Chieh: it follows by Poincare duality. On a closed orientable $4k+2$-manifold there is a nondegenerate antisymmetric pairing $H^{2k+1} \times H^{2k+1} \to H^{4k+2}$, and such things only exist on even-dimensional vector spaces (en.wikipedia.org/wiki/Symplectic_vector_space). $\endgroup$ – Qiaochu Yuan Dec 9 '14 at 22:11
  • $\begingroup$ @QiaochuYuan Got it. Thank you! $\endgroup$ – Chieh LIU Dec 9 '14 at 22:15

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