Is there a connected closed 5-manifold $M$ such that $\oplus_{i\geq 0} H^i(M, \mathbb{Z})$ is generated by $H^1(M, \mathbb{Z})\oplus H^2(M, \mathbb{Z})\oplus H^3(M, \mathbb{Z})$ but is not generated by $H^1(M, \mathbb{Z})\oplus H^2(M, \mathbb{Z})$ or $H^1(M, \mathbb{Z})\oplus H^3(M, \mathbb{Z})$ or $H^2(M, \mathbb{Z})\oplus H^3(M, \mathbb{Z})$?
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4$\begingroup$ The connected sum of $S^1\times\mathbb{CP}^2$ and $S^2\times S^3$ seems to do the job. $\endgroup$– Bertram ArnoldCommented Mar 30, 2021 at 12:39
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4$\begingroup$ With that said, your recent questions give the impression that there is some general statement that you want to check. The counterexamples that were given as answers can be easily obtained by first finding counterexamples in graded rings and then realizing them as cohomology algebras by hand. In fact, rationally it is understood which graded rings arise as cohomology algebras, cf. link.springer.com/article/10.1007/s10711-015-0135-z $\endgroup$– Bertram ArnoldCommented Mar 30, 2021 at 12:45
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5$\begingroup$ I think as written, this is an exaggeration: in this later paper with one overlapping author arxiv.org/pdf/1702.07892.pdf it is mentioned that the authors do not know whether there exists a manifold in dimension $2^{10}$ such that $b_i = 1$ for $i = 0, 2^9, 2^{10}$ and all other Betti numbers vanish. I believe this is still open. $\endgroup$– Jens ReinholdCommented Mar 30, 2021 at 13:12
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1$\begingroup$ @JensReinhold For this particular case of dimension $2^{10}$ you mention, Kreck and Zagier have ruled out the existence of such a manifold - see around 40:40 in this talk by Zagier youtube.com/watch?v=lLEWKDtCPTU (but in general the question of which dimensions support closed manifolds with cohomology $\mathbb{Q}[x]/(x^3)$ is indeed still open as far as I know). $\endgroup$– Aleksandar MilivojevićCommented Mar 30, 2021 at 17:48
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2$\begingroup$ Ah, thanks! Actually I once heared Kreck talking about this project, too (math.ku.dk/english/calendar/events/algtopseminar-10092018a), but the only thing I remember from this talk is that the prime 34511 seems to be peculiar, as it is the smallest odd middle Betti number of a manifold of dimension $2^{25}$ whose Betti numbers otherwise all vanish (except in bottom and top degree). Around this time, the very same prime showed up in my own work (arxiv.org/abs/1807.11539 Rem. 2.20) but in a rather different dimension and I still have no satisfactory explanation for this... $\endgroup$– Jens ReinholdCommented Mar 30, 2021 at 19:05
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