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== Solution == <wikitex>; This problem was in fact solved by Arkowitz and Lupton {{cite|Arkowitz&Lupton2000|Examples 5.1 & 5.2}}: ''There do exist inflexible closed simply connected manifolds''. Arkowitz and Lupton give examples of simply connected rational Poincaré differential graded algebras, $\mathcal{M}_1$ and $\mathcal{M}_2$ of dimensions 208 and 228. Both of these algebras have a finite set of homotopy classes of self-maps and so the corresponding rational Poincaré complexes are inflexible. They also indicate why the algebras $\mathcal{M}_1$ and $\mathcal{M}_2$ can be realised by closed simply connected manifolds $M_1$ and $M_2$. It follows that the manifolds $M_1$ and $M_2$ are inflexible. The realisation of $\mathcal{M}_1$ and $\mathcal{M}_2$ relies upon a theorem proven independently by Barge and Sullivan. A special case of this theorem is as follows: {{beginthm|Theorem|{{cite|Barge1976|Theorem 1}}{{cite|Sullivan1977|Theorem 13.2}}}} Let $\mathcal{M}$ be a simply connected rational Poincaré differential graded algebra of dimension n. If either * $n \neq 4k$ or * $n = 4k$ and the intersection form of $\mathcal{M}$ represents the trivial element of $W_0(\Qq)$, the Witt group of $\Qq$, then there is a closed simply connected smooth manifold $M$ with rational homotopy type given by $\mathcal{M}$. {{endthm}} {{beginthm|Remark}} Note that Arkowitz and Lupton {{cite|Arkowitz&Lupton2000}} state that the theorem of Barge-Sullivan applies and do not give detailed arguments why $\mathcal{M}_1$ or $\mathcal{M}_2$ satisfy the hypotheses of Barge-Sullivan. See the [[Talk:Self-maps of simply connected manifolds#Details for Arkowitz and Lupton's paper|discussion page]] for more information. {{endthm}} </wikitex>

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