Simple examples are usual even spheres. Also the product of $2n$-sphere with real euclidean space of dimension $2r$ where $r<n$, is the example of such kinds. In closed oriented case the rational homology spheres are the only examples due to Poincare duality. Can anybody share some examples of such kind of manifolds with rich rational cohomology.
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3$\begingroup$ What do you mean by "even dimensional connected manifolds whose all nonzero rational cohomology groups are greater than the half of the dimension of manifolds" ? Do mean each of their rank or number of such groups? $\endgroup$– Anubhav MukherjeeOct 14, 2017 at 18:21
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$\begingroup$ Sorry, Please do not consider zeroth cohomology group. Let dim(M)=d. All rational cohomology groups are trivial whose degrees between 0 and d/2+1. $\endgroup$– King KhanOct 15, 2017 at 5:12
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1$\begingroup$ If M has nontrivial rational cohomology group then its degree greater then the half of the dim(M). Rationally acyclic spaces are trivial examples of such kind. $\endgroup$– King KhanOct 15, 2017 at 5:39
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