Construction of connected closed oriented $4d-$dimensional manifold whose has only middle homology group is nontrivial.

Over integer see paper of J. Eells, Jr. & N.H. Kuiper, Manifolds which are like projective planes.


Theorem A of Smooth manifolds with prescribed rational cohomology ring by Fowler and Su shows that the only dimensions in which a smooth such manifold can exist is $n = 2^a$ or $8(2^a + 2^b)$ where $a \neq b$. Note, while the authors are concerned with simply connected such manifolds, the conclusion does not require any hypothesis on the fundamental group.

In the earlier paper Rational analogues of projective planes, Su showed that there are infinitely many homeomorphism types of such manifolds in dimension $32$.

In a recent paper, On dimensions supporting a rational projective plane, Kennard and Su show that there are examples in dimensions $128$ and $256$ (Theorem A). Moreover, there are infinitely many dimensions of the form $2^a$ and infinitely many of the form $8(2^a + 2^b)$ with $a \neq b$ for which there is no such manifold which is simply connected.

Note, all known examples have dimension of the form $2^a$.

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Manifolds

Algebraic Topology

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