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A space is acyclic if the reduced integral homology groups are all trivial.

I want to know whether there exists a closed manifold which is acyclic.

A necessary condition is that this manifold will not be orientable. Another necessary condition is that it will not be a surface because of the known results about classifications of closed surfaces.

So the question becomes: Is there a non-orientable acyclic closed manifold of dimension 3 or more?

Thank you in advance.

Djamel
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  • I assume you don't allow boundary? – Timmathy Oct 28 '21 at 13:12
  • yes I require a compact manifold without boundary. – Djamel Oct 28 '21 at 15:45
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    In some sense, this is a duplicate of https://math.stackexchange.com/questions/1606611/first-homology-group-of-non-orientable-manifold because that MSE question and answer establish that non-orientable implies non-acyclic. If you want, you can write up your own answer. – Jason DeVito - on hiatus Oct 28 '21 at 20:36

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