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I have learned that, if we have a connected, oriented and compact n-dimensional manifold, the top de Rham cohomology is isomorphic to $\mathbb{R}$, i.e $$ H_{\text{dR}}^n(M) \cong \mathbb{R}.$$ However, Poincaré's lemma states that for any star-shaped subset of $ \mathcal{U} \subset \mathbb{R}^n$, it holds that $$ {H_{\text{dR}}^k(\mathcal{U}) = 0} \qquad \text{for all }k \neq 0.$$

Consider now the ball of radius $1$, this is evidently star shaped. Furthermore, it seems to me that it also suffices the properties in the first result. What is wrong in this reasoning?

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    $\begingroup$ In just the topological setting, the idea is that because $\mathcal U$ is star-shaped, it is contractible – i.e. it can be continuously deformed to a point, and because (singular) cohomology is invariant under such continuous deformations, the (singular) cohomology of $\mathcal U$ must be that of a point, i.e. trivial for all $k \neq 0$. I suspect something similar is true in the de Rham setting? $\endgroup$ – Rylee Lyman Jan 24 at 16:20
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    $\begingroup$ In your first statement you listed the property 'connected' twice. Is that a typo and is there a property missing? $\endgroup$ – quarague Jan 24 at 16:20
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    $\begingroup$ one problem with the open ball is that although it is bounded, it is not compact? $\endgroup$ – Rylee Lyman Jan 24 at 16:22
  • $\begingroup$ Yes, it had to be compact. Thanks! $\endgroup$ – Greg Jan 24 at 16:23
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    $\begingroup$ There is an additional assumption needed to conclude $H^n(M)\cong \mathbb{R}$: $M$ must have no boundary. This fails in your ball example, if you take the closed ball. $\endgroup$ – Jason DeVito Jan 24 at 16:26
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I know the statement in the form: "top de-Rham cohomology of compact, connected, oriented manifold without boundary is isomorphic to $\mathbb{R}$".

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    $\begingroup$ That's where I made my mistake. I swapped books, where in the first one a manifold is defined without boundary, and in the second one it is defined with boundary. Thank you very much! $\endgroup$ – Greg Jan 25 at 16:09

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