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When I calculate the de Rham Cohomologies of point-punctured manifolds I come upon this apparent contradiction and was hoping somebody can resolve my confusion. Let $h^kM$ be the dimension of the $k$-th de Rham Cohomology space. Consider the following (are these true?) facts:

Fact I: Sphere Cohomologies: For the $n-1$-sphere $S^{n-1}$ we have $h^k S^{n-1}=1$ if $k=0$ or $n-1$ and $h^k S^{n-1}=0$ otherwise.

Fact II: Poincaré Duality: For every compact, connected orientable $n$-manifold $h^kM=h^{n-k}M$.

Fact III: If $M$ is compact $h^0M$ is the number of connected components.

Consider a compact, connected, orientable $n$-manifold $M$ and fix some $p \in M$. Let $U=M-\{p\}$ and let $V$ be a small open ball about $p$. Use the Mayer-Vietoris exact sequence: $$ 0 \to H^0M \to H^0U \oplus H^0V \to H^0 U \cap V \to H^1M \to H^1U \oplus H^1V \to H^1 U \cap V $$ As $U \cap V$ retracts onto an $n-1$-sphere, it has the $n-1$-sphere's cohomology. Further $V$ is contractible to a point so it's cohomology is the trivial space. We have using the above facts that the sequence translates to: $$ 0=0+1-1+1-h^1M+h^1U-0 $$ so that $h^1U=h^1M-1$. By Poincaré duality $h^{n-1}U=h^{n-1}M-1$. Now consider the exact sequence $$ H^{n-2}U \cap V \to H^{n-1}M \to H^{n-1}U\oplus H^{n-1}V \to H^{n-1} U \cap V \to H^n M \to H^nU \oplus H^nV \to H^n U \cap V $$ which translates down to $$ 0=0+h^{n-1}M-h^{n-1}U + 1-h^nM+h^nU -0 $$ Using Poincaré duality we conclude $h^nM=h^0M=1$ and $h^nU=h^0U=1$ as $U$ and $M$ are connected. Hence $h^{n-1}U= h^{n-1}M+1$, which is exactly in contradiction with the previous result. Something is terribly wrong here. Ideas?

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For you fact III, it is true for any topological space, not necessarily compact.

Why do you say that you can use Poincaré Duality on $\mathrm U$ (which is not compact) ?

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  • $\begingroup$ Actually, number of path components would be more accurate if we are working in the generality of all topological spaces... $\endgroup$ – Zhen Lin Apr 10 '13 at 8:44

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