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?


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) ?

  • $\begingroup$ 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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