Poincaré Duality Theorem

Theorem

Let $M$ be an $n$-manifold.

Let its $n$th homology group $\map {H_n} M$ be infinite and cyclic (that is, that $M$ is an orientable manifold).

Then $\map {H_r} M$ is homeomorphic to the $\paren {n - r}$th cohomology group $\map {H^{n - r} } M$ for all $r$.

This article needs proofreading. In particular: I am quoting Nelson who claims "Such a manifold is said to be orientable" but I've looked at that page and it seems there is a lot of work to do to establish that fact. If you believe all issues are dealt with, please remove {{Proofread}} from the code. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Proofread}} from the code.

Corollary

Let $V$ be an $n$-space.

Let $S$ and $T$ be subspaces of $M$ of dimension $r$ and $n - r$.

Then $S$ and $T$ usually meet at a single point.

Proof

This theorem requires a proof. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Source of Name

This entry was named for Jules Henri Poincaré.

Historical Note

Poincaré Duality Theorem/Historical Note

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Poincaré duality theorem
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Poincaré duality theorem