----- > [!proposition] Proposition. ([[middle cohomology of a compact oriented manifold]]) > Let $M$ be a [[compact]] $\mathbb{Z}$-[[(homological) orientation of a manifold|oriented]] [[manifold]] of dimension $2n$. Consider [[the perfect Poincare pairing]] $\langle -,- \rangle : H^{n}(M) \otimes H^{n}(M) \to \mathbb{Z},$ a [[bilinear map|bilinear form]] given by $\langle \varphi, \psi \rangle=(\varphi \smile \psi)[M]$. > For $n$ even, (so $4$ [[divides]] $\text{dim } M$) this is a [[symmetric multilinear map|symmetric form]] and for $n$ odd it is [[alternating multilinear map|skew-symmetric]]. ^proposition > [!proof]- Proof. ([[middle cohomology of a compact oriented manifold]]) > ~ ----- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```