Lagmental Vicfred

Poincaré Duality Pairs Complementary Degrees by Vicfred

tags: mathematics, cohomology, cup products, Poincare duality

A closed oriented n-manifold has a perfect pairing between degree k and degree n minus k cohomology. The point is to make the formal expression readable enough to audit line by line.

Statement

Cohomology applies \(\operatorname{Hom}(-,R)\) to chains and reverses arrows. The cup product makes \(H^\ast(X;R)\) a graded ring rather than only a graded group.

$$ M^n\ \text{closed and oriented} $$

I keep the defining relation \(\mathsf D\) above the derived relation \(\mathsf C\). This exposes whether cancellation used \(x\ne0\) and whether the conclusion is canonical.

$$ H^k(M;R)\times H^{n-k}(M;R)\to R,\qquad(\alpha,\beta)\mapsto\langle\alpha\smile\beta,[M]\rangle $$

Worked algebra

The following line is the smallest calculation that still exercises the mechanism. It keeps nested delimiters and the order of operations explicit.

$$ H^k(M;R)\xrightarrow{\ \cap[M]\ }H_{n-k}(M;R) $$

Conceptual compression

The compact alignment is a local map of the argument: assumptions on the first row, consequence on the second. Any generalisation must preserve that dependency.

$$ \begin{aligned} \mathsf{D}\;&:\quad M^n\ \text{closed and oriented},\\[5pt] \mathsf{C}\;&:\quad H^k(M;R)\times H^{n-k}(M;R)\to R,\qquad(\alpha,\beta)\mapsto\langle\alpha\smile\beta,[M]\rangle. \end{aligned} $$

Caveat

Isomorphic cohomology groups do not guarantee isomorphic cohomology rings. Products can distinguish spaces that additive invariants cannot.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] H^k(M;R)\times H^{n-k}(M;R)\to R,\qquad(\alpha,\beta)\mapsto\langle\alpha\smile\beta,[M]\rangle \end{gathered}} $$

The final box is a summary, not a new assumption; the proof still lives in the definitions and the intervening calculation. The source keeps each scope delimiter visible for later inspection.

This article was posted on Wed 23 October 2013. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.