Lagmental Vicfred

Coboundaries Reverse the Chain Direction by Vicfred

tags: mathematics, cohomology, cup products, Poincare duality

A cochain evaluates chains, and its coboundary is precomposition with the boundary map. A small computation will anchor the general statement before the abstraction takes over.

Notation

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.

$$ C^n(X;R)=\operatorname{Hom}(C_n(X),R) $$

A reliable calculation names domain and codomain. The notation \(\mathsf{data}\mapsto\mathsf{claim}\) is harmless only after both \(\operatorname{dom}\) and \(\operatorname{cod}\) have been fixed.

$$ (d\varphi)(c)=\varphi(\partial c),\qquad d^2=0 $$

Stress the formula

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

$$ C_{n+1}\xrightarrow{\partial_{n+1}}C_n\xrightarrow{\partial_n}C_{n-1}\quad\Longrightarrow\quad C^{n-1}\xrightarrow{d^{n-1}}C^n\xrightarrow{d^n}C^{n+1} $$

Interpretation

The formula is reusable precisely because it says which pieces are structural and which belong only to the worked example.

$$ \begin{aligned} \mathsf{D}\;&:\quad C^n(X;R)=\operatorname{Hom}(C_n(X),R),\\[5pt] \mathsf{C}\;&:\quad (d\varphi)(c)=\varphi(\partial c),\qquad d^2=0. \end{aligned} $$

Limit of the argument

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

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] (d\varphi)(c)=\varphi(\partial c),\qquad d^2=0 \end{gathered}} $$

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 Sun 03 July 2011. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.