Lagmental Vicfred

Homology Quotients Cycles by Boundaries by Vicfred

tags: mathematics, homology, chain complexes, exact sequences

The nth homology group retains closed n-chains after declaring filled cycles trivial. I want the notation, the mechanism, and the failure mode visible at the same time.

Set-up

A chain complex \((C_\bullet,\partial)\) satisfies \(\partial_{n-1}\partial_n=0\). Homology \(H_n=\ker\partial_n/\operatorname{im}\partial_{n+1}\) measures cycles not explained as boundaries.

$$ Z_n=\ker\partial_n,\qquad B_n=\operatorname{im}\partial_{n+1} $$

The formulas should not be merged too early. The datum \(\mathsf D\), the conclusion \(\mathsf C\), and the bridge \(\Longrightarrow\) have three different logical jobs.

$$ H_n(C_\bullet)=Z_n/B_n $$

The calculation

Now evaluate one representative case. The result should agree with the structural law above, but it is obtained without assuming the conclusion.

$$ 0\subseteq B_n\subseteq Z_n\subseteq C_n,\qquad\begin{cases}z\in Z_n&\Longleftrightarrow\partial z=0,\\z\in B_n&\Longleftrightarrow z=\partial c.\end{cases} $$

What survives abstraction

The invariant statement is the one that does not depend on a convenient choice of coordinates, representatives, basis, or enumeration.

$$ \begin{aligned} \mathsf{D}\;&:\quad Z_n=\ker\partial_n,\qquad B_n=\operatorname{im}\partial_{n+1},\\[5pt] \mathsf{C}\;&:\quad H_n(C_\bullet)=Z_n/B_n. \end{aligned} $$

The boundary

Homology depends on the chosen coefficient ring. Torsion may disappear over a field or change under reduction modulo a prime.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] H_n(C_\bullet)=Z_n/B_n \end{gathered}} $$

With the dependency made explicit, the same pattern can be recognised safely in nearby problems. A changed hypothesis should now be easy to spot.

This article was posted on Mon 26 January 2009. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.