Lagmental Vicfred

Mayer--Vietoris Computes Homology from Two Pieces by Vicfred

Last updated: Sun 30 March 2014

tags: mathematics, homology, chain complexes, exact sequences

An open cover yields a long exact sequence involving the intersection, pieces, and union. Keeping the exact identity in view prevents the geometric or probabilistic intuition from drifting.

Definitions first

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.

$$ X=U\cup V $$

The definition determines which expressions are legal; only then does the identity become meaningful. An equality in \(\mathcal A\) may change ambient meaning, so I keep \(\mathsf D\) separate from \(\mathsf C\).

$$ \cdots\to H_n(U\cap V)\to H_n(U)\oplus H_n(V)\to H_n(X)\xrightarrow{\delta}H_{n-1}(U\cap V)\to\cdots $$

A small case in full

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

$$ S^n=U\cup V,\quad U,V\simeq\ast,\quad U\cap V\simeq S^{n-1}\Longrightarrow\widetilde H_k(S^n)\cong\widetilde H_{k-1}(S^{n-1}) $$

The reusable statement

The aligned summary deliberately puts the datum and conclusion on different rows. Mathematically, this is the distinction between specifying an object and proving a property of it.

$$ \begin{aligned} \mathsf{D}\;&:\quad X=U\cup V,\\[5pt] \mathsf{C}\;&:\quad \cdots\to H_n(U\cap V)\to H_n(U)\oplus H_n(V)\to H_n(X)\xrightarrow{\delta}H_{n-1}(U\cap V)\to\cdots. \end{aligned} $$

A nearby false statement

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

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] \cdots\to H_n(U\cap V)\to H_n(U)\oplus H_n(V)\to H_n(X)\xrightarrow{\delta}H_{n-1}(U\cap V)\to\cdots \end{gathered}} $$

I would use the boxed line as a reference later, while returning to the full display whenever a hypothesis becomes uncertain. That division keeps compression from becoming ambiguity.

This article was posted on Tue 11 February 2014. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.