Exact Sequences & the Long Exact Sequence

Image equals kernel: the bookkeeping that turns unknown homology groups into arithmetic.

The idea

An exact sequence is the principal device for computing a homology group we do not know from groups we do.

Definition (Exact sequence).

A sequence of abelian groups and homomorphisms $\cdots \to A_{n+1} \xrightarrow{\ \alpha_{n+1}\ } A_{n} \xrightarrow{\ \alpha_{n}\ } A_{n-1} \to \cdots$ is exact at $A_{n}$ when $\operatorname{im} \alpha_{n+1} = \ker \alpha_{n}$, and exact when it is exact at every group.

A chain complex only requires the containment $\operatorname{im} \alpha_{n+1} \subseteq \ker \alpha_{n}$, and the homology group at that spot measures how far the containment falls short of equality; so an exact sequence is a chain complex with no homology anywhere.

Exactness constrains each group by its neighbours, and zeros say the most. Exactness of $0 \to A \xrightarrow{\ \alpha\ } B$ at $A$ says $\alpha$ is injective, and exactness of $A \xrightarrow{\ \alpha\ } B \to 0$ at $B$ says $\alpha$ is surjective. So a short exact sequence $0 \to A \xrightarrow{\ \alpha\ } B \xrightarrow{\ \beta\ } C \to 0$ says exactly that $\alpha$ embeds $A$ in $B$ as a subgroup and that $\beta$ identifies $C$ with the quotient $B/A$.

Topology supplies exact sequences in quantity. A subspace $A \subseteq X$ generates one long exact sequence relating the homology of $A$, of $X$, and of the pair through every degree — and we usually find an unknown group in it by locating a zero two places away.

Ways to work on it

• Walkthrough. Exactness, short exact sequences, relative homology, and the long exact sequence of a pair run on the disk and its boundary circle.
• Proof. The diagram chase showing the connecting homomorphism is well defined — the construction every long exact sequence is built from.
• Practice. Decide exactness at a spot, pin down the middle of a short exact sequence, and fill gaps in a long one.
• Hardest. Compute the homology of the Möbius band relative to its boundary.

Not sure where to start? Take the ten-question placement test.