Exact Sequences and Excision - Core Definitions

Exact sequences provide the algebraic framework for computing homology through decomposition, while excision enables cutting and pasting arguments.

Definition

A sequence of abelian groups and homomorphisms $\cdots \to A_{n+1} \xrightarrow{f_{n+1}} A_n \xrightarrow{f_n} A_{n-1} \to \cdots$ is exact at $A_n$ if $\text{im}(f_{n+1}) = \ker(f_n)$ . The sequence is exact if it's exact at every group.

Exactness means "the image equals the kernel": elements mapped to zero are precisely those coming from the previous map. This captures algebraic dependencies systematically.

Example

A short exact sequence $0 \to A \xrightarrow{f} B \xrightarrow{g} C \to 0$ means:

$f$ is injective (since $\ker(f) = \text{im}(0) = 0$ )
$g$ is surjective (since $\text{im}(g) = \ker(0) = C$ )
$\text{im}(f) = \ker(g)$ , so $C \cong B / \text{im}(f)$

Definition

For a pair $(X, A)$ , the long exact sequence of the pair is: $\cdots \to H_n(A) \xrightarrow{i_*} H_n(X) \xrightarrow{j_*} H_n(X, A) \xrightarrow{\partial} H_{n-1}(A) \to \cdots$

Here $i : A \hookrightarrow X$ is inclusion, $j : X \to (X, A)$ is the quotient map, and $\partial$ is the connecting homomorphism.

The connecting map $\partial : H_n(X, A) \to H_{n-1}(A)$ has degree $-1$ and is natural. It's constructed from the snake lemma applied to the short exact sequence of chain complexes $0 \to C_\bullet(A) \to C_\bullet(X) \to C_\bullet(X, A) \to 0$ .

Theorem

(Excision Theorem) Let $X = A \cup B$ where $\overline{Z} \subseteq \text{int}(A)$ for some $Z \subseteq A \cap B$ . Then excising $Z$ induces isomorphisms: $H_n(X \setminus Z, A \setminus Z) \xrightarrow{\cong} H_n(X, A)$ for all $n$ . Cutting out $Z$ doesn't affect relative homology.

Example

Write $S^n = D^n_+ \cup D^n_-$ (hemispheres). Excise a small disk around a point in $D^n_- \setminus S^{n-1}$ . Then: $H_n(S^n, D^n_-) \cong H_n(D^n_+, S^{n-1})$ The right side is $H_n(D^n_+/S^{n-1}) = H_n(S^n)$ since collapsing the boundary gives $S^n$ .

Definition

A split exact sequence is a short exact sequence $0 \to A \to B \to C \to 0$ where $B \cong A \oplus C$ (as groups, not canonically). Splitting means there's a section $C \to B$ or retraction $B \to A$ .

Remark

Exact sequences encode how algebraic objects decompose. The long exact sequence of a pair shows that $H_n(X, A)$ measures the "difference" between $H_n(X)$ and $H_n(A)$ . Excision says this difference is local, depending only on neighborhoods.