Singular Homology - Main Theorem

The Eilenberg-Steenrod axioms completely characterize singular homology, establishing it as the canonical homology theory for topological spaces.

Theorem

(Eilenberg-Steenrod Axioms) Singular homology satisfies:

Homotopy: Homotopic maps induce the same homomorphisms on homology
Exactness: For pairs $(X, A)$ , the sequence $H_n(A) \to H_n(X) \to H_n(X, A)$ is exact
Excision: $H_n(X \setminus Z, A \setminus Z) \cong H_n(X, A)$ when $\overline{Z} \subseteq \text{int}(A)$
Dimension: $H_n(\{*\}) = 0$ for $n \neq 0$ and $H_0(\{*\}) = \mathbb{Z}$

Theorem

(Uniqueness) On the category of CW pairs, any homology theory satisfying the Eilenberg-Steenrod axioms is naturally isomorphic to singular homology. This makes singular homology the unique "ordinary" homology theory.

Theorem

(Long Exact Sequence of a Pair) For any pair $(X, A)$ , there's a natural long exact sequence: $\cdots \to H_n(A) \xrightarrow{i_*} H_n(X) \xrightarrow{j_*} H_n(X, A) \xrightarrow{\partial} H_{n-1}(A) \to \cdots$

The connecting homomorphism $\partial$ has degree $-1$ and makes this sequence exact everywhere.

Construction of $\partial$ : Given a relative cycle $\alpha \in Z_n(X, A)$ , write $\alpha = c + c'$ where $c \in C_n(X)$ and $c' \in C_n(A)$ . Then $\partial\alpha = [\partial c] \in H_{n-1}(A)$ . This is well-defined and natural.

Theorem

(Excision Theorem) Let $X = A \cup B$ with $X = \text{int}(A) \cup \text{int}(B)$ . Then: $H_n(A, A \cap B) \cong H_n(X, B)$ for all $n$ . This allows "cutting and pasting" in homology computations.

Example

Computing $H_n(S^n)$ : Use excision with $S^n = D^n_+ \cup D^n_-$ where the disks overlap at the equator $S^{n-1}$ . By excision: $H_n(D^n_+, S^{n-1}) \cong H_n(S^n, D^n_-)$ The left side is $H_n(D^n/S^{n-1}) = H_n(S^n)$ (collapsing boundary gives $S^n$ ). The right side fits in the exact sequence of $(S^n, D^n_-)$ with $H_n(D^n_-) = 0$ , yielding $H_n(S^n) = \mathbb{Z}$ .

Theorem

(Universal Coefficients) For any abelian group $G$ : $0 \to H_n(X) \otimes G \to H_n(X; G) \to \text{Tor}(H_{n-1}(X), G) \to 0$ This splits (non-naturally), relating homology with different coefficient groups.

Remark

The axioms make homology computable: excision reduces problems to local data, exactness provides inductive tools, and homotopy invariance connects geometric spaces to algebraic invariants. This systematic approach is homological algebra.