Singular Homology - Key Properties

Singular homology satisfies fundamental axioms (Eilenberg-Steenrod axioms) that characterize it uniquely and make it a powerful computational tool.

Theorem

(Homotopy Invariance) If $f, g : X \to Y$ are homotopic continuous maps, then the induced homomorphisms $f_*, g_* : H_n(X) \to H_n(Y)$ are equal for all $n \geq 0$ .

Proof idea: A homotopy $H : X \times [0,1] \to Y$ induces a chain homotopy $P : C_n(X) \to C_{n+1}(Y)$ satisfying $\partial P + P\partial = g_\# - f_\#$ on chains. This forces $f_* = g_*$ on homology.

Theorem

(Functoriality) Singular homology defines a functor from topological spaces to graded abelian groups:

$(\text{id}_X)_* = \text{id}_{H_\bullet(X)}$
$(g \circ f)_* = g_* \circ f_*$ for continuous maps $f, g$

Definition

A continuous map $f : X \to Y$ induces homomorphisms $f_\# : C_n(X) \to C_n(Y)$ by composition: $f_\#(\sigma) = f \circ \sigma$ for singular simplices $\sigma : \Delta^n \to X$ . Since $\partial(f_\# \sigma) = f_\#(\partial \sigma)$ , this descends to homology: $f_* : H_n(X) \to H_n(Y)$ .

Theorem

(Dimension Axiom) For a one-point space $\{*\}$ : $H_n(\{*\}) = \begin{cases} \mathbb{Z} & n = 0 \\ 0 & n > 0 \end{cases}$

Theorem

(Exactness) For a pair $(X, A)$ where $A \subseteq X$ , there is a long exact sequence: $\cdots \to H_n(A) \xrightarrow{i_*} H_n(X) \xrightarrow{j_*} H_n(X, A) \xrightarrow{\partial} H_{n-1}(A) \to \cdots$ where $H_n(X, A) = H_n(C_\bullet(X)/C_\bullet(A))$ is the relative homology.

Definition

Relative homology $H_n(X, A)$ measures cycles in $X$ that are boundaries when restricted to $A$ . The long exact sequence relates absolute and relative homology systematically.

Theorem

(Excision) If $Z \subseteq A \subseteq X$ with $\overline{Z} \subseteq \text{int}(A)$ , then the inclusion $(X \setminus Z, A \setminus Z) \hookrightarrow (X, A)$ induces isomorphisms: $H_n(X \setminus Z, A \setminus Z) \cong H_n(X, A)$ for all $n$ . "Cutting out" $Z$ doesn't affect relative homology.

Remark

The Eilenberg-Steenrod axioms (homotopy invariance, exactness, excision, dimension) completely characterize singular homology on the category of CW pairs. Any other theory satisfying these axioms is naturally isomorphic to singular homology.

Example

For spheres $S^n$ , singular homology gives: $H_k(S^n) = \begin{cases} \mathbb{Z} & k = 0, n \\ 0 & \text{otherwise} \end{cases}$ This can be proven using Mayer-Vietoris or by recognizing $S^n$ as a CW complex.