Singular Homology - Core Definitions

Singular homology extends simplicial homology to arbitrary topological spaces, not requiring triangulations. It is the most general and flexible homology theory.

Definition

The standard $n$ -simplex is $\Delta^n = \{(t_0, \ldots, t_n) \in \mathbb{R}^{n+1} : \sum t_i = 1, t_i \geq 0\}$ . A singular $n$ -simplex in a topological space $X$ is a continuous map $\sigma : \Delta^n \to X$ .

Unlike simplicial homology where simplices are actual subsets of the space, singular simplices are just continuous images of standard simplices. This allows defining homology for any topological space.

Definition

The singular $n$ -chain group $C_n(X)$ is the free abelian group generated by all singular $n$ -simplices in $X$ : $C_n(X) = \mathbb{Z}\langle \sigma : \sigma \text{ is a singular } n\text{-simplex in } X \rangle$

Elements are formal finite sums $\sum n_i \sigma_i$ where $n_i \in \mathbb{Z}$ and $\sigma_i : \Delta^n \to X$ are continuous.

Example

For a point space $X = \{*\}$ , there is exactly one singular $n$ -simplex for each $n$ (the constant map), so $C_n(\{*\}) \cong \mathbb{Z}$ for all $n \geq 0$ .

Definition

The $i$ -th face map $d^i : \Delta^{n-1} \to \Delta^n$ is the inclusion $(t_0, \ldots, t_{n-1}) \mapsto (t_0, \ldots, t_{i-1}, 0, t_i, \ldots, t_{n-1})$ . It embeds $\Delta^{n-1}$ as the face of $\Delta^n$ opposite the $i$ -th vertex.

The boundary operator $\partial_n : C_n(X) \to C_{n-1}(X)$ is defined on a singular simplex $\sigma : \Delta^n \to X$ by: $\partial_n \sigma = \sum_{i=0}^n (-1)^i \sigma \circ d^i$

Each term $\sigma \circ d^i$ is a singular $(n-1)$ -simplex: the restriction of $\sigma$ to the $i$ -th face of $\Delta^n$ .

Theorem

As in simplicial homology, $\partial_{n-1} \circ \partial_n = 0$ . This makes $(C_\bullet(X), \partial)$ a chain complex.

The proof is identical to the simplicial case: each $(n-2)$ -face appears twice with opposite signs when computing $\partial^2$ .

Definition

The singular homology groups of $X$ are: $H_n(X) = \ker(\partial_n) / \text{im}(\partial_{n+1}) = Z_n(X) / B_n(X)$ where $Z_n(X)$ are $n$ -cycles and $B_n(X)$ are $n$ -boundaries.

Remark

Singular homology works for all topological spaces: manifolds, CW complexes, fractals, even pathological spaces. It requires no triangulation or special structure. The price is that $C_n(X)$ is typically enormous (uncountably generated), making direct computation challenging.

For "nice" spaces (CW complexes, manifolds), singular homology agrees with simplicial homology, but it's defined much more generally.