Singular Homology

Overview

Singular homology associates abelian groups to a space by comparing cycles with boundaries. It can be defined on every topological space without choosing a triangulation. The construction uses many parametrized simplices, often infinitely many possible generators, but each individual chain remains a finite formal sum. Smaller models are used for calculations when justified.

Prerequisites

Understand free abelian groups, kernels, images, and quotient groups. A chain complex is a sequence of groups and boundary maps whose consecutive composites vanish. Geometric pictures help, but a cycle is an algebraic condition on a formal sum, not necessarily one embedded closed surface.

Definition and conventions

The standard simplex $\Delta^n$ has ordered vertices $v_0,\ldots,v_n$. A singular simplex is a continuous map $\sigma:\Delta^n\to X$. With integral coefficients, $C_n(X)$ is the free abelian group generated by these maps. Its boundary is

The face restrictions use the induced vertex order. Set $C_n=0$ for negative degrees and define $H_n(X;\mathbb Z)=\ker\partial_n/\operatorname{im}\partial_{n+1}$.

This entry uses ordinary, unreduced homology. Other coefficient groups and reduced homology must be stated explicitly. Hausdorff and countability conditions are unnecessary for the construction.

Intuition

A cycle has no remaining boundary after cancellation. A boundary already arises as the edge of a higher-dimensional chain, so it contributes no new homology class. A nonzero class is an obstruction to filling a cycle within the space. Different chains represent the same class when their difference can be filled.

Examples and non-examples

For a point, every positive-dimensional homology group vanishes and $H_0\cong\mathbb Z$. The circle has $H_0\cong\mathbb Z$ and $H_1\cong\mathbb Z$, with higher groups zero. A single traversal represents a generator in degree one.

For the disk, the boundary-circle cycle becomes a boundary and vanishes in $H_1(D^2)$. It remains nonzero when considered in the circle itself. Thus the same visible curve may be trivial or nontrivial according to the ambient space. For a space with two path components, $H_0$ is the direct sum of two copies of $\mathbb Z$.

Important properties

Every codimension-two face appears twice in $\partial^2\sigma$, with opposite signs, proving $\partial^2=0$. Continuous maps induce chain maps by composition and hence homology homomorphisms.

Homotopic maps induce the same homomorphism. Triangulating a simplex times an interval gives a prism operator whose boundary identity is the required chain homotopy. Consequently contractible nonempty spaces have the homology of a point. Homology alone does not classify homotopy types.

Connections and further reading

Cellular homology computes the same invariant on CW complexes using their cells. Cohomology reverses the direction of induced maps and adds products. Read Hatcher, §§2.1–2.2 for the inspected definition, signs, and comparison.