Singular Homology

Homology for every space at once, with no triangulation to choose and nothing left to check.

The idea

Singular homology defines the homology groups $H_{n}(X)$ of an arbitrary topological space, with no triangulation chosen in advance. Simplicial homology has two defects: it applies only to a space someone has already cut into simplices, and nothing in it explains why the answer does not depend on the cutting. Singular homology removes both at once by using every simplex there is.

Definition (Singular homology).

A singular $n$-simplex in a space $X$ is any continuous map $\sigma \colon \Delta^{n} \to X$ from the standard $n$-simplex. Let $C_{n}(X)$ be the free abelian group on all of them, with the alternating-sum boundary map $\partial_{n}$ of the simplicial theory. The singular homology groups of $X$ are $H_{n}(X) = \ker \partial_{n} \,/\, \operatorname{im} \partial_{n+1}.$

Nothing else is required of a singular simplex: it may fold over itself, crush a face to a point, or be constant.

Because no triangulation was chosen, there is nothing to check: homeomorphic spaces have isomorphic groups, and so do homotopy equivalent ones. The price is that $C_{n}(X)$ is free abelian on a set that is usually uncountable, so we compute the groups from properties of the space rather than from the definition: path components split the groups into pieces, a contractible space has the homology of a point, and a continuous map induces a homomorphism between the groups of its two spaces.

Ways to work on it

• Walkthrough. Singular simplices, chain groups for an arbitrary space, the homology of a point, induced maps, and homotopy invariance.
• Proof. Why a path-connected space has H_0 ℤ — the augmentation, and the paths that turn a vanishing coefficient sum into a boundary.
• Practice. Count path components, transport homology along a deformation retraction, evaluate the augmentation, and read off an induced map.
• Hardest. No retraction of the disk onto its boundary circle, why the argument needs degree one, and the fixed-point theorem it delivers.

Not sure where to start? Take the ten-question placement test.