singular homology

Overview

The fundamental group works well to identify “holes” of low dimension, but it doesn’t work well for “bigger” holes. Instead we can use continuous maps from a simplex to the topological space. The goal is to use homological algebra on a chain complex of groups of these continuous maps. To do so, we need a little bit of work to get the necessary parts.

Necessary prerequisite constructions

Singular simplices

Let be a topological space. A (singular) n-simplex in X is a continuous map

where is a n-simplex.

Note the terminology of the word singular is meant to show that this might not be an embedding. It is only a continuous map, so the image might not “look” like a simplex, certain singularities are allowed (@hatcher2002).

Let be the set of all singular n-simplices in .

Singular n-simplex in X for small n

The th face map induces a function

Singular chains

Now we want to build a chain complex using .

The free abelian group on is called the group of (singular) n-chains on . It is denoted .

Since is a free abelian group with as a basis, we can write uniquely as the finite sum

We set for .

In order to make the boundary map for the chain complex, we can use the one from . Using the universal property of free groups since there is a map

we can build a unique group homomorphism

that respects the boundary map. This is shown in this diagram:

Explicitly

Collecting these together gives us the nth boundary operator

Singular homology

Now that we have the chain complex , we can define the singular homology.

The group of singular n-cycles in is

The group of singular n-boundaries in is

The nth singular homology of is the group

Important properties

Induced chain map

A continuous function induces a chain map

Moreover,

and

If is a homeomorphism, then

Proof

todo

0th homology

Let be a topological space. Then the 0th homology group is the free abelian group of path components of .

Proof

todo

Resources

• @hatcher2002 - Chapter 2