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

$$\sigma :{\Delta }^n⟶X$$

where ${\Delta }^n$ 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 ${\text{Sing}}_n(X)$ be the set of all singular n-simplices in $X$.

Singular n-simplex in X for small n

${\text{Sing}}_0(X)\simeq X\text{bijectively}$

${\text{Sing}}_1(X)=\{\text{paths in }X\}$

The $i$th face map $d^i:{\Delta }^{n-1}\to {\Delta }^n$ induces a function

$$\begin{array}{rl}{\partial }_i:{\text{Sing}}_n(X)& ⟶{\text{Sing}}_{n+1}(X)\\ \sigma & ⟼\sigma \circ d^i\end{array}$$

Singular chains

Now we want to build a chain complex using ${\text{Sing}}_n(X)$.

The free abelian group on ${\text{Sing}}_n(X)$ is called the group of (singular) n-chains on $X$. It is denoted $C_n(X)$.

Since $C_n(X)$ is a free abelian group with ${\text{Sing}}_n(X)$ as a basis, we can write $\sigma \in C_n(X)$ uniquely as the finite sum

$$\sigma =\sum _{{\sigma }_i\in {\text{Sing}}_n(X)}{a}_i{\sigma }_i{a}_i\in \mathbb{Z}.$$

We set $C_n(X)=0$ for $n\le -1$.

In order to make the boundary map for the chain complex, we can use the one from ${\text{Sing}}_n(X)$. Using the universal property of free groups since there is a map

$${\text{Sing}}_n(X)\stackrel{{\partial }_i}{\to }{\text{Sing}}_{n-1}(X)\hookrightarrow C_{n-1}(X)$$

we can build a unique group homomorphism

$$C_n(X)⟶C_{n-1}(X)$$

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

Explicitly

$${\partial }_k\sum _j{a}_j{\sigma }_j=\sum _j{a}_j{\partial }_k{\sigma }_j.$$

Collecting these together gives us the nth boundary operator

$$\begin{array}{rl}\partial :C_n(X)& ⟶C_{n-1}(X)\\ \sigma & ⟼\sum (-1{)}^i{\partial }_i\sigma \end{array}$$

Singular homology

Now that we have the chain complex $C_{\bullet }(X)$, we can define the singular homology.

The group of singular n-cycles in $X$ is

$$Z_n(X)=\ker (\partial :C_n(X)⟶C_{n-1}(X))⊴C_n(X)$$

The group of singular n-boundaries in $X$ is

$$B_n(X)=\text{im}(\partial :C_{n+1}(X)⟶C_n(X))⊴C_n(X)$$

The nth singular homology of $X$ is the group

$$H_n(X)=Z_n(X)/B_n(X).$$

Important properties

Induced chain map

A continuous function $f:X\to Y$ induces a chain map

$$C_{\bullet }(f):C_{\bullet }(X)⟶C_{\bullet }(Y).$$

Moreover,

$$C_{\bullet }(f\circ g)=C_{\bullet }(f)\circ C_{\bullet }(g)$$

and

$$C_{\bullet }(i{d}_X)=i{d}_{C_{\bullet }(X)}.$$

If $f$ is a homeomorphism, then

$$H_{\bullet }(f):H_{\bullet }(X)\stackrel{\sim }{\to }{H}_{\bullet }(Y).$$

Proof

todo

0th homology

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

$$H_0(X)\simeq \mathbb{Z}{\pi }_0(X)={\left({\prod _{\alpha \in {\pi }_0(x)}}^{\ast }{\mathbb{Z}}_{\alpha }\right)}_{ab}$$

Proof

todo

Resources

• @hatcher2002 - Chapter 2