This document is a seminar note on singular homology.

Before reading this article, make sure you have known the Chapter 1-3 of *An Introduction to Algebraic Topology*, written by Rotman.

You can download the pdf version from this link.

**Reference**

*An Introduction to Algebraic Topology*, Rotman*Topology and Geometry*, Bredon*代数学方法 卷一：基础架构*, 李文威- nlab

**Why should we learn homology?**

When we use calculus to solve physical problems, we often encounter functions like

where

It can be noticed that

Homology was initially defined as closed curves or surfaces with the same integral values, but this definition cannot adapt to general topological spaces. From a geometric intuition, a hole should be a type of topological invariant that remains unchanged under homeomorphism. Therefore, we need to find some new methods to define homology, and thereby study the properties of these holes. Next, we will begin to learn a definition method of homology, which is the singular homology.

## 1. Definition of Singular Homology

Let's begin with some basic definitions.

1.1 Definition.Let

be a set and . We can define a binary operation

It's easy to check that

is an abelian group. And we call it the free abelian groupon. We usually denote

as . For example, we can use

to denote We also use

to denote the free abelian group on .

The free abelian group has a universal property.

1.2 Theorem (Universal Property).Let

be a set and be the free abelian group on it.

is an abelian group and there is a function . Then there exists a unique group homomorphism making diagram commute, where

is the inclusion. And is called extending by linearity.

Proof.Define

. It's easy to check that is a group homomorphism. Then let's prove

is unique. Let

is another function fit the theorem. Then we have Therefore,

and is unique.

1.3 Definition.For

, the standardis-simplex We can see their graphs in this picture:

We can denote

as , where are vertices of . Sometimes, we will see

as with .

When we perform integrations on some geometric shapes, these shapes often have an orientation. Similarly, we can also define an orientation for simplices.

1.4 Definition.An

orientationofis a linear ordering of its vertices. An orientation thus gives a tour of the vertices. For example, the orientation

of gives a counterclockwise tour. Two orientations of

are the sameif, as permutations of, they have the same parity (i.e., both are even or both are odd); otherwise the orientations are opposite.

1.5 Definition.Let

be a topological space, . A (singular)in-simplex is a continuous function .

denotes the free abelian group generated by all singular -simplices in ; and we define for all integers . And we call an element in

is an on-chain .

Let's consider what the elements in

Let

How to use

When X is a plane with some holes, we can notice that the holes in the plane can be surrounded by a 1-chain, whose simplices precisely form some "closed curves".

Therefore, we need a method to find "closed curves" and try to use them to describe holes.

As we can see, "closed curves" have no boundary while "open curves" always have at least two points as its boundary.

Thus, we need to describe the boundary of an

1.6 Definition.Define the

of-th face map

We can consider that the usage of these maps is to take out the face opposite to the

1.7 Definition.If

is continuous and , then its boundaryisif

, define . As shown in the following figure, defining the boundary in this way is consistent with our geometric intuition.

Then we extend

by linearity, and hence we get a group homomorphism , for all . The homomorphisms are called boundary operators.Strictly speaking, we ought to write

instead of since these homomorphisms do depend on ; however, this is rarely done. For each

, we have constructed a sequence of free abelia groups and homomorphisms , called the

singular complexof; it is denoted by or, more simply, by .

Then, we can find all "closed curves".

1.8 Definition.The group of

(singular)in-cycles , denoted by , is .

Although we can find all "closed curves" in a plane

Therefore, we know that there are some "closed curves", boundaries of

1.9 Definition.The group of

(singular)in-boundaries , denoted by , is .

Wait, are

1.10 Lemma.If

, the face maps satisfy It's easy to prove by using the definition of

.

1.11 Theorem.For all

, we have ; and thus .

Proof.Since

is generated by all -simplices , it suffices to show that for each such .

Eventually, we can describe holes by using the quotient group.

1.12 Definition.For each

, the -th (singular) homology groupof a spaceis The coset

, where is an -cycle, is called the homology classof, and it is denoted by .

## 2. Functorialities of and

When we view

2.1 Definition.A

(chain) complexis a sequence of abelian groups and homomorphisms, where

, such that for each . The homomorphism is called the differentiationofdegree, and is called the termofdegree. The complex above is denoted by

or, more simply, by .

2.2 Definition.If

and are complexes, a chain mapis a sequence of homomorphisms such that the diagram commutes.

2.3 Theorem.All complexes and chain maps form a category, denoted by

, when one defines composition of chain maps coordinatewise: . It's easy to prove this is a category.

2.4 Theorem.Let

be a continuous function and , where and is an -simplex for all . Define , and then we get a sequence of homomorphisms . (Note: does depend on , but we usually don't give it a subscript.) Define

. And then are functors.

Proof.Let

and are continuous functions. Then, for any -simplex in , we have By the universal property of free abelian groups, we have

. Let

be the identity map of . Then, for any -simplex in , we have Similarly, because of the universal property of free abelian groups, we have

. Consequently,

is a functor. Next, to check

is a functor, we need to verity the commutation of diagram ; for any

-simplices in , we have

2.5 Definition.Let

be a chain complex. Then, there are Similarly,

is called the group of ,-cycles is called the group of , and-boundaries is called .-th homology group Let

be a chain map. Define We also denote

as . If is continuous. We denote as , too.

2.6 Theorem.

is a functor. And thus, the composition

is also a functor. We always write as .

Proof.Let

be chain complexes, and be chain maps. Then we have

and

Therefore,

is a functor.

2.7 Lemma.Let

be a subspace of with inclusion . Then is an injection for every .

Proof.Let

, where are distinct -simplices. Then we have

. Since

, we have are distinct. Therefore, for each , .

**Recall: Coproduct**

Let

Then the **coproduct** is an object

commute.

Coproduct is a specific example of colimits (direct limits). Next, I will show you that

2.8 Lemma.Functor

preserve coproduct; that is is a set of topological spaces and let , and then we have .

Proof.Let

be inclusions. It's sufficient to prove

has the same universal property as ; that is, for all complexes and chain maps , there is a unique chain map such that the diagrams commute.

Existence of

: For each

-simplex , since is path connected, there is a such that