Singular Homology

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 are constants.

It can be noticed that is a singularity of it, and when integrating this function, we always choose an integration path or integration surface that bypasses the singularity. Moreover, from Maxwell's equations (or Stokes' theorem), we know that if we choose a closed surface in space for integration, the value of the integration only relates to the singularities inside. Here, the singularities can be considered as holes in the space defined by the function. This is why we are interested in the holes in space and consequently study their properties.

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 group on ​​.

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 standard ​-simplex is

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 orientation of is 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 same if, 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) -simplex in 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 -chain on .

Let's consider what the elements in are.

Let be an -chain, where and is continuous. We can regard as a -dimensional geometric shape on (this may not be accurate), and the sign of denote the orientation of . Then we can consider as a combination of some -dimensional geometric shapes on .

How to use -chains to describe holes in ?

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 ​-chain.

1.6 Definition.

Define the -th face map of

We can consider that the usage of these maps is to take out the face opposite to the -th vertex in the n-simplex. With these maps, we can define the boundary of -chains.

1.7 Definition.

If is continuous and , then its boundary is

if , 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 complex of ; it is denoted by or, more simply, by .

Then, we can find all "closed curves".

1.8 Definition.

The group of (singular) -cycles in , denoted by , is .

Although we can find all "closed curves" in a plane , we still can't describe holes in . We can note that the holes in the plane can be surrounded by 1-chains, but cannot be covered by 2-chains.

Therefore, we know that there are some "closed curves", boundaries of -chains, surrounding no holes!

1.9 Definition.

The group of (singular) -boundaries in , denoted by , is .

Wait, are -boundaries really -cycles?

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 group of a space is

The coset , where is an -cycle, is called the homology class of , and it is denoted by .

2. Functorialities of and

When we view and ​ from the perspective of category theory, we can find that they are actually functors.

2.1 Definition.

A (chain) complex is a sequence of abelian groups and homomorphisms

, where , such that for each . The homomorphism is called the differentiation of degree , and is called the term of degree .

The complex above is denoted by or, more simply, by .

2.2 Definition.

If and are complexes, a chain map is 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 -boundaries, and 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 be a category, is a set of some objects.

Then the coproduct is an object with inclusions such that for all morphisms , there is a unique morphism make all diagrams like

commute.

Coproduct is a specific example of colimits (direct limits). Next, I will show you that ​ preserve coproduct. Actually, is co-continuous; that is preserve colimits.

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 .

    Therefore, for each with are -simplices, we have , where .

    Define

    It's easy to know are group homomorphisms. We should show that ​ is a chain map.

    For and are chain maps, we have commutative diagrams

    and

    ; hence

  • Uniqueness of :

    If there are and both satisfy the properties, then we have

    Therefore, .

2.9 Theorem.

Functor preserve coproduct; that is is a set of chain complexes, and then we have .

Proof.

Let and be inclusions.

It's sufficient to prove has the same universal property as ; that is, for all abelian groups and homomorphisms , there is a unique homomorphism , such that the diagrams

commute.

  • Existence of :

Each , where , can be write as with ​.

Define

Next, let's show that is well-defined.

Let , and then there is with ​.

Because of the commutative diagram

, we have

Thus, ​.

Consequently, we have ; hence .

  • Uniqueness of :

If there are and both satisfy the properties, then we have

Therefore, .

2.10 Corollary.

If is the set of path components of , then we have

3. Dimension Axiom and Homotopy Axiom

In this section, we will introduce two axioms of homoblogy theory.

3.1 Theorem (Dimension Axiom).

If is a one-point space, then for all .

Proof.

For each , there is only one singular -simplex , where is a constant map.

Therefore, . Then, let's compute the boundary operators:

It follows that, for all ,

Thus, for each ,we have two conditions:

  • If is odd, we have and . Hence, .
  • If is even, we have .

Therefore, .

3.2 Definition.

A space is called acyclic if for all .

3.3 Proposition.

  • If , then for all .
  • If is a one-point space, then ​​.
  • If is the Cantor set, then

3.4 Theorem.

If is a non-empty path-connected space, then .

Hence, for any topological space , , where are path components of .

Proof.

For , we have .

(We will use to denote the map .)

Define . It's obvious that is a subgroup of .

Next, let's prove .

  • If , then there is with .

    And then, we have .

    Since are generators, .

    Therefore, .

  • Let , and fix a point .

    Let be a path from to (consider as ).

    Then let . We can find that

    Therefore, .

Define . It's trivial that is a homomorphism with .

Consequently, .

Next, we will learn the homotopy axiom; that is if are two homotopic continuous maps, then we have . However, we will not prove this theorem directly. Let us prove it step by step.

3.5 Lemma.

If be a bounded convex subset of , then for all .

Proof.

Fix a point . Let be a -simplex.

For all , define

And then define

It's easy to check ​ is a homomorphism.

When , we claim that, for a -chain ,

If is true, we have, for all ,​

Consequently, ; hence, the lemma has been proved.

Next, we will prove .

Let be a -simplex. And then we have

; and

Thus, we have .

By using the universal property of free abelian groups, we proved .

3.6 Definition.

Let be two chain complexes, and and ​ be chain maps. We call and are (chain) homotopic if there are homomorphisms such that ​​ for each .

The following diagram is NOT commutative.

3.7 Theorem.

If and are chain homotopic, then we have for each .

Proof.

Let , and then we have

3.8 Lemma.

Let be a space and, for , define

Let be two homotopic continuous maps.

If , then we have .

Proof.

Let be the homotopy. Then we have

3.9 Definition.

Let be two categories, and be two (covariant) functors. A natural transformation is a collection of morphisms ​ such that the diagrams

commute.

3.10 Theorem (Homotopy Axiom).

If be two homotopic continuous maps, then

Notations:

Let are maps. Define

Let is a map. Define

Proof.

As what we discussed above (3.5, 3.6 and 3.7), it's sufficient to prove and are chain homotopic.

Define , and , where is a continuous map.

It's easy to check is a functor.

Actually, we will construct natural transformations such that

Since is a natural transformation, we also have a commutative diagram

. Let be the identity map and be a -simplex on . (We will use in short for in the remaining part of this proof.)

If exists, we can notice that

Therefore, in order to construct , we only need to construct .

Next, let's construct by induction on .

  • Base step for induction (​):

    We have for only has one homomorphism.

    Let ​ be the identity map.

    Define

    and, for -simplex ,

    By extending by linearity, we get homomorphisms .

    Then we should verify the inductive hypothesis.

    • For -simplex , we have

      Consequently,

      And by the universal property of the abelian groups, we know is true.

    • (Naturality of ​)

      Let be a continuous map. Then we have

      Similarly, by the universal property, we know is a natural transformation.

  • Assume that ​​.

    Let ​ be the identity map.

    If is true, we have

    Therefore, we need to prove .

    Since is a bounded convex subset of , by lemma 3.4, we have ; that is .

    Consequently, we only need to prove .

    Since

    we have ; that is , such that .

    Define

    and

    By extending by linearity, we get homomorphisms .

    Next, we need to verify inductive hypothesis.

    • For -simplex ​, we have

      And by the universal property of the abelian groups, we know is true.

    • (Naturality of )

      Let be a continuous map. Then we have

      Similarly, by the universal property, we know is a natural transformation.

Therefore, we can also see as a functor from to .

3.11 Corollary.

If and have the same homotopic type, then we have .

3.12 Corollary.

If is contractible, then for all , and .

4. Hurewicz Theorem

In this section, let's talk about the relationship between and . Additionally, all the topological spaces in this section are path-connected, and we always choose a base point .

We will denote the abelianized fundamental group of at as ; that is , where is the commutator subgroup of .

We will use stands for a homotopy class; stands for the coset of in ; and stands for " is homotopic to ", while " stands for is homologous to ​".

4.1 Lemma.

If and are paths in such that then the -chain is a boundary.

Proof.

Define a continuous map ​ as indicated by the following picture.

In more detail, first define on :

Now define on all of by setting it constant on the line segment with endpoints and , and constant on the line segments with endpoints and .

Then we can find that .

4.2 Lemma.

If is a constant, then is a boundary.

Proof.

By lemma 3.1, we have is a boundary.

4.3 Lemma.

If and are paths with , then ​​.

And thus is a boundary; hence, is a boundary by lemma 3.2.

Proof.

Let be the homotopy.

Define

The definition of this map is illustrated in the following picture.

Then, it's obvious that , where is a constant map.

Therefore, by lemma 3.2, .

4.4 Definition.

The Hurewicz map is

By lemma 4.1, 4.2 and 4.3, we can know that is a well-defined homomorphism.

As we can see, is a abelian group, while may not. Therefore, induces a homomorphism

In fact, ​ is an isomorphism, I will show you soon.

To prove ​ is an isomorphism, we can try to construct its inverse.

Since is path-connected, for all there is a path from to ​.

For each -simplex , which is also a path, define ​​.

Then by extending by linearity, we get a homomorphism .

To ensure can induce a map from to , we should check .

4.5 Lemma.

The map takes the group into .

Proof.

Let be a -simplex. Let for and ​.

Then

For is a boundary of -simplex, which is nullhomotopic rel (Rotman, Exercise 3.4), .

Therefore, by the universal property of the abelian group, we get .

Consequently, induces the homomorphism .

If is a loop, then we can easily find .

Thus, to show is the inverse of , it's sufficient to prove .

4.6 Lemma.

If is a -cycle, then .

Proof.

Let , where are -simplices.

Then we have ; hence .

Therefore, by lemma 4.1-4.3, we have

Finally, we proved the Hurewicz theorem.

4.7 Theorem (Hurewicz).

If is a path-connected space with base point , then

Proof.

From what we talked above, is the isomorphism.

4.8 Corollay.

  • .
  • If is simply connected, then .

4.9 Theorem.

Let ​ be the Hurewicz map.

Then is a natural transformation from to ; that is the diagram

commutes, for each continuous map .

Proof.

If be a loop in with base point , then

$$
$$
暂无评论

发送评论 编辑评论


				
|´・ω・)ノ
ヾ(≧∇≦*)ゝ
(☆ω☆)
(╯‵□′)╯︵┴─┴
 ̄﹃ ̄
(/ω\)
∠( ᐛ 」∠)_
(๑•̀ㅁ•́ฅ)
→_→
୧(๑•̀⌄•́๑)૭
٩(ˊᗜˋ*)و
(ノ°ο°)ノ
(´இ皿இ`)
⌇●﹏●⌇
(ฅ´ω`ฅ)
(╯°A°)╯︵○○○
φ( ̄∇ ̄o)
ヾ(´・ ・`。)ノ"
( ง ᵒ̌皿ᵒ̌)ง⁼³₌₃
(ó﹏ò。)
Σ(っ °Д °;)っ
( ,,´・ω・)ノ"(´っω・`。)
╮(╯▽╰)╭
o(*////▽////*)q
>﹏<
( ๑´•ω•) "(ㆆᴗㆆ)
😂
😀
😅
😊
🙂
🙃
😌
😍
😘
😜
😝
😏
😒
🙄
😳
😡
😔
😫
😱
😭
💩
👻
🙌
🖕
👍
👫
👬
👭
🌚
🌝
🙈
💊
😶
🙏
🍦
🍉
😣
Source: github.com/k4yt3x/flowerhd
颜文字
Emoji
小恐龙
花!
上一篇