sign in sign up
<<
Home , Simplicial homology

TOPOLOGY 2 – UNIVERSITYOF PITTSBURGH,SPRING 2020

READINGGUIDE 1: FOR CHAPTER 2, THROUGH SECTION 2.1

Your reading assignment, due Friday March 20 at 1pm, is to ask two questions about the textbook material in Chapter 2 up to the end of Section 2.1. You can ask any kind of question. These questions might naturally

• ask about something that you read and struggled to understand • expresses a thought about something you understood and want to know more about.

The main point of this document is to give you some ideas of what the important concepts are in this reading. As always, not all things you read will be equally important, and I will emphasize some things differently than the textbook. Mainly, I will emphasize certain algebraic structures, especially cochain complexes, more than the textbook. After reading this document, you should be able to understand what chain complexes and are on a general level. Then after reading the textbook as well, this should give you a good sense of the definitions of simplicial homology and singluar homology of a .

1. FREEABELIANGROUPSONASET

⊕X 1.1. Definition. Let X be a set. Then there is a on X, denoted Z , which can be written down in several ways. I think that the most natural is to defined as

⊕X X Z = { ax · x : ax = 0 for all but finitely many x ∈ X}. x∈X ⊕X ⊕X We consider x ∈ X to be an element of Z by sending it to 1 · x ∈ Z . ⊕X ⊕X Elements of Z are summed in a coordinate-wise way with respect to the basis X of Z . That is, ! ! X X X ax · x + bx · x = (ax + bx) · x. x∈X x∈X x∈X

When X is the finite set X = {x1, . . . , xn}, then this is just like a vector space on the basis xn – the only difference is that Z is not a field. ⊕X Exercise. Check that you understand that Z is an abelian group.

1.2. The universal property of the free abelian group on a set. These groups are called free because a ⊕X group homomorphism from Z to an abelian group G is characterized by where X is sent. That is,

⊕X • any group homomorphism Z → G is characterized by where this homomorphism sends X ⊕X • conversely, any set map f : X → G induces a unique group homomorphism Z → G that sends ⊕X the subset X ⊂ Z according to f.

In symbols, we have the universal property of free abelian groups on a set, ⊕X HomAbGp(Z ,G) = HomSet(X,G). It is not an exaggeration to say that everything you want to do with a free abelian group comes from the universal property. 1 1.3. Appearance in the reading. These free abelian groups are used constantly starting at the bottom of page 99. For instance, on page 100, we see that “C2 is the infinite cyclic group generated by A.” This means {A} that C2 = Z , the free abelian group on the singleton set consisting of the 2-cell A.

Likewise, on page 100, when we read that “∂2 sends both A and B to a − b,” this is implicitly using the {A,B} universal property: a homomorphism Z → G is characterized by where A and B are sent.

Then there is the crucial statement on page 101: “For a cell complex X, one has chain groups Cn(X) which are free abelian groups with basis the n-cells of X, ...” The main reason I have emphasized free abelian groups in this generality is their appearance on page 108, when the discussion of begins. Here, observe that when X is a topological space, Cn(X) is defined to be the free abelian group on the set of n-simplices, and an n- is simply a continuous map σ : ∆n → X. The set of n-simplices is huge in general – that is why it is important to have a clear idea of what “the free abelian group on a set” is.

2. CHAINCOMPLEXESANDTHEIRHOMOLOGY

Chain complexes and homology are algebraic concepts that can be introduced without reference to topology. Here, unlike the textbook, we will first introduce these notions abstractly. Then we will point out a few places where these appear in the text.

2.1. Definitions. A of abelian groups C• is the data of

• Abelian groups Cn for n ∈ Z – another term for this is a graded abelian group or Z-graded abelian group; often Cn is called the group of n-chains. • Group homomorphisms ∂n : Cn → Cn−1, for n ∈ Z, often known as the boundary homomorphisms satisfying the property that

• for all n ∈ Z, the group homomorphism ∂n−1 ◦ ∂n : Cn → Cn−2 is the zero map. Thinking of ∂ as L ∂ , this is sometimes expressed as “∂2 = 0. Sometimes we call ∂ a “differential” n∈Z n because it obeys this rule (in reference to differential forms). Let (C, ∂) be notation1 for a chain complex M M (C, ∂) = ( Cn, ∂ = ∂n). n∈Z n∈Z Another way of expressing the “∂2 = 0” rule is that

image ∂n+1 ⊂ ker ∂n for all n ∈ Z. Since this inclusion is an inclusion of abelian groups, this allows for the definition of the homology of a chain complex. The homology of a chain complex (C, ∂) is the graded abelian group written H(C), where

ker ∂n Hn(C) = . image ∂n+1

Remark. The textbook will often state what C2,C1, and C0 are, and will not define Ci for other i ∈ Z. Then the text will only define ∂2 : C2 → C1 and ∂1 : C1 → C0. You should think of all of these other Ci as the trivial abelian group 0, and also think of all of the rest of the ∂i as the only possible option, which is

1 Or you could write (C•, ∂•). the zero map. For example, see the chain complex written down at the top of page 106, which writes down a map “∂0 : C0 → 0.” Really, this is saying that C−1 = 0, but the textbook does not write down “C−1.” X Example. Let C1 = Z and C0 = Z, where X = {x1, x2}. Let ∂1 be the map defined by sending x1 7→ 2 and x2 7→ 0. (At this point, implicitly, all other Ci and ∂i are zero.) Then observe that there are isomorphisms of abelian groups H0 ' Z/2Z,H1 ' Z.

2.2. More terminology. The image of ∂n+1 : Cn+1 → Cn is called the group of n-boundaries. The kernel of ∂n : Cn → Cn−1 is called the group of n-cycles. Thus it is common to say that “homology is defined to be cycles modulo boundaries.” The terms “cycle” and “boundary” come from the topological root that Section 2.1 discusses,2 but they are not exclusive to this context.

2And the case of simplicial homology is where these cycles and boundaries have especially nice pictures to motivate them.