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Facts about Homology
Melissa the Doll
May 20, 2007
Hi there, folks! My name is Melissa, a sort of friend of a friend of Corey’s. That sort
of makes Corey my friend, and I don’t know how I feel about that. I mean, it’s nice to sit
down and have an intellectual discussion with someone, and that’s certainly NOT what I
get when I call up Corey. I want to talk about homology, and he wants to talk about how
bad the officiating was during the Suns/Spurs basketball series. So I’ve given up, and I
decided to write this handout for you all, just to give you a concise list of facts that we’ve
developed so far in our study of homology. Rock on!
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Definitions
Let {e0 , e1 , . . .} be an ordered basis for R∞ . Let
( n
)
n
X
X
∞
∆n :=
ti ei ∈ R |ti ≥ 0, and
ti = 1 .
i=0
i=0
1
The ∆n live in Zürich, and a map σ : ∆n → X is called a singular n-simplex. As with
any map, one may consider restrictions of this map to certain subsets of ∆n . For this
reason, we may express a singular n-simplex as (P0 , . . . , Pn ) where Pi = σ(ei ), with the
understanding that removal of any of the points Pi from this list designates a restriction
of σ to a subset of ∆n . For example, if σ : ∆2 → X, with σ = (P0 , P1 , P2 ), then (P0 , P2 )
would be the map σ with restriction to the (e0 , e2 )-plane in R∞ .
Let Cn (X) be the free abelian group generated by the singular n-simplices. Thus, an
arbitrary element
P θ ∈ Cn (X) is a finite linear combination of integers multiplied by nsimplices: θ =
ni σi , where ni ∈ Z, and σi : ∆n → X. If we construct a map with
domain Cn (X) that we wish to be a group homomorphism, then we may simply describe
what the map does to each of the singular n-simplices, and then extend by linearity.
For example, we wish to define a function ∂n : Cn (X) → Cn−1P
(X). We will
P describe
what ∂n σ is (where σ is a singular n-simplex), then set ∂n θ = ∂n ( ni σi ) =
ni (∂n σi ).
My fingers are sort of getting tired of typing the n on the ∂. From now on, I’ll leave it out
unless there is a possible misunderstanding as to what ∂ I may be talking about.
So we define
∂σ = ∂(P0 , . . . , Pn ) =
n
X
(−1)i (P0 , . . . , P̌i , . . . , Pn ) .
i=0
Here, the notation indicates that we’re talking about ∂n (it’s the only one of these maps
that I could apply to such an object, so leaving the n out is unambiguous), and that
(P0 , . . . , P̌i , . . . , Pn ) = (P0 , . . . , Pi−1 , Pi+1 , . . . , Pn ). It’s σ with the ith entry removed.
So we have a homomorphism ∂n : Cn (X) → Cn−1 (X), and it satisfies the following
property.
Theorem 1.1 The ∂ homomorphisms satisfy ∂n−1 ∂n θ = 0 for all θ ∈ Cn (X). More
concisely, we could say ∂ 2 = 0.
The situation we find ourselves in is a sequence of abelian groups, with homomorphisms
between them that compose twice to 0. The diagram below is sort of the way people write
such a situation down on paper (but not always! We will be writing down other sequences
denoted similarly), with fn−1 fn = 0:
fn+1
fn
fn−1
· · · −→ An −→ An−1 −→ · · ·
The sequence of groups with their homomorphisms (that satisfy fn−1 fn = 0 is called a
chain complex. The chain complex (C∗ (X), ∂∗ ) is called the singular chain complex of the
topological space X, and is regarded as a somewhat special chain complex (the ∗ sort of
tells you where the index is. I wouldn’t want to write (Cn (X), ∂n ) since that would be a
pair of just one of the chain groups and one of the maps. A chain complex is a collection
of these objects).
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Definition 1.2 For a given topological space X, the nth homology group of X, denoted
Hn (X) is defined as
ker ∂n
.
Hn (X) :=
Im ∂n+1
2
Properties
We have a short list of properties, but here are some of them.
Proposition 2.1 Use the notation and objects we have defined above.
1. If X is the topological space consisting of only one point, then Hn (X) = 0 (the trivial
group) if n > 0, and H0 (X) ∼
= Z.
2. If X is nonempty and path connected, then H0 (X) ∼
= Z.
3. If X = A t B (the disjoint union of A and B), then Hn (X) ∼
= Hn (A) ⊕ Hn (B).
Quite possibly our most important property that we’ve proved so far in class is the
homotopy property. We develop the machinery involved, and then state the important
result, along with what one of its important consequences is.
Let f : X → Y be continuous. Then we can push forward singular simplices with an
σ
f
associated map f] as follows. If σ : ∆n → X, then f ◦ σ : ∆n → X → Y maps from Zürich
into Y . So f ◦σ is a singular simplex on Y . Denote this composition as
◦σ. Then,
Pf] (σ) = fP
we extend by linearity to define the map f] : Cn (X) → Cn (Y ) as f] ( ni σi ) = ni f] (σi ).
We have the following properties for f] .
Proposition 2.2 The following properties hold for the morphism f] :
1. ∂nY f] = f] ∂nX .
2. id] = the identity map.
3. (f ◦ g)] = f] ◦ g] .
This gives rise to a well-defined map f∗ : Hn (X) → Hn (Y ) defined as f∗ ([α]) = [f] (α)].
Proposition 2.3 The morphism f∗ satisfies (id)∗ = the identity map, and (f ◦g)∗ = f∗ ◦g∗ .
And so we can state the homotopy property:
Theorem 2.4 Let f, g : X → Y be continuous. If f ' g, then f∗ = g∗ as maps on
homology.
The important corollary is
Corollary 2.5 If X ' Y , then Hn (X) ∼
= Hn (Y ).
In particular, if Y is a deformation retract of X, then the above would state that their
homology groups are the same. Rock on!
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