Math 757 Homology theoryHomology theory Lecture 10 - 1/24/2011 Axiom 4: Additivity Computations...

Homology theory

Lecture 10 -1/24/2011

Axiom 4:Additivity

Computations

Lecture 12 -1/26/2011

Naturality of theconnectinghomomorphism

Lecture 13 -1/27/2011

CellularHomology

Math 757Homology theory

January 27, 2011

Homology theory

Lecture 10 -1/24/2011

Axiom 4:Additivity

Computations

Lecture 12 -1/26/2011


Lecture 13 -1/27/2011

CellularHomology

Axiom 4: Additivity

Definition 90 (Direct sum in Ab)

Given abelian groups {Aα}, the direct sum is an abelian group Awith homomorphisms

γα : Aα → A

satifying the following universal property of direct sums:

Given homomorphisms fα : Aα → K there is a unique f : A→ K s.t.∀α fα = f ◦ γα

Definition 91 (Direct sum in Chain)

The direct sum for chain complexes {Cα} is the direct sum asabelian groups with the added stipulation that all homomorphismsmust be chain maps

Homology theory

Lecture 10 -1/24/2011

Axiom 4:Additivity

Computations

Lecture 12 -1/26/2011


Lecture 13 -1/27/2011

CellularHomology

Theorem 92 (Sing. simp. homology satisfies Axiom A4)

Xα disjoint spaces. Then

Hn

(∐Xα)

=⊕α

Hn(Xα)

Exercise 93 (HW4 - Problem 1)

Verify that the homology functors Hn : Chain→ Ab commute withdirect sums.

In particular, show that if Cα are chain complexes then

Hn

(⊕α

Cα

)∼=⊕α

Hn (Cα)

This may be done by verifying that Hn (⊕

α Cα) satisfies theuniversal property for direct sums.

Homology theory

Lecture 10 -1/24/2011

Axiom 4:Additivity

Computations

Lecture 12 -1/26/2011


Lecture 13 -1/27/2011

CellularHomology

Definition 94 (Good pair)

(X ,A) is a good pair if there is a neighborhood V of A in X suchthat V deformation retracts to A.

Theorem 95 (Relative homology and quotient spaces)

If (X ,A) is a good pair then we have an isomorphism

q∗ : Hn(X ,A)→ H̃n(X/A)

where q∗ is induced by the compositionC (X ,A)→ C (X/A,A/A)→ C̃ (X/A)

Homology theory

Lecture 10 -1/24/2011

Axiom 4:Additivity

Computations

Lecture 12 -1/26/2011


Lecture 13 -1/27/2011

CellularHomology

Proof of Theorem 95.

Hn(X ,A)α−−−−−→ Hn(X ,V )

β←−−−−− Hn(X − A,V − A)

q∗

y y q′∗

yHn(X/A,A/A)

γ−−−−−→ Hn(X/A,V /A)η←−−−−− Hn(X/A− A/A,V /A− A/A)

• β and η are isomorphisms by excision.

• The long exact sequence of the triple (X ,V ,A) gives

Hn(V ,A) −→ Hn(X ,A)α−→ Hn(X ,V ) −→ Hn−1(V ,A)

Deformation retaction of V to A shows

Hn(V ,A) ∼= Hn(A,A) ∼= 0

So α is an isomorphism

• Same argument for (X/A,V /A,A/A) shows γ is isomorphism.

• q′∗ is an isomorphism since(X − A,V − A)→ (X/A− A/A,V /A− A/A) is a homeo

• q∗ = γ−1ηq′∗β−1α

Homology theory

Lecture 10 -1/24/2011

Axiom 4:Additivity

Computations

Lecture 12 -1/26/2011


Lecture 13 -1/27/2011

CellularHomology

Corollary 96 (Long exact sequence for a good pair)

If (X ,A) is a good pair then

· · · −→ H̃n(A) −→ H̃n(X ) −→ H̃n(X/A) −→ H̃n−1(A) −→ · · ·

is exact.

Proof.

By Proposition 61 (Week 2) we have a long exact sequence

· · · −→ H̃n(A) −→ H̃n(X ) −→ Hn(X ,A) −→ H̃n−1(A) −→ · · ·

By Theorem 95 above Hn(X ,A) ∼= H̃(X/A)

Homology theory

Lecture 10 -1/24/2011

Axiom 4:Additivity

Computations

Lecture 12 -1/26/2011


Lecture 13 -1/27/2011

CellularHomology

Proposition 97 (Homology groups of spheres)

H̃k(Sn) =

{Z, k = n0, k 6= n

Proof.

• True for S−1 = ∅.

• True for S0 = {−1, 1}.• Suppose true for Sn−1

• Consider good pair (Dn,Sn−1)

H̃k(Dn) −→ H̃k(Dn/Sn−1) −→ H̃k−1(Sn−1) −→ H̃k−1(Dn)

• Dn/Sn−1 ∼= Sn

•

H̃k(Sn) ∼= H̃k−1(Sn−1) =

{Z, k − 1 = n − 10, k − 1 6= n − 1

Homology theory

Lecture 10 -1/24/2011

Axiom 4:Additivity

Computations

Lecture 12 -1/26/2011


Lecture 13 -1/27/2011

CellularHomology


Show that for X nonempty

H̃n(X ) ∼= H̃n+1(SX )


Let X be the n-point set. Compute H̃k(X )


Let Y = Rn =∨n

i=1 S1 be the rose with n-petals (see Hatcher pg.

10). Compute H̃k(X )

Homology theory

Lecture 10 -1/24/2011

Axiom 4:Additivity

Computations

Lecture 12 -1/26/2011


Lecture 13 -1/27/2011

CellularHomology

Definition 101 (Local homology groups)

x ∈ X the local homology of X at x is

Hn(X ,X − x)

• Let V be any neighborhood of x

• By excision with (X ,A,U) = (X ,X − x ,X − V ) we have

Hn(X ,X − x) = Hn(V ,V − x)

• Hence local homology depends only on topology near x

• Can use local homology show X not homeomorphic to Y

Homology theory

Lecture 10 -1/24/2011

Axiom 4:Additivity

Computations

Lecture 12 -1/26/2011


Lecture 13 -1/27/2011

CellularHomology

Example 102

Rn is contractible so the long exact sequence of the pair (Rn,Rn − 0)yeilds

H̃k(Rn) −→ Hk(Rn,Rn − 0) −→ H̃k−1(Rn − 0) −→ H̃k−1(Rn)

Rn − 0 ' Sn−1 so the local homology of Rn at 0 is

Hk(Rn,Rn − 0) ∼= H̃k−1(Rn − 0) ∼= H̃k−1(Sn−1) =

{Z, k = n0, k 6= n

So Rn � Rm if n 6= m.

Definition 103

An n-dimensional homology manifold is a space X such that for allx ∈ X

Hk(X ,X − x) ∼={

Z, k = n0, k 6= n

Homology theory

Lecture 10 -1/24/2011

Axiom 4:Additivity

Computations

Lecture 12 -1/26/2011


Lecture 13 -1/27/2011

CellularHomology

Naturality of the connectinghomomorphism

• Suppose we have a commutative diagram of chain complexesand chain maps with exact rows.

0 // C

γ

��

//

D

δ

��

//

E

ε

��

// 0

0 // C ′ // D ′ // E ′ // 0

• The Snake Lemma gives

· · · //

Hn+1(E )

ε∗

��

// Hn(C )

γ∗

��

//

Hn(D)

δ∗

��

//

Hn(E )

ε∗

��

// Hn−1(C )

γ∗

��

//

· · ·

· · · // Hn+1(E ′) // Hn(C ′) // Hn(D ′) // Hn(E ′) // Hn−1(C ′) // · · ·

• Fact that Hn : Chain→ Ab is a functor shows some squarescommute.

Homology theory

Lecture 10 -1/24/2011

Axiom 4:Additivity

Computations

Lecture 12 -1/26/2011


Lecture 13 -1/27/2011

CellularHomology

Theorem 104 (Naturality of connecting homomorphism)

Given a commutative diagram of chain complexes and chain mapswith exact rows.

0 // C

γ

��

//

D

δ

��

//

E

ε

��

// 0

0 // C ′ // D ′ // E ′ // 0

The following diagram commutes:

Hn+1(E )

ε∗

��

∂ //

Hn(C )

γ∗

��

Hn+1(E ′)∂ // Hn(C ′)

Homology theory

Lecture 10 -1/24/2011

Axiom 4:Additivity

Computations

Lecture 12 -1/26/2011


Lecture 13 -1/27/2011

CellularHomology

Proof of Theorem 104.

0 −−−−→ Cnin−−−−→ Dn

qn−−−−→ En −−−−→ 0y∂Cn

y∂Dn

y∂En

0 −−−−→ Cn−1in−1−−−−→ Dn−1

qn−1−−−−→ En−1 −−−−→ 0

• We must verify that ∂ε∗ = γ∗∂.

• By definition ∂[en] = [cn−1] where in−1(cn−1) = ∂Dn dn for somedn ∈ Dn with qn(dn) = en

• Note that εen = εqn(dn) = q′nδ(dn)

• And ∂D′

n δ(dn) = δ∂Dn (dn) = δin−1(cn−1) = i ′n−1γ(cn−1)

• so ∂ε∗[en] = ∂[εen] = [γ(cn−1)] = γ∗[cn−1] = γ∗∂[en]

Homology theory

Lecture 10 -1/24/2011

Axiom 4:Additivity

Computations

Lecture 12 -1/26/2011


Lecture 13 -1/27/2011

CellularHomology

Note on category theory:

Definition 105 (Natural transformation)

• Given two functors F ,G : C → D• A natural transformation η from F to G assigns a morphismηC : F (C )→ G (C ) for each C ∈ Ob(C) such that the followingdiagram commute for all morphisms f ∈ MorC(C1,C2)

F (C1)

ηC1

��

F (f )//

F (C2)

ηC2

��

G (C1)G(f )

// G (C2)

Homology theory

Lecture 10 -1/24/2011

Axiom 4:Additivity

Computations

Lecture 12 -1/26/2011


Lecture 13 -1/27/2011

CellularHomology

How does our naturality fit into this picture?

• Let C = SESChain be the category of short exact sequences ofchain complexes C ↪→ D � E with morphisms given bycommuting diagrams of chain maps from one short exactsequence to another.

• Let D = Ab

• Let F (C ↪→ D � E ) = Hn(E )

• Let G (C ↪→ D � E ) = Hn−1(C )

• Let η(C ↪→D�E) = ∂ : Hn(E )→ Hn−1(C )

Or

• Let C = TopPair

• Let D = Ab

• Let F (X ,A) = Hn(X ,A)

• Let G (X ,A) = Hn−1(A)

• Let η(X ,A) = ∂ : Hn(X ,A)→ Hn−1(A)

Homology theory

Lecture 10 -1/24/2011

Axiom 4:Additivity

Computations

Lecture 12 -1/26/2011


Lecture 13 -1/27/2011

CellularHomology


Show that if P is the set of path components of X then H0(X ) isisomorphic to the free abelian group on P and that if X is nonemptythen H̃0(X ) is the free abelian group on P − {c0} where c0 is a pathcomponent of X .

Homology theory

Lecture 10 -1/24/2011

Axiom 4:Additivity

Computations

Lecture 12 -1/26/2011


Lecture 13 -1/27/2011

CellularHomology

Cellular Homology

Definition 107 (Chain complex for cellular homology)

Let X be a CW complex (see Hatcher pg. 5). Let X−1 = ∅ and

CCW

n (X ) =

{Hn(X n,X n−1), n ≥ 00, n < 0

Let∂CW : CCW

n (X )→ CCW

n−1(X )

be the connecting homomorphism

Hn(X n,X n−1)∂−→ Hn−1(X n−1,X n−2)

from the long exact sequence of the triple (X n,X n−1,X n−2)

Proposition 108

CCW(X ) is a chain complex

Homology theory

Lecture 10 -1/24/2011

Axiom 4:Additivity

Computations

Lecture 12 -1/26/2011


Lecture 13 -1/27/2011

CellularHomology

Proof of Proposition 108.

• We must verify that (∂CW)2 = 0.

• We have the following commutative diagram of chain maps

0 // C (X n)

j]

��

// C (X n+1)

��

// C (X n+1,X n)

1

��

// 0

0 // C (X n,X n−1) // C (X n+1,X n−1) // C (X n+1,X n) // 0

• Naturality of the connecting homomorphism says the followingsquare from the long exact sequence of homology commutes:

Hn+1(X n+1,X n)

1

��

∂ // Hn(X n)

j∗

��

Hn+1(X n+1,X n)∂CW

// Hn(X n,X n−1)

• So ∂CW = j∗∂

Homology theory

Lecture 10 -1/24/2011

Axiom 4:Additivity

Computations

Lecture 12 -1/26/2011


Lecture 13 -1/27/2011

CellularHomology

Proof of Proposition 108 (continued).

• We have commutative

Hn(X n)

j∗

��

Hn+1(X n+1,X n)

∂

66mmmmmmmmmmmmm∂CW

// Hn(X n,X n−1)

∂

��

∂CW// Hn−1(X n−1,X n−2)

Hn−1(X n−1)

j∗66lllllllllllll

• Vertical composition

Hn(X n)j∗−→ Hn(X n,X n−1)

∂−→ Hn−1(X n−1)

is part of long exact sequence of the pair (X n,X n−1)

• So(∂CW)

2= (j∗∂) (j∗∂) = j∗ (∂j∗) ∂ = j∗ ◦ 0 ◦ ∂ = 0

Homology theory

Lecture 10 -1/24/2011

Axiom 4:Additivity

Computations

Lecture 12 -1/26/2011


Lecture 13 -1/27/2011

CellularHomology

Definition 109 (Cellular Homology)

If X is a CW complex then the nth cellular homology group of X is

HCW

n (X ) = Hn(CCW(X ))

For a CW complex X how do HCWn (X ) and Hn(X ) relate?

Theorem 110 (Cellular and sing. simp. homology agree)

If X is a CW complex then

HCW

n (X ) ∼= Hn(X )

Advantages of HCW

1 If X is finite then CCW(X ) is finitely generated

2 boundary maps in CCW(X ) can be easily understood.

Homology theory

Lecture 10 -1/24/2011

Axiom 4:Additivity

Computations

Lecture 12 -1/26/2011


Lecture 13 -1/27/2011

CellularHomology

The proof of Theorem 110 will use the following properties ofhomology for CW complexes.

Lemma 111 (Homology of CW complexes)

Let X is a CW complex. Let Γn be the set of n-cells of X

1

Hk(X n,X n−1) ∼={

Z[Γn] k = n0, k 6= n

2 If k > nHk(X n) = 0

3 If k < n and i : X n → X is the inclusion map then

Hk(X n)∼=−→i∗

Hk(X )

Homology theory

Lecture 10 -1/24/2011

Axiom 4:Additivity

Computations

Lecture 12 -1/26/2011


Lecture 13 -1/27/2011

CellularHomology

Proof of Lemma 111 Claim 1.

• We have a commuting quotient maps of good pairs

(∐α∈Γn Dn,

∐α∈Γn Sn−1

) r //

p

((

(X n,X n−1)q

// (X n/X n−1, {∗})

• By Theorem 95 we get that

p∗ : Hn

(∐α∈Γn

Dn,∐α∈Γn

Sn−1

)→ H̃n(X n/X n−1)

andq∗ : Hn(X n,X n−1)→ H̃n(X n/X n−1)

are isomorphisms.

• So we get an isomorphism

r∗ : Hn

(∐α∈Γn

Dn,∐α∈Γn

Sn−1

)→ Hn(X n,X n−1)

Homology theory

Lecture 10 -1/24/2011

Axiom 4:Additivity

Computations

Lecture 12 -1/26/2011


Lecture 13 -1/27/2011

CellularHomology

Proof of Lemma 111 Claim 1 (continued).

• Applying the additivity axiom

Hn(X n,X n−1) ∼= Hn

(∐α∈Γn

Dn,∐α∈Γn

Sn−1

)∼=⊕α∈Γn

Hn(Dn,Sn−1)

∼=⊕α∈Γn

H̃n(Sn)

∼=⊕α∈Γn

Z

∼= Z[Γn]

Math 757 Homology theoryHomology theory Lecture 10 - 1/24/2011 Axiom 4: Additivity Computations...

Documents

Transcript of Math 757 Homology theoryHomology theory Lecture 10 - 1/24/2011 Axiom 4: Additivity Computations...