math 757 homology theoryhomology theory lecture 1 - 1/3/2011 category theory eilenberg-steenrod...

Homology theory

Lecture 1 -1/3/2011

Category Theory

Eilenberg-SteenrodAxioms

Lecture 2 -1/4/2011

Existence ofhomologytheories

Construction ofsingularsimplicialhomology

Lecture 3 -1/5/2011

Axiom A5(Dimension)

Homology as afunctor

Lecture 4 -1/6/2011

Homology forpairs

Axiom A2(Exactness)

Math 757Homology theory

January 6, 2011

Homology theory

Lecture 1 -1/3/2011

Category Theory


Lecture 2 -1/4/2011



Lecture 3 -1/5/2011

Axiom A5(Dimension)


Lecture 4 -1/6/2011

Homology forpairs

Axiom A2(Exactness)

Category Theory

Definition 1 (Category)

A category C consists of

1 objects Ob(C) of C.

2 For all X ,Y ∈ Ob(C) a set Mor(X ,Y ) called morphisms fromX to Y

3 For all X ,Y ,Z ∈ Ob(C) a “composition”◦ : Mor(X ,Y )×Mor(Y ,Z )→ Mor(X ,Z )

Satisfying

1 For all X ∈ Ob(C) there is distinguished 1 = 1X ∈ Mor(X ,X )s.t. f ◦ 1 = 1 ◦ f = f for all morphisms f .

2 For all morphisms f , g , h we have f ◦ (g ◦ h) = (f ◦ g) ◦ h

Homology theory

Lecture 1 -1/3/2011

Category Theory


Lecture 2 -1/4/2011



Lecture 3 -1/5/2011

Axiom A5(Dimension)


Lecture 4 -1/6/2011

Homology forpairs

Axiom A2(Exactness)

Categories

Example 2 (PTop)

C = PTop

• Ob(PTop) pointed topological spaces (X , x0)

• Mor((X , x0), (Y , y0)) (continuous) maps f : X → Y withf (x0) = y0

• 1 ∈ Mor((X , x0), (X , x0)) is identity function (1(x) = x).

Example 3 (Group)

C = Group

• Ob(Group) is all groups

• Mor(X ,Y ) homomorphisms from X to Y with composition.

• 1 ∈ Mor(X ,X ) is identity function (1(x) = x).

Homology theory

Lecture 1 -1/3/2011

Category Theory


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Axiom A5(Dimension)


Lecture 4 -1/6/2011

Homology forpairs

Axiom A2(Exactness)

Categories

Example 4 (Paths in X )

X a topological space.

• Ob(C) points of X

• Mor(x , y) = {[f ]|f a path from x to y} with concatenation

• 1x ∈ Mor(x , x) is class of constant path [cx ]

Homology theory

Lecture 1 -1/3/2011

Category Theory


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Axiom A5(Dimension)


Lecture 4 -1/6/2011

Homology forpairs

Axiom A2(Exactness)

Functors

Definition 5 (Functor)

A functor F from a category C to a category D consists of

1 For each object X ∈ Ob(C) an object F (X ) ∈ Ob(D)

2 For each morphism f : X → Y in Mor(C) a morphismF (f ) : F (X )→ F (Y ) in Mor(D)

Satisfying

1 For all X ∈ Ob(C) we have F (1X ) = 1F (X )

2 F (f ◦ g) = F (f ) ◦ F (g)

Homology theory

Lecture 1 -1/3/2011

Category Theory


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Axiom A5(Dimension)


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Homology forpairs

Axiom A2(Exactness)

Functors

Example 6 (Fundamental Group)

π1 : PTop→ Group is a functor. (Easy proof)

Example 7

f : X → Y continuous

• C is points of X with morphisms classes of paths in X

• D is points of Y with morphisms classes of paths in Y

• F (x) = f (x) and F ([γ]) = [f γ]

F : C → D is a functor.

Homology theory

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Category Theory


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Axiom A5(Dimension)


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Homology forpairs

Axiom A2(Exactness)

Definition 8 (Abelian Groups)

Ab is category with

• Ob(Ab) is all abelian groups

• Mor(A,B) all homomorphisms h : A→ B

Definition 9 (Topological pairs)

TopPair category with

• (X ,A) ∈ Ob(TopPair) if X a space and A ⊂ X a subspace.

• Mor((X ,A), (Y ,B)) is set of continuous f : X → Y s.t.f (A) ⊂ B

Abusing notation X means (X ,∅)

Homology theory

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Category Theory


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Axiom A5(Dimension)


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Homology forpairs

Axiom A2(Exactness)

Definition 10 (homotopy)

f , g : (X ,A)→ (Y ,B) are homotopic if there is continuous

F : (X × I ,A× I )→ (Y ,B)

with F (x , 0) = f (x) and F (x , 1) = g(x)

Definition 11 (relative homotopy)

X ′ ⊂ X then f , g : (X ,A)→ (Y ,B) are homotopic rel X ′ if there iscontinuous

F : (X × I ,A× I )→ (Y ,B)

with F (x , 0) = f (x) and F (x , 1) = g(x) and F (x ′, t) = f (x ′) for allx ′ ∈ X ′, t ∈ I

Homology theory

Lecture 1 -1/3/2011

Category Theory


Lecture 2 -1/4/2011



Lecture 3 -1/5/2011

Axiom A5(Dimension)


Lecture 4 -1/6/2011

Homology forpairs

Axiom A2(Exactness)

Eilenberg-Steenrod Axioms

Axiomatic approach:

1 List axioms satisfied by homology theory

2 Assume such a theory exists

3 Compute homology groups using axioms

4 Define singular homology theory

5 Show it satisfies the axioms (very technical)

Homology theory

Lecture 1 -1/3/2011

Category Theory


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Axiom A5(Dimension)


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Homology forpairs

Axiom A2(Exactness)

Analog

Axiomatic approach to reals R:

1 F is a field.(F ,+, ·, 0, 1) a setwith elements 0, 1 ∈ Fand binary functions +, · : F × F → F

1 associativity of + (∀a, b, c ∈ F a+ (b + c) = (a+ b) + c.)2 associativity of ·3 etc.

2 F is an ordered field

1 trichotomy (∀a, b ∈ F either a = b, a < b, or a > b.)2 0 < 13 etc.

3 KEY: LUB axiomEvery nonempty bounded subset of F has a sup

Homology theory

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Category Theory


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Axiom A2(Exactness)

Analog

Example 12 (Ordered fields)

Q, Q(r1, · · · , rn), R, ∗R (the hyperreals)

Whereas

Theorem 13

There is an ordered field R satisfying LUB.

Proof.

Construct (R,+, ·, <) using Cauchy sequences.

Theorem 14

Any ordered field satisfying LUB is isomorphic to R.

Homology theory

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Category Theory


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Axiom A5(Dimension)


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Homology forpairs

Axiom A2(Exactness)


Definition 15 (Homology theory)

A homology theory (h∗, ∂) is a series of functorshn : TopPair→ Ab and for each pair (X ,A) natural transformations

∂n(X ,A) : hn(X ,A)→ hn−1(A)

called “boundary maps” satisfying

A1. Homotopy Axiom

A2. Exactness Axiom

A3. Excision Axiom

A4. Additivity Axiom

We will also assume

A5. Dimension Axiom

Homology theory

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Category Theory


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Axiom A5(Dimension)


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Homology forpairs

Axiom A2(Exactness)


Axiom A1. Homotopy

If f , g : (X ,A)→ (Y ,B) are homotopic maps

then homorphisms

hn(f ) : hn(X ,A)→ hn(Y ,B)

andhn(g) : hn(X ,A)→ hn(Y ,B)

are equal.

In other words modifying f : (X ,A)→ (Y ,B) by a homotopypreserves hn(f ).

Homology theory

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Category Theory


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Axiom A5(Dimension)


Lecture 4 -1/6/2011

Homology forpairs

Axiom A2(Exactness)


Axiom A2. Exactness

For any pair (X ,A) let

i : A→ X

j : X → (X ,A)

be the inclusion maps.

The sequence

· · · ∂n+1(X ,A)−−−−−−→ hn(A)hn(i)−−−→ hn(X )

hn(j)−−−→ hn(X ,A)

∂n(X ,A)−−−−−→ hn−1(A)hn−1(i)−−−−→ hn−1(X )

hn−1(j)−−−−→ · · ·

is exact.

Homology theory

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Category Theory


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Axiom A5(Dimension)


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Homology forpairs

Axiom A2(Exactness)


Axiom A3. Excision

For any pair (X ,A) if U ⊂ A is open with U ⊂ int A then theinclusion

i : (X − U,A− U)→ (X ,A)

induces an isomorphism

hn(i) : hn(X − U,A− U) ∼= hn(X ,A)

Allows us to “excise” U from homology calculations

Homology theory

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Category Theory


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Axiom A5(Dimension)


Lecture 4 -1/6/2011

Homology forpairs

Axiom A2(Exactness)


Axiom A4. Additivity

Xα disjoint spaces. Then

hn

(∐Xα)

=⊕α

hn(Xα)

Homology theory

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Category Theory


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Axiom A5(Dimension)


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Homology forpairs

Axiom A2(Exactness)


Axiom A5. Dimension (not required for a homology theory)

For the one point space {∗}

hn({∗}) =

{Z, n = 00, n 6= 0

Later we will relax Axiom A5 slightly to allow:

Axiom A5′

For the one point space {∗}

hn({∗}) =

{A, n = 00, n 6= 0

For some abelian group A.

Homology theory

Lecture 1 -1/3/2011

Category Theory


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Axiom A5(Dimension)


Lecture 4 -1/6/2011

Homology forpairs

Axiom A2(Exactness)

Existence of homology theories

In this course we show:

Theorem 16

Singular simplicial homology theory (H∗, ∂) satisfies 1-5.

Homology theory

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Category Theory


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Axiom A5(Dimension)


Lecture 4 -1/6/2011

Homology forpairs

Axiom A2(Exactness)

Stable homotopy theory

Supension gives

πn(X ,A)S−→ πn+1(SX ,SA)

S−→ πn+2(S2X ,S2A)S−→ · · ·

Definition 17 (Stable homotopy group)

The stable homotopy group is

πSn (X ,A) = lim−→

k

πn+k(SkX ,SkA)

Theorem 18

(Relative) stable homotopy groups (πS∗ , ∂

S) satisfy 1-4.

Homology theory

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Category Theory


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Axiom A5(Dimension)


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Homology forpairs

Axiom A2(Exactness)

Construction of singular simplicialhomology

Outline

1 Define chain complexes (easy)

2 Give chain complex for singular simplicial homology (easy)

3 Define homology of a chain complex (easy)

4 Show homology of this chain complex satisfies axioms (hard)

Homology theory

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Category Theory


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Axiom A5(Dimension)


Lecture 4 -1/6/2011

Homology forpairs

Axiom A2(Exactness)

Exactness

Definition 19 (Exact)

A,B,C groups

g : A→ B, h : B → C homomorphisms.

Ag−→ B

h−→ C

is exact at B if Im(g) = ker(h)

Definition 20 (Exact Sequence)

A (possibly bi-infinite) sequence of homomorphisms

· · · ϕi+3−−→ Ai+2ϕi+2−−→ Ai+1

ϕi+1−−→ Aiϕi−1−−−→ · · ·

is exact if Im(ϕi+1) = ker(ϕi )

Homology theory

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Category Theory


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Axiom A5(Dimension)


Lecture 4 -1/6/2011

Homology forpairs

Axiom A2(Exactness)

Chain Complexes

Definition 21 (Chain Complex)

A chain complex is a pair (C , ∂) of abelian groups Cn andhomomophisms ∂n+1 : Cn+1 → Cn

· · · ∂3−→ C2∂2−→ C1

∂1−→ C0∂0−→ C−1

∂−1−−→ · · ·

satisfying∂i ◦ ∂i+1 = 0.

Often suppress ∂ and call C a chain complex.

Note: All chain complexes we consider will have Cn = 0 for n ≤ −2(and almost always C−1 = 0 too)

Homology theory

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Category Theory


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Homology forpairs

Axiom A2(Exactness)

Chain Complexes

Definition 22 (Chain Complex (alt. def.))

A chain complex is a pair (C , ∂) where

C =⊕n∈Z

Cn

is a Z-graded abelian group with degree -1 homomorphism

∂ : C → C

satisfying∂2 = 0.

Homology theory

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Category Theory


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Axiom A2(Exactness)

Chain Complex for SingularSimplicial Homology

Goal: Given a top. space X give a chain complex C (X )

Definition 23 (n-simplex)

The n-simplex is

∆n =

{(x0, x1, · · · , xn) ∈ Rn+1

∣∣∣∣xi ≥ 0, and∑i=1

xi = 1

}.

Given vectors v0, v1, · · · , vn ∈ Rm we have a map

[v0, v1, · · · , vn] : ∆n → Rm

where [v0, v1, · · · , vn](x0, x1, · · · , xn) =∑n

i=1 xivi

Homology theory

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Definition 24 (singular n-simplices of X )

The singular n-simplices of the top. space X is the set

Sn(X ) = {σ : ∆n → X}

Definition 25 (singular simplicial n-chains of X )

The singular simplicial n-chains of the space X is the free abeliangroup on Sn(X )

Cn(X ) =

{Z[Sn(X )], n ≥ 00, n < 0

Formal Z-linear combinations of maps σ : ∆n → X .

Homology theory

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Axiom A2(Exactness)


What about ∂ for C (X )?

Definition 26 (Face of a singular simplex)

Given a singular simplex σ : ∆n → X the ith face of σ is

∂ iσ : ∆n−1 → X

where

(∂ iσ)(x0, x1, · · · , xn−1) = σ(x0, x1, · · · , xi−1, 0, xi , · · · , xn−1)

Homology theory

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Category Theory


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Definition 27 (Boundary map)

Define ∂n : Sn(X )→ Cn−1(X ) as follows:

For a 0-simplex σ : ∆0 → X set ∂0σ = 0.

For n ≥ 1, given a singular simplex σ : ∆n → X let

∂nσ =n∑

i=0

(−1)i∂ iσ

Extend Z-linearly to get a homomorphism

∂n : Cn(X )→ Cn−1(X )

(Recall Cn(X ) = Z[Sn(X )]).

Homology theory

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Category Theory


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Homology forpairs

Axiom A2(Exactness)

Recall

X a space.

• Sn(X ) = {σ|σ : ∆n → X} singular simplices of X

• Cn(X ) = Z[Sn(X )] n-chains of X for n ≥ 0

• ∂n : Cn(X )→ Cn−1(X ) is

∂nσ =n∑

i=0

(−1)i∂ iσ

for n ≥ 1

Homology theory

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Category Theory


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Axiom A2(Exactness)

Lemma 28

Let σ ∈ Sn(X ).If j ≤ i then ∂ i∂jσ = ∂j∂ i+1σ.

Proof.

Given σ : ∆n → X . If j ≤ i then

(∂ i∂jσ)(x0, x1, · · · , xn−2)

= (∂jσ)(x0, x1, · · · , xi−1, 0, xi , · · · , xn−2)

= σ(x0, x1, · · · , xj−1, 0, xj , · · · , xi−1, 0, xi , · · · , xn−2)

= (∂ i+1σ)(x0, x1, · · · , xj−1, 0, xj , · · · , xi−1, xi , · · · , xn−2)

= (∂j∂ i+1σ)(x0, x1, · · · , xj−1, xj , · · · , xi−1, xi , · · · , xn−2)

Homology theory

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Category Theory


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Axiom A2(Exactness)

Lemma 29 (Boundary map)

If X is a space then (C (X ), ∂) is a chain complex.

Example 30 (3-simplex)

Proof.

Given σ : ∆n → X

∂n−1∂nσ = ∂n−1

n∑j=0

(−1)j∂jσ

=n−1∑i=0

(−1)i∂ in∑

j=0

(−1)j∂jσ

=n−1∑i=0

n∑j=0

(−1)i+j∂ i∂jσ

Homology theory

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Axiom A2(Exactness)



Proof (continued).

=n−1∑i=0

i∑j=0

(−1)i+j∂ i∂jσ +n∑

j=i+1

(−1)i+j∂ i∂jσ

=

n−1∑i=0

i∑j=0

(−1)i+j∂ i∂jσ

+

n−1∑i=0

n∑j=i+1

(−1)i+j∂ i∂jσ

=

n−1∑i=0

i∑j=0

(−1)i+j∂j∂ i+1σ

+

n−1∑i=0

n∑j=i+1

(−1)i+j∂ i∂jσ

Homology theory

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Proof (continued).

=

∑0≤j≤i≤n−1

(−1)i+j∂j∂ i+1σ

+

∑0≤i<j≤n

(−1)i+j∂ i∂jσ

=

∑0≤i≤j≤n−1

(−1)i+j∂ i∂j+1σ

+

∑0≤i<j≤n

(−1)i+j∂ i∂jσ

=

∑0≤i<j≤n

(−1)i+j−1∂ i∂jσ

+

∑0≤i<j≤n

(−1)i+j∂ i∂jσ

Homology theory

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Axiom A2(Exactness)

Homology of Chain Complexes

Definition 30 (cycles and boundaries)

Let (C , ∂) be a chain complex.The n-cycles of C is the subgroup

Zn(C ) = ker ∂n ⊂ Cn

The n-boundaries of C is the subgroup

Bn(C ) = Im ∂n+1 ⊂ Cn

Note: ∂n∂n+1 = 0 so Bn(C ) ⊂ Zn(C ).

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Homology of Chain Complexes

Definition 31 (homology of a chain complex)

Let (C , ∂) be a chain complex. The nth homology group of C isthe group

Hn(C ) := Zn(C )/Bn(C ) = ker ∂n/ Im ∂n+1

Lemma 32

Let (C , ∂) be a chain complex. The chain complex is exact at Cn ifand only if

Hn(C ) = 0

Homology theory

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Homology of a space

Definition 33 (homology of a space)

Let X be a space and (C (X ), ∂) its chain complex of singularsimplicial chains.

· · · ∂3−→ C2(X )∂2−→ C1(X )

∂1−→ C0(X )∂0−→ 0

∂−1−−→ 0∂−2−−→ · · ·

The nth (singular simplicial) homology group of X is the group

Hn(X ) := Hn(C (X ))

= Zn(C (X ))/Bn(C (X ))

= ker ∂n/ Im ∂n+1

Homology theory

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Axiom A5 (Dimension)

Exercise 34 (HW 1 - Problem 1)

Prove Axiom A5 (Dimension) for singular simplicial homology. Thatis, show that if X = {∗} is the one-point space then

Hn(X ) =

{Z, n = 00, n 6= 0

.

Homology theory

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Homology and maps

Theorem 35

f : X → Y a homeomorphism. Then Hn(X ) ∼= Hn(Y )

Proof?

Our guide will be functoriality.

Observation

Given a map f : X → Y we have graded group homomorphism

f] : C (X )→ C (Y )

where a singular simplex σ : ∆n → X is sent to

f]σ = f ◦ σ (extend linearly)

Homology theory

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Homology as a functor

Question

(C , ∂C ) and (D, ∂D) chain complexes and

γ : C → D

a graded group homomorphism. What property of γ ensures that weget induced

γ∗ : Hn(C )→ Hn(D)?

Homology theory

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Category of Chain Complexes

Definition 36 (chain map)

(C , ∂C ) and (D, ∂D) chain complexes. A chain map is a gradedgroup homomorphism (of degree 0)

τ : C → D

satisfyingτ ◦ ∂C = ∂D ◦ τ.

That is the following diagram commutes:

· · ·∂Cn+2−−−−→ Cn+1

∂Cn+1−−−−→ Cn

∂Cn−−−−→ Cn−1

∂Cn−1−−−−→ · · ·yτn+1

yτn yτn−1

· · ·∂Dn+2−−−−→ Dn+1

∂Dn+1−−−−→ Dn

∂Dn−−−−→ Dn−1

∂Dn−1−−−−→ · · ·

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Definition 37 (Category Chain)

The category Chain has chain complexes as objects and chain mapsas morphisms.

Proposition 38

Taking singular simplicial chains

C : Top→ Chain

is a functor where f : X → Y induces the chain map

C (f ) : C (X )→ C (Y )

given byC (f )(σ) = f](σ) = f ◦ σ

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Homology forpairs

Axiom A2(Exactness)



Show that iff : X → Y

is a map thenC (f ) : C (X )→ C (Y )

is a chain map.

Homology theory

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Category Theory


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Axiom A5(Dimension)


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Homology forpairs

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Proposition 40

Taking nth homology

Hn : Chain→ Ab

is a functor where the chain map

τ : C → D

induces a well-defined homorphism

Hn(τ) : Hn(C )→ Hn(D)


Prove Proposition 40

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Homology forpairs

Axiom A2(Exactness)


Prove Theorem 35 (Homology groups are topological invariants).

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Axiom A5(Dimension)


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Homology forpairs

Axiom A2(Exactness)

Homology for pairs

Note: in order to satisfy homology theory axioms (Definition 15) weneed homology for pairs.

Note: If A ⊂ X is a subspace then inclusion induces an inclusionC (A) ⊂ C (X ).

Definition 43 (Chain complex for a pair)

If (X ,A) is a topological pair (A ⊂ X ) then the singular simplicialn-chains of (X ,A) is the free abelian group

Cn(X ,A) = Cn(X )/Cn(A)

where if q : Cn(X )→ Cn(X ,A) is the quotient map and ∂X is theboundary map for C (X ) then the boundary map∂ : Cn(X ,A)→ Cn−1(X ,A) is given by

∂(q(σ)) = q(∂Xσ)

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For any pair of spaces (X ,A)

∂n : Cn(X ,A)→ Cn−1(X ,A)

is well-defined and ∂n ◦ ∂n+1 = 0

Note: By definition (and Exercise 44) the quotient mapq : C (X )→ C (X ,A) is a chain map.

Definition 45 (Homology of a pair)

If (X ,A) is a topological pair (A ⊂ X ) then the nth singularsimplicial homology group of (X ,A) is the abelian group

Hn(X ,A) := Zn(X ,A)/Bn(X ,A)

= ker ∂n/ Im ∂n+1

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Proposition 46

If f : (X ,A)→ (Y ,B) is a map (f : X → Y is a map and f (A) ⊂ B)then f induces a chain map

C (f ) : C (X ,A)→ C (Y ,B)


Prove Proposition 46

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Axiom A2(Exactness)

Axiom A2 (Exactness)

Definition 48 (Exact sequence of chain complex)

An exact sequence of chain complexes is a sequence of chaincomplexes and chain maps which is exact in the category of abeliangroups.

Example 49

By definition of C (X ,A) the following is an exact sequence of chaincomplexes

0→ C (A)i−→ C (X )

q−→ C (X ,A)→ 0

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Snake Lemma

Lemma 50 (Snake Lemma)

A short exact sequence of chain complexes

0→ Ci−→ D

q−→ E → 0

induces a long exact sequence of homology groups

· · · ∂−→ Hn(C )i∗−→ Hn(D)

q∗−→ Hn(E )∂−→ Hn−1(C )

i∗−→ · · ·

where the connecting homomorphism ∂ is defined below.

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Homology forpairs

Axiom A2(Exactness)

Proof of A2

Proof of Exactness (A2) for singular simplicial homology.

We have exact sequence of chain complexes

0→ C (A)i−→ C (X )

q−→ C (X ,A)→ 0

By Snake Lemma we get long exact sequence of homology groups

· · · ∂−→ Hn(A)i∗−→ Hn(X )

q∗−→ Hn(X ,A)∂−→ Hn−1(A)

i∗−→ · · ·

where the boundary map

∂n(X ,A) : Hn(X ,A)→ Hn−1(A)

of is defined to be ∂n.

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Homology forpairs

Axiom A2(Exactness)

Proof of Snake Lemma.

The short exact sequence of chain complexes

0→ Ci−→ D

q−→ E → 0

Gives commutative diagram

0 −−−−→ Cn+1in+1−−−−→ Dn+1

qn+1−−−−→ En+1 −−−−→ 0y∂Cn+1

y∂Dn+1

y∂En+1

0 −−−−→ Cnin−−−−→ Dn

qn−−−−→ En −−−−→ 0y∂Cn

y∂Dn

y∂En

0 −−−−→ Cn−1in−1−−−−→ Dn−1

qn−1−−−−→ En−1 −−−−→ 0

First we define ∂n+1 : Hn+1(E )→ Hn(C )

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Homology forpairs

Axiom A2(Exactness)

Proof of Snake Lemma (continued).

0 −−−−→ Cn+1in+1−−−−→ Dn+1

qn+1−−−−→ En+1 −−−−→ 0y∂Cn+1

y∂Dn+1

y∂En+1

0 −−−−→ Cnin−−−−→ Dn

qn−−−−→ En −−−−→ 0y∂Cn

y∂Dn

y∂En

0 −−−−→ Cn−1in−1−−−−→ Dn−1

qn−1−−−−→ En−1 −−−−→ 0

Each class Hn+1(E ) has n-cycle rep en+1 ∈ ker ∂En+1 ⊂ En+1.

qn+1 surjects so ∃dn+1 ∈ Dn+1 s.t. qn+1(dn+1) = en+1

qn∂Dn+1(dn+1) = ∂En+1qn+1(dn+1) = ∂En+1(en+1) = 0 so

∂Dn+1(dn+1) ∈ ker qn

∃!cn ∈ Cn s.t. in(cn) = ∂Dn+1(dn+1).

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0 −−−−→ Cn+1in+1−−−−→ Dn+1

qn+1−−−−→ En+1 −−−−→ 0y∂Cn+1

y∂Dn+1

y∂En+1

0 −−−−→ Cnin−−−−→ Dn

qn−−−−→ En −−−−→ 0y∂Cn

y∂Dn

y∂En

0 −−−−→ Cn−1in−1−−−−→ Dn−1

qn−1−−−−→ En−1 −−−−→ 0

in−1∂Cn (cn) = ∂Dn in(cn) = ∂Dn ∂

Dn+1(dn+1) = 0 so

in−1 injective so ∂Cn (cn) = 0.

Define ∂n+1 : Hn+1(E )→ Hn(C ) by setting

∂n+1([en+1]) = [cn]

Homology theory

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0 −−−−→ Cn+1in+1−−−−→ Dn+1

qn+1−−−−→ En+1 −−−−→ 0y∂Cn+1

y∂Dn+1

y∂En+1

0 −−−−→ Cnin−−−−→ Dn

qn−−−−→ En −−−−→ 0y∂Cn

y∂Dn

y∂En

0 −−−−→ Cn−1in−1−−−−→ Dn−1

qn−1−−−−→ En−1 −−−−→ 0

Claim: ∂n+1 : Hn+1(E )→ Hn(C ) independent of dn+1.

Suppose qn+1(d ′n+1) = en+1 and in(c ′n) = ∂Dn+1(d ′n+1)

qn+1(d ′n+1 − dn+1) = 0 so ∃cn+1 s.t. in+1(cn+1) = d ′n+1 − dn+1

in∂Cn+1(cn+1) = ∂Dn+1in+1(cn+1) = ∂Dn+1d ′n+1 − ∂Dn+1dn+1

in injective so ∂Cn+1(cn+1) = c ′n − cn. Hence [c ′n] = [cn].

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0 −−−−→ Cn+2in+2−−−−→ Dn+2

qn+2−−−−→ En+2 −−−−→ 0y∂Cn+2

y∂Dn+2

y∂En+2

0 −−−−→ Cn+1in+1−−−−→ Dn+1

qn+1−−−−→ En+1 −−−−→ 0y∂Cn+1

y∂Dn+1

y∂En+1

0 −−−−→ Cnin−−−−→ Dn

qn−−−−→ En −−−−→ 0

Claim: ∂n+1 : Hn+1(E )→ Hn(C ) independent of rep en+1.

Suppose [e′n+1] = [en+1]. Then ∃en+2 ∈ En+2 s.t.e′n+1 = en+1 + ∂En+2en+2 and dn+2 s.t. qn+2(dn+2) = en+2.

qn+1(dn+1 + ∂Dn+2dn+2) = en+1 + ∂En+2en+2 = e′n+1

But, ∂Dn+1(dn+1 + ∂Dn+2dn+2) = ∂Dn+1(dn+1) = in(cn).

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Thus ∂n+1 : Hn+1(E )→ Hn(C ) is well-defined.

Claim: ∂n+1 : Hn+1(E )→ Hn(C ) is a homomorphism.

If ∂n+1([e1n+1]) = [c1

n ] and ∂n+1([e2n+1]) = [c2

n ]

Then ∃d1n+1, d

2n+1 s.t. qn+1(d1

n+1) = e1n+1 and qn+1(d2

n+1) = e2n+1

and in(c1n ) = ∂Dn+1d1

n+1 and in(c2n ) = ∂Dn+1d2

n+1

So qn+1(d1n+1 + d2

n+1) = e1n+1 + e2

n+1 andin(c1

n + c2n ) = ∂Dn+1(d1

n+1 + d2n+1)

∂n+1([e1n+1] + [e2

n+1]) = [c1n + c2

n ] = ∂n+1([e1n+1]) + ∂n+1([e2

n+1])

Homology theory

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Axiom A2(Exactness)


Claim:

· · · ∂−→ Hn(C )i∗−→ Hn(D)

q∗−→ Hn(E )∂−→ Hn−1(C )

i∗−→ · · ·

is exact.

I. Im i∗ ⊂ ker q∗

q ◦ i : C → E is 0 so q∗ ◦ i∗ = 0.

II. Im q∗ ⊂ ker ∂

Suppose [dn+1] ∈ Hn+1(D) then dn+1 ∈ ker ∂D so ∂D(dn+1) = 0.in(0) = 0 so ∂q∗[dn+1] = 0.

III. Im ∂ ⊂ ker i∗

Suppose [en+1] ∈ Hn+1(E ) then ∃dn+1 ∈ Dn+1 s.t.qn+1(dn+1) = en+1. i∗∂[en+1] = [∂D(dn+1)] = 0.

Homology theory

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Axiom A2(Exactness)


Claim:

· · · ∂−→ Hn(C )i∗−→ Hn(D)

q∗−→ Hn(E )∂−→ Hn−1(C )

i∗−→ · · ·

is exact.

IV. ker q∗ ⊂ Im i∗

Suppose [dn] ∈ ker q∗. Then q(dn) = ∂Een+1. By surjectivity of qthere is dn+1 s.t. q(dn+1) = en+1.

q(dn − ∂Ddn+1) = q(dn)− ∂Eq(dn+1) = q(dn)− ∂Een+1 = 0

∃cn s.t. i(cn) = dn − ∂Ddn+1. cn is a cycle since:

i∂Ccn = ∂D icn = ∂D(dn − ∂Ddn+1) = ∂D(dn) = 0

Hence i∗[cn] = [dn − ∂Ddn+1] = [dn].

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Axiom A2(Exactness)


Claim:

· · · ∂−→ Hn(C )i∗−→ Hn(D)

q∗−→ Hn(E )∂−→ Hn−1(C )

i∗−→ · · ·

is exact.

V. ker ∂ ⊂ Im q∗

Suppose [en+1] ∈ ker ∂. Then there is dn+1 ∈ dn+1 and cn ∈ Cn s.t.qdn+1 = en+1 and i(cn) = ∂Ddn+1. Further there is cn+1 ∈ Cn+1 s.t.∂Ccn+1 = cn

∂D(dn+1 − i(cn+1)) = ∂Ddn+1 − ∂Ddn+1 = 0 so dn+1 − i(cn+1) is acycle.

q(dn+1 − i(cn+1)) = en+1 − 0 so q∗[dn+1 − i(cn+1)] = [en+1].

Homology theory

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Category Theory


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Homology forpairs

Axiom A2(Exactness)


Claim:

· · · ∂−→ Hn(C )i∗−→ Hn(D)

q∗−→ Hn(E )∂−→ Hn−1(C )

i∗−→ · · ·

is exact.

VI. ker i∗ ⊂ Im ∂

Suppose [cn] ∈ ker i∗ then ∃dn+1 ∈ Dn+1 s.t. ∂D(dn+1) = i(cn).

∂Eq(dn+1) = q∂D(dn+1) = qi(cn) = 0 so q(dn+1) is a cycle.

∂[q(dn+1)] = [cn].

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