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
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)
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
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)
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
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)
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
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)
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
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)
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
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)
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
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)
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
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)
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
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)
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
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)
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
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)
Eilenberg-Steenrod Axioms
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
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)
Eilenberg-Steenrod Axioms
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
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)
Eilenberg-Steenrod Axioms
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
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)
Eilenberg-Steenrod Axioms
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
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)
Eilenberg-Steenrod Axioms
Axiom A4. Additivity
Xα disjoint spaces. Then
hn
(∐Xα)
=⊕α
hn(Xα)
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)
Eilenberg-Steenrod Axioms
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
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)
Existence of homology theories
In this course we show:
Theorem 16
Singular simplicial homology theory (H∗, ∂) satisfies 1-5.
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)
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
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)
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
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)
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
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)
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
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)
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
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)
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
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)
Chain Complex for SingularSimplicial Homology
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
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)
Chain Complex for SingularSimplicial Homology
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
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)
Chain Complex for SingularSimplicial Homology
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
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)
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
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)
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
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)
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
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)
Lemma 29 (Boundary map)
If X is a space then (C (X ), ∂) is a chain complex.
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
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)
Lemma 29 (Boundary map)
If X is a space then (C (X ), ∂) is a chain complex.
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
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)
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 ).
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)
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
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)
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
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)
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
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)
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
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)
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
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)
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−−−−→ · · ·
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)
Category of Chain Complexes
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
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)
Category of Chain Complexes
Exercise 39 (HW 1 - Problem 2)
Show that iff : X → Y
is a map thenC (f ) : C (X )→ C (Y )
is a chain map.
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)
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)
Exercise 41 (HW 1 - Problem 3)
Prove Proposition 40
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)
Exercise 42 (HW 1 - Problem 4)
Prove Theorem 35 (Homology groups are topological invariants).
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)
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)
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)
Exercise 44 (HW 1 - Problem 5)
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
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)
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)
Exercise 47 (HW 1 - Problem 6)
Prove Proposition 46
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)
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
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)
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.
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)
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.
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)
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 )
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)
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).
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)
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
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
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)
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
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].
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)
Proof of Snake Lemma (continued).
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).
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)
Proof of Snake Lemma (continued).
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
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)
Proof of Snake Lemma (continued).
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
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)
Proof of Snake Lemma (continued).
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].
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)
Proof of Snake Lemma (continued).
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
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)
Proof of Snake Lemma (continued).
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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