Math 757 Homology Theory

Homology theory

Lecture 10 - 1/24/2011 Axiom 4: Additivity Computations Lecture 12 - 1/26/2011 Naturality of the connecting Math 757 homomorphism Lecture 13 - Homology theory 1/27/2011 Cellular Homology

January 27, 2011 Homology theory Axiom 4: Additivity Lecture 10 - 1/24/2011 Axiom 4: Additivity Computations Deﬁnition 90 (Direct sum in Ab) Lecture 12 - 1/26/2011 Naturality of the Given abelian groups {Aα}, the direct sum is an abelian group A connecting homomorphism with homomorphisms Lecture 13 - 1/27/2011 γα : Aα → A Cellular Homology 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 {C α} is the direct sum as abelian groups with the added stipulation that all homomorphisms must be chain maps Homology theory Theorem 92 (Sing. simp. homology satisﬁes Axiom A4) Lecture 10 - 1/24/2011 Axiom 4: Xα disjoint spaces. Then Additivity Computations a M Lecture 12 - Hn Xα = Hn(Xα) 1/26/2011 Naturality of the α connecting homomorphism Lecture 13 - 1/27/2011 Cellular Exercise 93 (HW4 - Problem 1) Homology

Verify that the homology functors Hn : Chain → Ab commute with direct sums. In particular, show that if C α are chain complexes then ! M α ∼ M α Hn C = Hn (C ) α α

L α This may be done by verifying that Hn ( α C ) satisﬁes the universal property for direct sums. Homology theory

Lecture 10 - 1/24/2011 Axiom 4: Additivity Computations Deﬁnition 94 (Good pair) Lecture 12 - 1/26/2011 (X , A) is a good pair if there is a neighborhood V of A in X such Naturality of the connecting that V deformation retracts to A. homomorphism Lecture 13 - 1/27/2011 Cellular Theorem 95 (Relative homology and quotient spaces) Homology If (X , A) is a good pair then we have an isomorphism

q∗ : Hn(X , A) →H˜n(X /A)

where q∗ is induced by the composition C(X , A) → C(X /A, A/A) →C˜(X /A) Homology theory Proof of Theorem 95.

Lecture 10 - α β 1/24/2011 Hn(X , A) −−−−−→ Hn(X , V ) ←−−−−− Hn(X − A, V − A) Axiom 4: Additivity    q   0  Computations ∗y y q∗y Lecture 12 - γ η 1/26/2011 Hn(X /A, A/A) −−−−−→ Hn(X /A, V /A) ←−−−−− Hn(X /A − A/A, V /A − A/A) Naturality of the connecting homomorphism Lecture 13 - • β and η are isomorphisms by excision. 1/27/2011 Cellular • The long exact sequence of the triple (X , V , A) gives Homology α Hn(V , A) −→ Hn(X , A) −→ 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. 0 • q∗ is an isomorphism since (X − A, V − A) → (X /A − A/A, V /A − A/A) is a homeo −1 0 −1 • q∗ = γ ηq∗β α Homology theory

Lecture 10 - 1/24/2011 Axiom 4: Additivity Corollary 96 (Long exact sequence for a good pair) Computations Lecture 12 - If (X , A) is a good pair then 1/26/2011 Naturality of the connecting ˜ ˜ ˜ ˜ homomorphism · · · −→Hn(A) −→Hn(X ) −→Hn(X /A) −→Hn−1(A) −→· · · Lecture 13 - 1/27/2011 Cellular is exact. Homology Proof. By Proposition 61 (Week 2) we have a long exact sequence

· · · −→Hñ(A) −→Hñ(X ) −→ Hn(X , A) −→Hñ−1(A) −→· · · ∼ By Theorem 95 above Hn(X , A) =H˜(X /A) Homology theory Proposition 97 (Homology groups of spheres)

Lecture 10 - 1/24/2011 ˜ n Z, k = n Axiom 4: Hk (S ) = Additivity 0, k 6= n Computations Lecture 12 - 1/26/2011 Naturality of the Proof. connecting homomorphism −1 Lecture 13 - • True for S = . 1/27/2011 ∅ Cellular 0 Homology • True for S = {−1, 1}. • Suppose true for S n−1 • Consider good pair (Dn, S n−1)

n n n−1 n−1 n H˜k (D ) −→H˜k (D /S ) −→H˜k−1(S ) −→H˜k−1(D )

• Dn/S n−1 =∼ S n • Z, k − 1 = n − 1 H˜ (S n) =∼H˜ (S n−1) = k k−1 0, k − 1 6= n − 1 Homology theory

Lecture 10 - 1/24/2011 Axiom 4: Exercise 98 (HW4 - Problem 2) Additivity Computations Lecture 12 - Show that for X nonempty 1/26/2011 Naturality of the ∼ connecting H˜n(X ) =H˜n+1(SX ) homomorphism Lecture 13 - 1/27/2011 Cellular Homology Exercise 99 (HW4 - Problem 3)

Let X be the n-point set. Compute H˜k (X )

Exercise 100 (HW4 - Problem 4) n Wn 1 Let Y = R = i=1 S be the rose with n-petals (see Hatcher pg. 10). Compute H˜k (X ) Homology theory

Lecture 10 - 1/24/2011 Axiom 4: Additivity Deﬁnition 101 (Local homology groups) Computations Lecture 12 - x ∈ X the local homology of X at x is 1/26/2011 Naturality of the connecting homomorphism Hn(X , X − x) Lecture 13 - 1/27/2011 Cellular Homology • 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 Example 102 Lecture 10 - 1/24/2011 Rn is contractible so the long exact sequence of the pair (Rn, Rn − 0) Axiom 4: Additivity yeilds Computations Lecture 12 - n n n n n 1/26/2011 H˜k (R ) −→ Hk (R , R − 0) −→H˜k−1(R − 0) −→H˜k−1(R ) Naturality of the connecting homomorphism Rn − 0 ' S n−1 so the local homology of Rn at 0 is Lecture 13 - 1/27/2011 Cellular Z, k = n Homology H (Rn, Rn − 0) =∼H˜ (Rn − 0) =∼H˜ (S n−1) = k k−1 k−1 0, k 6= n

So Rn Rm if n 6= m.

Deﬁnition 103 An n-dimensional homology manifold is a space X such that for all x ∈ X Z, k = n H (X , X − x) =∼ k 0, k 6= n Homology theory Naturality of the connecting Lecture 10 - 1/24/2011 Axiom 4: homomorphism Additivity Computations • Suppose we have a commutative diagram of chain complexes Lecture 12 - 1/26/2011 and chain maps with exact rows. Naturality of the connecting homomorphism 0 / C / D / E / 0 Lecture 13 - 1/27/2011 Cellular γ δ ε Homology 0 / C 0 / D0 / E 0 / 0

• The Snake Lemma gives

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

γ γ ε∗ ∗ δ∗ ε∗ ∗ 0 0 0 0 0 ··· / Hn+1(E ) / Hn(C ) / Hn(D ) / Hn(E ) / Hn−1(C ) / ···

• Fact that Hn : Chain → Ab is a functor shows some squares commute. Homology theory

Lecture 10 - Theorem 104 (Naturality of connecting homomorphism) 1/24/2011 Axiom 4: Additivity Given a commutative diagram of chain complexes and chain maps Computations with exact rows. Lecture 12 - 1/26/2011 Naturality of the connecting 0 / C / D / E / 0 homomorphism Lecture 13 - γ δ ε 1/27/2011 Cellular Homology 0 / C 0 / D0 / E 0 / 0

The following diagram commutes:

∂ Hn+1(E) / Hn(C)

ε∗ γ∗ 0 ∂ 0 Hn+1(E ) / Hn(C ) Homology theory

Lecture 10 - 1/24/2011 Proof of Theorem 104. Axiom 4: Additivity Computations in qn Lecture 12 - 0 −−−−→ Cn −−−−→ Dn −−−−→ En −−−−→ 0 1/26/2011    Naturality of the C D E connecting ∂n ∂n ∂n homomorphism y y y

Lecture 13 - in−1 qn−1 1/27/2011 0 −−−−→ Cn−1 −−−−→ Dn−1 −−−−→ En−1 −−−−→ 0 Cellular Homology • We must verify that ∂ε∗ = γ∗∂. D • By deﬁnition ∂[en] = [cn−1] where in−1(cn−1) = ∂n dn for some dn ∈ Dn with qn(dn) = en 0 • Note that εen = εqn(dn) = qnδ(dn) D0 D 0 • And ∂n δ(dn) = δ∂n (dn) = δin−1(cn−1) = in−1γ(cn−1) • so ∂ε∗[en] = ∂[εen] = [γ(cn−1)] = γ∗[cn−1] = γ∗∂[en] Homology theory

Lecture 10 - 1/24/2011 Note on category theory: Axiom 4: Additivity Computations Deﬁnition 105 (Natural transformation) Lecture 12 - 1/26/2011 Naturality of the • Given two functors F , G : C → D connecting homomorphism • A natural transformation η from F to G assigns a morphism Lecture 13 - 1/27/2011 ηC : F (C) → G(C) for each C ∈ Ob(C) such that the following Cellular Homology diagram commute for all morphisms f ∈ MorC(C1, C2)

F (f ) F (C1) / F (C2)

η η C1 C2 G(f ) G(C1) / G(C2) Homology theory How does our naturality ﬁt into this picture? Lecture 10 - 1/24/2011 • Let C = SESChain be the category of short exact sequences of Axiom 4: Additivity chain complexes C ,→ D E with morphisms given by Computations commuting diagrams of chain maps from one short exact Lecture 12 - 1/26/2011 sequence to another. Naturality of the connecting homomorphism • Let D = Ab Lecture 13 - • Let F (C ,→ D E) = H (E) 1/27/2011 n Cellular Homology • Let G(C ,→ D E) = Hn−1(C) • Let η = ∂ : H (E) → H (C) (C,→DE) n n−1 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 Naturality of the connecting homomorphism Exercise 106 (HW4 - Problem 5) Lecture 13 - 1/27/2011 Show that if P is the set of path components of X then H0(X ) is Cellular Homology isomorphic to the free abelian group on P and that if X is nonempty then H˜0(X ) is the free abelian group on P − {c0} where c0 is a path component of X . Homology theory Cellular Homology Lecture 10 - 1/24/2011 Axiom 4: Additivity Deﬁnition 107 (Chain complex for cellular homology) Computations −1 Lecture 12 - Let X be a CW complex (see Hatcher pg. 5). Let X = and 1/26/2011 ∅ Naturality of the connecting n n−1 homomorphism H (X , X ), n ≥ 0 C CW(X ) = n Lecture 13 - n 0, n < 0 1/27/2011 Cellular Homology Let CW CW CW ∂ : Cn (X ) → Cn−1(X ) be the connecting homomorphism

n n−1 ∂ n−1 n−2 Hn(X , X ) −→ Hn−1(X , X )

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

Proposition 108 C CW(X ) is a chain complex Homology theory Proof of Proposition 108.

2 Lecture 10 - • We must verify that (∂CW) = 0. 1/24/2011 Axiom 4: Additivity • We have the following commutative diagram of chain maps Computations Lecture 12 - n n+1 n+1 n 1/26/2011 0 / C(X ) / C(X ) / C(X , X ) / 0 Naturality of the connecting homomorphism j] 1 Lecture 13 - 1/27/2011 n n−1 n+1 n−1 n+1 n Cellular 0 / C(X , X ) / C(X , X ) / C(X , X ) / 0 Homology • Naturality of the connecting homomorphism says the following square from the long exact sequence of homology commutes:

n+1 n ∂ n Hn+1(X , X ) / Hn(X )

1 j∗

CW n+1 n ∂ n n−1 Hn+1(X , X ) / Hn(X , X )

CW • So ∂ = j∗∂ Homology theory Proof of Proposition 108 (continued).

Lecture 10 - 1/24/2011 • We have commutative Axiom 4: Additivity Computations n Hn(X ) Lecture 12 - 6 1/26/2011 mm ∂ mmm Naturality of the mmm j∗ connecting mmm homomorphism mm CW CW n+1 n ∂ n n−1 ∂ n−1 n−2 Lecture 13 - Hn+1(X , X ) / Hn(X , X ) / Hn−1(X , X ) 1/27/2011 l6 Cellular j∗ lll Homology ∂ lll lll lll n−1 Hn−1(X )

• Vertical composition

n j∗ n n−1 ∂ n−1 Hn(X ) −→ Hn(X , X ) −→ Hn−1(X )

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 - Deﬁnition 109 (Cellular Homology) 1/24/2011 Axiom 4: Additivity If X is a CW complex then the nth cellular homology group of X is Computations

Lecture 12 - CW CW 1/26/2011 Hn (X ) = Hn(C (X )) Naturality of the connecting homomorphism Lecture 13 - CW 1/27/2011 For a CW complex X how do Hn (X ) and Hn(X ) relate? Cellular Homology Theorem 110 (Cellular and sing. simp. homology agree) If X is a CW complex then

CW ∼ Hn (X ) = Hn(X )

Advantages of H CW 1 If X is ﬁnite then C CW(X ) is ﬁnitely generated 2 boundary maps in C CW(X ) can be easily understood. Homology theory

Lecture 10 - The proof of Theorem 110 will use the following properties of 1/24/2011 Axiom 4: homology for CW complexes. Additivity Computations Lecture 12 - Lemma 111 (Homology of CW complexes) 1/26/2011 Naturality of the n connecting Let X is a CW complex. Let Γ be the set of n-cells of X homomorphism 1 Lecture 13 - n 1/27/2011 n n−1 ∼ Z[Γ ] k = n Cellular Hk (X , X ) = Homology 0, k 6= n

2 If k > n n Hk (X ) = 0 n 3 If k < n and i : X → X is the inclusion map then

n ∼= Hk (X ) −→ Hk (X ) i∗ Homology theory Proof of Lemma 111 Claim 1.

Lecture 10 - • We have a commuting quotient maps of good pairs 1/24/2011 Axiom 4: Additivity p Computations Lecture 12 - 1/26/2011 q ( ` n ` n−1 r n n−1 n n−1 Naturality of the n D , n S / (X , X ) / (X /X , {∗}) connecting α∈Γ α∈Γ homomorphism Lecture 13 - 1/27/2011 • By Theorem 95 we get that Cellular Homology ! a n a n−1 n n−1 p∗ : Hn D , S →H˜n(X /X ) α∈Γn α∈Γn

and n n−1 n n−1 q∗ : Hn(X , X ) →H˜n(X /X ) are isomorphisms. • So we get an isomorphism ! a n a n−1 n n−1 r∗ : Hn D , S → Hn(X , X ) α∈Γn α∈Γn Homology theory

Lecture 10 - Proof of Lemma 111 Claim 1 (continued). 1/24/2011 Axiom 4: Additivity Computations • Applying the additivity axiom Lecture 12 - 1/26/2011 ! Naturality of the n n−1 ∼ a n a n−1 connecting Hn(X , X ) = Hn D , S homomorphism α∈Γn α∈Γn Lecture 13 - 1/27/2011 ∼ M n n−1 Cellular = Hn(D , S ) Homology α∈Γn ∼ M n = H˜n(S ) α∈Γn M =∼ Z α∈Γn =∼ Z[Γn]