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 fAαg, 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. 8α fα = f ◦ γα Definition 91 (Direct sum in Chain) The direct sum for chain complexes fC αg 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 · · · −!H~n(A) −!H~n(X ) −! Hn(X ; A) −!H~n−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; 1g. • 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 2 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 2 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 " 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 "∗ ∗ δ∗ "∗ ∗ 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 " 1/27/2011 Cellular Homology 0 / C 0 / D0 / E 0 / 0 The following diagram commutes: @ Hn+1(E) / Hn(C) "∗ 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 @"∗ = γ∗@. D • By definition @[en] = [cn−1] where in−1(cn−1) = @n dn for some dn 2 Dn with qn(dn) = en 0 • Note that "en = "qn(dn) = qnδ(dn) D0 D 0 • And @n δ(dn) = δ@n (dn) = δin−1(cn−1) = in−1γ(cn−1) • so @"∗[en] = @["en] = [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 2 Ob(C) such that the following Cellular Homology diagram commute for all morphisms f 2 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 − fc0g where c0 is a path component of X .