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The long exact sequence of a pair and excision 10 November Definition 1. A chain complex (C∗, ∂) is a sequence of abelian groups Cp indexed by the integers, together with maps ∂ Cp − → Cp−1 satisfying ∂∂ = 0. The homology of (C∗, ∂) is the sequence of abelian groups Hp(C∗, ∂) = Ker ∂ : Cp → Cp−1 . Img ∂ : Cp+1 → Cp If C∗ and D∗ are chain complexes, then a map f ∗ : C∗ − →D∗ is a sequence of group homomorphism fp : Cp → Dp, compatible with the boundary maps ∂. A sequence of maps of chain complexes f g A∗ − → B∗ − → C∗ is exact if, for each p the sequence f g Ap − → Bp − → Cp is exact. Let f g 0− →A∗ − → B∗ − → C∗ − →0 be a short exact sequence of chain complexes. We define a map δ HpC − → Hp−1A as follows. Suppose c ∈ Cp with ∂c = 0. Choose a lift b ∈ Bp. Then g∂b = ∂gb = ∂c = 0, so there is a unique element a ∈ Ap−1 such that f a = ∂b. Note that f ∂a = f ∂∂b = 0, so ∂a = 0. We set δ([c]) = [a]. Proposition 2. The sequence δ Hp f Hp g δ ... − → HpA −−−→ HpB −−→ HpC − → ... is exact. Proof. Easy; see Bredon or better, do it yourself. Corollary 3. If (X, A) is a pair of spaces, then there is a long exact sequence in homology δ hbd ... − → HpA − →HpX − →Hp(X, A) −−→ . . . . Moreover if f : (X, A) → (Y, B) is a pair, the diagram Hp(X, A) −→ Hp−1A y y Hp(Y, B) −→ Hp−1B commutes. A few drawings of the map δ in the case of relative homology suggest that, while Hp(X, A) is not equal to Hp(X)/Hp(A), it is closely related to Hp(X/A); in particular, Hp(X, A) doesn’t much care about the topology of X inside of A. The way this is expressed in the axioms is The Excision axiom Given a pair (X, A) and an open set U ⊂ X such that Ū ⊂ int(A), the inclusion (X − U, A − U ) ,→ (X, A) induces an isomorphism ∼ H (X, A). H∗(X − U, A − U ) = ∗ We will prove this after Thanksgiving. Lemma 4 (Barratt-Whitehead). Let f h g . . . −→ Cp+1 −→ Ap −→ Bp −→ Cp −→ ∼ γ = y α y β y ∼ γ = y f0 g0 h0 0 0 0 . . . −→ Cp+1 −→ Ap −→ Bp −→ Cp0 −→ be a commutative diagram in which the rows are long exact, and the map γ is always an isomorphism. Then the sequence f 0 −β hγ −1 g 0 0 0 − →Ap −−→ Ap ⊕ Bp −−−→ Bp −−−−→ Ap−1 − → α,f is exact. Proof. This is an exercise, due Friday December 1. Now let X be a space, and suppose that X =U ∪V with U, V open. We have a long exact sequence · · · → Hp(U ) → Hp(X) → Hp(X, U ) → . . . We also have a long exact sequence · · · → Hp(U ∩ V ) → Hp(V ) → Hp(V, U ∩ V ). The inclusion (V, U ∩ V ) = (X − U ∩ V c, U − U ∩ V c) ,→ (X, U ) is an excision, and so induces an isomorphism ∼ H (X, U ). Hp(V, U ∩ V ) = p We therefore have a ladder Hp(U ∩ V ) −→ Hp(V ) −→ Hp(V, U ∩ V ) y HpU −→ y HpX −→ ∼ = y Hp(X, U ). The Barratt-Whitehead lemma gives the MayerVietoris sequence. Summary: The Eilenberg-Steenrod axioms. An (ordinary) homology theory is a functor H from the category of pairs of topological spaces to the category of graded abelian groups, together with maps ∂ : Hp(X, A) → Hp−1A, natural in the pair (X, A), with the following properties. 1. (homotopy) If f ' g : (X, A) → (Y, B), then H∗f = H∗g. 2. If (X, A) is a pair, and i and j are the inclusions i A− →X j (X, ∅) − → (X, A) then the sequence i j∗ ∂ ∗ HpX −→ Hp(X, A) − → Hp−1A − → − →HpA −→ is exact. 3. Excision 4. (Dimension) For the one-point space P , H0P = G HiP = 0 i 6= 0 5. (Milnor) The natural map is an isomorphism Hp( a i a ∼ Xi) = HpXi. i
