Algebraic Topology [Lecture notes] by Christoph Schweigert

By Christoph Schweigert

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6 (Relative Mayer-Vietoris sequence). If A, B ⊂ X are open in A ∪ B, then the following sequence is exact: ... 9 δ G Hn (X, A ∩ B) G Hn (X, A) ⊕ Hn (X, B) G Hn (X, A ∪ B) δ G ... Reduced homology and suspension For any path-connected space, the zeroth homology is isomorphic to the integers, so this copy of Z is superfluous information and we want to get rid of it. Let pt denote the one-point topological space. Then for any space X there is a unique continuous map ε : X → pt. 1 We define Hn (X) := ker(Hn (ε) : Hn (X) → Hn (pt)) and call it the reduced nth homology group of the space X.

We can extend such results to the full relative case. Let A ⊂ X be a closed subspace and assume that A is a deformation retract of an open neighbourhood A ⊂ U . Let π : X → X/A be the canonical projection and b = {A} ∈ X/A the image of A. Then X/A is well-pointed with respect to the point b ∈ X/A. 8. In the situation above Hn (X, A) ∼ = Hn (X/A), 0 n. Proof. The canonical projection π : X → X/A induces a homeomorphism of pairs (X \ A, U \ A) ∼ = (X/A \ {b}, π(U ) \ {b}). Consider the following diagram: Hn (X, A) ∼ = G ∼ = Hn (X, U ) o Hn (X \ A, U \ A) ∼ = Hn (π) Hn (π)  Hn (X/A, b) ∼ = G Hn (X/A, π(U )) o ∼ =  Hn (X/A \ {b}, π(U ) \ {b}) The upper and lower left arrows are isomorphisms because A is a deformation retract of U , the isomorphism in the upper right is a consequence of excision, because A = Aˉ ⊂ U , cf.

We have to control what is going on in small degrees and dimensions. 2. We know from the Hurewicz isomorphism that H1 (Sm ) is trivial for m > 1, cf. 9. Here, we show this directly via the Mayer-Vietoris sequence: ... → 0 ∼ = H1 (X + ) ⊕ H1 (X − ) → H1 (Sm ) δ →Z∼ = H0 (X + ∩ X − ) → H0 (X + ) ⊕ H0 (X − ) ∼ = Z ⊕ Z. We have to understand the map in the second line. Let 1 be a base point of X + ∩ X − . Then the map on H0 is [1] → ([1], [1]). This map is injective and therefore the connecting homomorphism δ : H1 (Sm ) → H0 (X + ∩ X − ) is zero.

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