Algebraic Topology [Lecture notes] by Christoph Schweigert

By Christoph Schweigert

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18 implies that any subcomplex of a CW complex has the homotopy extension property. Indeed, two maps g: X→Y h : A × [0, 1] → Y and such that h|A×{0} = g can be combined to a single map g˜ : X × {0} ∪ A × [0, 1] → Y ; a retraction r : X × [0, 1] → X × {0} ∪ A × [0, 1] provides the homotopy extension ˜g ◦ r : X ×I →Y. 20. 1. For any subcomplex A ⊂ X, there is an open neighborhood U of A in X together with a strong deformation retract to A. In particular, for each skeleton X n there is an open neighborhood U in X (and as well in X n+1 ) of X n such that X n is a strong deformation retract of U .

3. For CW complexes X, Y we have 1. If the n-skeleta X n and Y n are homeomorphic, then Hq (X) ∼ = Hq (Y ), for all q < n. 2. If X has no q-cells, then Hq (X) ∼ = 0. 3. In particular, if q exceeds the dimension of X, then Hq (X) ∼ = 0. Proof. 55 1. 2 which asserts that Hn (X n+1 ) ∼ = Hn (X). 2. By assumption in 2. 2 onto Hq (X). Hence 2. is reduced to the statement in 3. 3. We show that Hn (X r ) ∼ = 0 for n > r. Consider the long exact sequence o relative homology → Hn+1 (X i , X i−1 ) → Hn (X i−1 ) → Hn (X i ) → Hn (X i , X i−1 ) → .

Consider the two right-most non-trivial columns in this diagram. Each gives a long exact sequence in homology and we focus on five terms: Hn (S∗U (A ∪ B)) Hn (X) G GH U n (S∗ (X)/S∗ (A Hn (ϕ) ∪ B)) δ Hn (ψ)  Hn (A ∪ B) G Hn (X) G G Hn−1 (S∗U (A ∪ B)) G Hn−1 (X) G Hn−1 (X) Hn−1 (ϕ)  Hn (X, A ∪ B) δ G  Hn−1 (A ∪ B) Then by the five-lemma, as Hn (ϕ) and Hn−1 (ϕ) are isomorphisms, so is Hn (ψ). 6 (Relative Mayer-Vietoris sequence). If A, B ⊂ X are open in A ∪ B, then the following sequence is exact: ...

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