By Christoph Schweigert
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Extra info for Algebraic Topology [Lecture notes]
Then W 2 /W 1 ∼ = 6 2 i=1 S . 12 Cellular homology In the following, X will always be a CW complex. 1. For the relative homology of the skeleta, we have Hq (X n , X n−1 ) = 0 for all q = n 1. Proof. Using the identification of relative homology and reduced homology of the quotient gives ˜ q (X n /X n−1 ) ∼ Hq (X n , X n−1 ) ∼ =H = ˜ q (Sn ). 8. 7. 2. Consider the inclusion in : X n → X of the n-skeleton X n into X. 1. The induced map Hn (in ) : Hn (X n ) → Hn (X) is surjective. 2. On the (n + 1)-skeleton we get an isomorphism Hn (in+1 ) : Hn (X n+1 ) ∼ = Hn (X).
The cone over an n-simplex is an (n+1)-simplex. The cone over Sn is a closed (n+1)-ball. 2. Note that for any topological space X, the cone CX is contractible to its apex. Thus ˜ n (CX, CA) = 0 ˜ n (CX) = 0 for all n 0. Similarly, for A ⊂ X, we have CA ⊂ CX and H H for all n 0. 3. The suspension of Sn is ΣSn ∼ = Sn+1 . 4. We have natural embeddings X → CX and CX → ΣX. We can see the suspension as two cones, glued together at their bases. 11 (Suspension isomorphism). Let A ⊂ X be a closed subspace and assume that A is a deformation retract of an open neighbourhood A ⊂ U .
Similarly, for A ⊂ X, we have CA ⊂ CX and H H for all n 0. 3. The suspension of Sn is ΣSn ∼ = Sn+1 . 4. We have natural embeddings X → CX and CX → ΣX. We can see the suspension as two cones, glued together at their bases. 11 (Suspension isomorphism). Let A ⊂ X be a closed subspace and assume that A is a deformation retract of an open neighbourhood A ⊂ U . Then ˜ n−1 (X, A), Hn (ΣX, ΣA) ∼ =H for all n > 0. Proof. 1. We first note two equivalences: X ∪ CA/CA A ⊂ CA: X/A, where the cone CA is attached to X by identifying A ⊂ X and ✁✄❈❆ ✁✄ ❈❆ ✁✄ ❈❆ CA ✁ ✄ ❈ ❆ ✁ ✄ ❈ ❆ ✁ ✄ ❈ ❆ A and CX/(CA ∪ X) ΣX/ΣA: r r r✁❆❅ r r✁ r ❆❅ r r r r✁ r rr❆r ❅ CX r✁r rr rr rr ❆rr ❅ r✁r rr rr rr rr r❆r r ❅ ✁r r r r r r r❆ ❅ X r r r✁❆❅r r✁ r❆❅ r✁r rr ❆rr ❅ ΣX r✁r rr rr ❆rr ❅ r✁r rr rr rr ❆rr ❅ ✁r r r r r ❆r ❅ ❅ ❆ rr rr rr rr r✁r ❅ ❆ rr rr rr r✁r ❅ ❆ rr rr r✁r ❅ ❆ r ✁r ❅❆r r✁r ❅ ❆✁ r CA ΣA 40 2.
Algebraic Topology [Lecture notes] by Christoph Schweigert