By Christoph Schweigert

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**Example text**

The map d is a boundary operator: the composition d2 is δ◦ ρ ◦ δ ◦ ◦ δ, but δ : Hn−1 (X n−1 ) → Hn−1 (X n−1 , X n−2 ) → Hn−2 (X n−2 ) is a composition in a long exact sequence and thus vanishes. 5 Let X be a CW complex. The cellular chain complex C∗ (X) consists of the free abelian groups Cn (X) := Hn (X n , X n−1 ) with boundary operator d : Hn (X n , X n−1 ) where δ GH n−1 (X n−1 ) GH n−1 (X n−1 , X n−2 ) is the map induced by the projection map Sn−1 (X n−1 ) → Sn−1 (X n−1 , X n−2 ). 6 (Comparison of cellular and singular homology).

Thus, ∂B k (e) − ∂e = ∂ ψ˜n−1 (c) c = ∂e = ∂(B k (e) − ψ˜n−1 (e)) is a boundary in SnU (X) as well. We remark that this isomorphism actually comes from a homotopy equivalence of chain complexes. 14 (Excision). ˉ ⊂ A. ˚ Then the inclusion i : (X \ W, A \ W ) → (X, A) induces Let W ⊂ A ⊂ X such that W an isomorphism Hn (i) : Hn (X \ W, A \ W ) ∼ = Hn (X, A) for all n 0. Proof. • We first prove that Hn (i) is surjective. e. ∂c ∈ Sn−1 (A). Consider the open covering ˚X\W ˉ } =: {U, V } of X. Now subdivide and find k such that c := B k c is a U = {A, U chain in Sn (X).

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 . Then ˜ n−1 (X, A), Hn (ΣX, ΣA) ∼ =H for all n > 0.