By Morgan J.W., Lamberson P.J.

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Let X = S p ∨ S q . compute H∗ (X). 3 Cellular (CW) Homology Let ∅ = X −1 ⊂ X 0 ⊂ X 1 ⊂ · · · ⊂ X N = X be a finite CW complex (see appendix C). 1. ∂cw Proof. The cellular boundary map fits into a commutative diagram involving portions of the long exact sequences of the pairs (X n+1 , X n ) and (X n , X n−1 ). 43 0 Hn (X n+1 ) ∼ = Hn (X) · · · −−−−→ Hn (X n ) ∂n+1 Hn+1 (X n+1 , X n ) 0 Hn (X n ) i ∗ ∂ ∂ cw cw −−− −→ Hn (X n , X n−1 ) −−− −→ Hn−1 (X n−1 , X n−2 ) −−−−→ · · · j∗ ∂n Hn−1 (X n−1 ) Hn−1 (X n−1 ) 0 2 factors through the 0 If we trace through this diagram we see that the composition ∂cw map, and is thus 0.

Vn ) we let τ i be the face of ∆n with vertices v0 , v1 , . . , vi . Given a sequence of faces, ∆n = τ 0 ⊂ τ 1 ⊂ · · · ⊂ τ n , we obtain an ordered list of vertices by letting v0 = τ0 , v1 the vertex of τ1 not in τ0 and so on. Then given a permutation p ∈ Σn+1 , we have the sequence of faces corresponding to the ordered list of vertices (vp(0) , vp(1) , . . , vp(n) ). So we have the following three ways of thinking of the n-simplex: Σn+1 ⇔ {ordered lists of vertices} ⇔ {sequences of inclusions of faces} For a face τ i of an n-simplex define τˆi to be the image under τ i of (1/(i+1), .

Urβ(j) , Usβ(j) , . . , Usβ(i) , . . , Usβ(k) ) i (−1) = j=0 i=0 k (−1)j−1 φ(Urβ(0) , . . , Urβ(i) , . . , Urβ(j) , Usβ(j) , . . , Usβ(k) ) . + j=i+1 and, k (−1)j δφ(Urβ(0) , . . , Urβ(j) , Usβ(j) , . . , Usβ(k) ) (Hδ)(φ)(Vβ(0) , . . , Vβ(k) ) = j=0 j k (−1)i φ(Urβ(0) , . . , Urβ(i) , . . , Urβ(j) , Usβ(j) , . . , Usβ(k) ) (−1)j = i=0 j=0 k (−1)i+1 φ(Urβ(0) , . . , Urβ(j) , Usβ(j) , . . , Usβ(i) , . . , Usβ(k) ) . + i=j Thus, k (−1)j (−1)j φ(Urβ(0) , . . , Urβ(j−1) , Usβ(j) , .

### Algebraic topology by Morgan J.W., Lamberson P.J.

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