Mayer-Vietoris long exact sequence

Let C denote the cone over A, which has been added to X, and consider the reduced version of the long exact sequence of the pair (Q, C)  

Our last application concerns the suspension construction, which was defined in Example 2.32(1). Let X be an arbitrary CW complex. Its suspension is suspX = X S, and there is a straightforward way to extend the cellular structure of X to the entire suspX. Namely, we add two new vertices, one for each apex, and each cell a of X gives rise to two new cells cr+ and cr, one inside each cone, such that dim = dim (t = dim a + 1. 

Let C denote one of the cones over X this is a half of suspX. By Proposition 5.15 we have iL (suspX) = H C,X). On the other hand, the reduced version of the long exact sequence for the pair C, X) coupled with the fact that Hn C) = 0 gives the equality Hn X) = Hn+i C,X) for all n. This implies (5.9).  

Another standard long exact sequence describes what happens when we glue together two CW complexes. Assume that X is a CW complex that we have represented as a union of two of its subcomplexes X = A V B. We have four cellular inclusion maps 

Theorem 5.17. (Mayer Vietoris long exact sequence) 

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Using the reduced version of the Mayer-Vietoris long exact sequence we once again confirm the equahties (5.9). 

Proposition 5.20. Assume that the CW complexes X and X are represented as unions of CW suhcornplexes X = AU B and X = AU B. Assume furthermore that we have a cellular map ip X X satisfying the additional conditions

chain complex map between the corresponding Mayer-Vietoris homology long exact sequences. 

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