Homology Q: Hatcher Top Page 125 - Isomorphism Explanation

[quasar987]

In Hatcher top of page 125, I do not understand where the first isomorphism comes from. He says it comes from the exact sequence of the pair, using the fact that CA is contractible, but what is the precise line of reasoning here?

Hatcher's book: http://www.math.cornell.edu/~hatcher/AT/AT.pdf

Thanks!

 

[yyat]

Since CA is contractible \tilde{H}_n(CA)=0 for all n (there is a tilde over the H, but it's hard to see). Thus, if you write down the exact sequence of the pair (X\cup CA,CA) every third group in the sequence vanishes, which makes the two other groups isomorphic.

 

[quasar987]

I see, thanks!