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Here are two fragments out of Hatcher's book (see http://pi.math.cornell.edu/~hatcher/AT/AT.pdf for the full book).

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Questions: (1) Does this mean we get a long exact sequence which has the form $$\dots \to\tilde{H}_n(A) \to \tilde{H}_n(X) \to H_n(X,A) \to \tilde{H}_{n-1}(A) \to \tilde{H}_{n-1} \to \dots$$

Or do I need to modify the $H_n(X,A)$ by $\tilde{H}_n(X,A)$ or something like that?

(2) Can someone explain the second paragraph in the picture in some detail? I was not able to understand what Hatcher was trying to explain.

Thanks in advance for any help.

  • The second paragraph explains what happens for reduced homology. In particular, it answers your first question. – Michael Albanese Jun 24 '20 at 21:37
  • @MichaelAlbanese So is the answer to the first question yes? –  Jun 24 '20 at 22:03
  • The question https://math.stackexchange.com/q/3297906 is not the same as the present one, but its answer contains all necessary details. Question (1): Yes. – Paul Frost Jun 24 '20 at 22:45

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