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I'm confused about Hatcher's Theorem 3.5 on page 203. The theorem states:

$H^n(X;G) \approx$ ker$d_n/$ Im$d_{n-1}$. Furthermore, the cellular cochain complex $\{H^{n}(X^n,X^{n-1};G),d_n\}$ is isomorphic to the dual of the cellular chain complex, obtained by applying Hom$(-,G)$.

I am particularly confused what it means for the cellular cochain complex to be isomorphich to the dual of the cellular chain complex.

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Well, there is the cohomology complex with maps $$\delta:H^i(X^{i},X^{i-1}) \to H^{i+1}(X^{i+1},X^i)$$ and there is also the complex with maps $$d^*:\mathrm{Hom}(H_i(X),G) \to \mathrm{Hom}(H_{i+1}(X),G)$$ where $d^*(f)=f \circ d$.

To show an isomorphism, you want to exhibit a family of maps $$f_i:H^i(X^{i},X^{i-1}) \to \mathrm{Hom}(H_i(X),G)$$ where $f_{i+1} \circ \delta=d^* f_{i}$ (so that it commutes with the maps.)

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  • $\begingroup$ I see, this makes sense. It's nice to have a proper definition in mathematics. $\endgroup$ – TuoTuo Aug 9 '18 at 1:40
  • $\begingroup$ no problem. I think the problem is that this is quite deep into the book, and it this point Hatcher assumes the reader can work out the relevant statements, so he is pretty loose here. Glad to help! $\endgroup$ – Andres Mejia Aug 10 '18 at 1:38
  • $\begingroup$ I don't think this is a good choice on his part. I can make a guess as what the definition is, but having to guess definitions is not very satisfactory. $\endgroup$ – TuoTuo Aug 12 '18 at 4:29

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