I have been learning about cohomology in algebriac topology from Hatcher's book. I am unsure about some proofs on the cohomology of cellular complexes. Please check, if the arguments are correct.
- $H^k(X^n,X^{n-1};G)=0$ for $k\neq n$
Proof: We know that the homologies satisfy $H_k(X^n,X^{n-1})=0$ for $k\neq n$. From the universal coefficient theorem we know that $H^k(X^n,X^{n-1};G) \cong \mathrm{Ext}(H_{k-1}(X^n,X^{n-1}), G)\oplus \mathrm{Hom}(H_{k}(X^n,X^{n-1}), G) \cong \mathrm{Ext}(0, G)\oplus \mathrm{Hom}(0, G)\cong 0.$
I now realise there is a problem in the case $k=n+1$ since $\mathrm{Ext}$ won't be $0$ automatically.
How would one prove $H^k(X,X^{n-1};G)=0$ for $k\leq n+1$?
Actually, as Hatcher points out when he introduces cellular homology, $H_n(X^n, X^{n-1})$ is free abelian. Therefore $\mathrm{Ext}(H_n(X^n, X^{n-1}), G) = 0$ as well, and the Ext term vanishes in all cases.