I don't really know how to use excision theorem to calculate homology groups at all and need to see som examples of applying excision theorem, for example how should I use excision to compute the homology groups of $ S^n$ ? For $S^2$ I removed a point of Subset of $S^2$ homeomorphic to $D^2$ but I got H2 is zero which is not true.
Compute homology groups of $S^n$ using excision theorem
algebraic-topologyhomology-cohomology
Best Answer
HINT: Suppose you use the pair $(S^2,D^2)$, where $D^2$ is the closed lower hemisphere, and you excise the point $p$, the south pole. Then Excision tells you that $H_*(S^2-\{p\},D^2-\{p\}) \cong H_*(S^2,D^2) \cong \tilde H_*(S^2)$. But the pair $(S^2-\{p\},D^2-\{p\})$ is homotopy equivalent to the pair $(D^2,S^1)$. Can you finish?