The excision theorem states that under suitable conditions, given subspaces $Z\subset A\subset X$, then we have isomorophisms of homology groups $H_n(X-Z,A-Z)\rightarrow H_n(X,A)$ for all $n$.
Intuitively, it seems to me that since relative chain groups $C_n(X,A)$ involve throwing out of all of the chains in $A$, then we should have $C_n(X-Z,A-Z)$ is isomorphic to $C_n(X,A)$, since at the end of the day, we are throwing out the same stuff, just in a different order.
But it seems like this is not the case (that is, the relative chain groups being isomorphic); is there a counterexample? It seems like the border groups play a role here, but I'm not exactly sure how that works…
Best Answer
Let $X=\mathbb R^d, d>1$, and $A=Z=\mathbb R^d\setminus\{0\}$, then $H_n(X,A)$ is the local homology which is isomorphic to $\mathbb Z$ for $n=d$, while $H_n(X-Z, A-Z)\simeq H_n(\text{point})$ is trivial for $n=d$.