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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...

Vasting
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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 $n=d$.

Just a user
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    My question had poor phrasing (sorry about that); I updated the wording. I am wondering about the relative chain groups, not the homology groups. – Vasting Oct 23 '21 at 20:20
  • Ah, okay, I think I'm having trouble with the intuition. Geometrically, what is the nontrivial cycle in $H_2(\mathbb{R}^2,\mathbb{R}^2\setminus{0})$? – Vasting Oct 23 '21 at 20:36
  • When $n=2$, it's the (class of) closed unit disk: Its boundary lies in $\mathbb R^2\setminus{0}$ which is $0$ in the relative chain group, and it's not a boundary of any higher dimensional singular chain. – Just a user Oct 23 '21 at 20:39
  • Okay, thanks! I see that now. So the idea then is that (using the same example) $C_2(\text{point},0)$ has only one element, while $C_2(\mathbb{R}^2,\mathbb{R}^2\setminus{0})$ has many, many elements, since $C_2(\mathbb{R}^2)$ is the free abelian group on maps from $2$-simplices. – Vasting Oct 23 '21 at 20:43
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    Yes. I assume you mean $C_2(pt, \emptyset)$. $C_2(\mathbb R^2, \mathbb R^2\setminus{0})$ has many nonzero simplices, as long as the image contains ${0}$, so it's not trivialized in the relative chain group. – Just a user Oct 23 '21 at 20:51