- #1

- 4,807

- 32

So we let V be one of the components of S^n\S and let x be in V. Then Bredon writes that excision implies the isomorphism

[tex]H_{i+1}(V,V\backslash \{x\})\cong H_{i+1}(\mathbb{D}^n,\mathbb{D}^n\backslash\{0\})[/tex]

(where

**D**^n is the closed n-disk).

It's like he said "V u S is homeomorphic to

**D**^n (with x being sent to 0), so excising the boundary [tex]\partial\mathbb{D}^n\cong S[/tex] gives the above isomorphism".

But for n>2, it is not true in general that V u S is homeomorphic to

**D**^n as the (counter-) example of the Alexander horned disk shows...

So what is it that he's doing in that step?P.S. Don't bother with Google Book, page 234 is missing from the preview. :(