453 First, a lemma: Let $G''_*$ and $G'_*$ be defined similarly to $K''_*$ and $K'_*$, except with |
453 First, a lemma: Let $G''_*$ and $G'_*$ be defined similarly to $K''_*$ and $K'_*$, except with |
454 $S^1$ replaced by some neighborhood $N$ of $* \in S^1$. |
454 $S^1$ replaced by some neighborhood $N$ of $* \in S^1$. |
455 ($G''_*$ and $G'_*$ depend on $N$, but that is not reflected in the notation.) |
455 ($G''_*$ and $G'_*$ depend on $N$, but that is not reflected in the notation.) |
456 Then $G''_*$ and $G'_*$ are both contractible |
456 Then $G''_*$ and $G'_*$ are both contractible |
457 and the inclusion $G''_* \sub G'_*$ is a homotopy equivalence. |
457 and the inclusion $G''_* \sub G'_*$ is a homotopy equivalence. |
458 For $G'_*$ the proof is the same as in (\ref{bcontract}), except that the splitting |
458 For $G'_*$ the proof is the same as in Lemma \ref{bcontract}, except that the splitting |
459 $G'_0 \to H_0(G'_*)$ concentrates the point labels at two points to the right and left of $*$. |
459 $G'_0 \to H_0(G'_*)$ concentrates the point labels at two points to the right and left of $*$. |
460 For $G''_*$ we note that any cycle is supported away from $*$. |
460 For $G''_*$ we note that any cycle is supported away from $*$. |
461 Thus any cycle lies in the image of the normal blob complex of a disjoint union |
461 Thus any cycle lies in the image of the normal blob complex of a disjoint union |
462 of two intervals, which is contractible by (\ref{bcontract}) and (\ref{disj-union-contract}). |
462 of two intervals, which is contractible by Lemma \ref{bcontract} and Corollary \ref{disj-union-contract}. |
463 Finally, it is easy to see that the inclusion |
463 Finally, it is easy to see that the inclusion |
464 $G''_* \to G'_*$ induces an isomorphism on $H_0$. |
464 $G''_* \to G'_*$ induces an isomorphism on $H_0$. |
465 |
465 |
466 Next we construct a degree 1 map (homotopy) $h: K'_* \to K'_*$ such that |
466 Next we construct a degree 1 map (homotopy) $h: K'_* \to K'_*$ such that |
467 for all $x \in K'_*$ we have |
467 for all $x \in K'_*$ we have |