48 Note that if there is no splitting $s$, we can let $h_0 = 0$ and get a homotopy |
48 Note that if there is no splitting $s$, we can let $h_0 = 0$ and get a homotopy |
49 equivalence to the 2-step complex $U(B^n; c) \to \cC(B^n; c)$. |
49 equivalence to the 2-step complex $U(B^n; c) \to \cC(B^n; c)$. |
50 |
50 |
51 For fields based on $n$-categories, $H_0(\bc_*(B^n; c)) \cong \mor(c', c'')$, |
51 For fields based on $n$-categories, $H_0(\bc_*(B^n; c)) \cong \mor(c', c'')$, |
52 where $(c', c'')$ is some (any) splitting of $c$ into domain and range. |
52 where $(c', c'')$ is some (any) splitting of $c$ into domain and range. |
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53 |
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54 \begin{cor} \label{disj-union-contract} |
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55 If $X$ is a disjoint union of $n$-balls, then $\bc_*(X; c)$ is contractible. |
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56 \end{cor} |
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57 |
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58 \begin{proof} |
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59 This follows from \ref{disjunion} and \ref{bcontract}. |
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60 \end{proof} |
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61 |
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62 Define the {\it support} of a blob diagram to be the union of all the |
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63 blobs of the diagram. |
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64 Define the support of a linear combination of blob diagrams to be the union of the |
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65 supports of the constituent diagrams. |
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66 For future use we prove the following lemma. |
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67 |
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68 \begin{lemma} \label{support-shrink} |
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69 Let $L_* \sub \bc_*(X)$ be a subcomplex generated by some |
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70 subset of the blob diagrams on $X$, and let $f: L_* \to L_*$ |
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71 be a chain map which does not increase supports and which induces an isomorphism on |
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72 $H_0(L_*)$. |
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73 Then $f$ is homotopic (in $\bc_*(X)$) to the identity $L_*\to L_*$. |
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74 \end{lemma} |
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75 |
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76 \begin{proof} |
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77 We will use the method of acyclic models. |
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78 Let $b$ be a blob diagram of $L_*$, let $S\sub X$ be the support of $b$, and let |
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79 $r$ be the restriction of $b$ to $X\setminus S$. |
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80 Note that $S$ is a disjoint union of balls. |
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81 Assign to $b$ the acyclic (in positive degrees) subcomplex $T(b) \deq r\bullet\bc_*(S)$. |
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82 note that if a diagram $b'$ is part of $\bd b$ then $T(B') \sub T(b)$. |
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83 Both $f$ and the identity are compatible with $T$ (in the sense of acyclic models), |
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84 so $f$ and the identity map are homotopic. |
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85 \end{proof} |
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86 |
53 |
87 |
54 \medskip |
88 \medskip |
55 |
89 |
56 \nn{Maybe there is no longer a need to repeat the next couple of props here, since we also state them in the introduction. |
90 \nn{Maybe there is no longer a need to repeat the next couple of props here, since we also state them in the introduction. |
57 But I think it's worth saying that the Diff actions will be enhanced later. |
91 But I think it's worth saying that the Diff actions will be enhanced later. |