29 |
29 |
30 For the next proposition we will temporarily restore $n$-manifold boundary |
30 For the next proposition we will temporarily restore $n$-manifold boundary |
31 conditions to the notation. |
31 conditions to the notation. |
32 |
32 |
33 Suppose that for all $c \in \cC(\bd B^n)$ |
33 Suppose that for all $c \in \cC(\bd B^n)$ |
34 we have a splitting $s: H_0(\bc_*(B^n, c)) \to \bc_0(B^n; c)$ |
34 we have a splitting $s: H_0(\bc_*(B^n; c)) \to \bc_0(B^n; c)$ |
35 of the quotient map |
35 of the quotient map |
36 $p: \bc_0(B^n; c) \to H_0(\bc_*(B^n, c))$. |
36 $p: \bc_0(B^n; c) \to H_0(\bc_*(B^n; c))$. |
37 For example, this is always the case if the coefficient ring is a field. |
37 For example, this is always the case if the coefficient ring is a field. |
38 Then |
38 Then |
39 \begin{prop} \label{bcontract} |
39 \begin{prop} \label{bcontract} |
40 For all $c \in \cC(\bd B^n)$ the natural map $p: \bc_*(B^n, c) \to H_0(\bc_*(B^n, c))$ |
40 For all $c \in \cC(\bd B^n)$ the natural map $p: \bc_*(B^n; c) \to H_0(\bc_*(B^n; c))$ |
41 is a chain homotopy equivalence |
41 is a chain homotopy equivalence |
42 with inverse $s: H_0(\bc_*(B^n, c)) \to \bc_*(B^n; c)$. |
42 with inverse $s: H_0(\bc_*(B^n; c)) \to \bc_*(B^n; c)$. |
43 Here we think of $H_0(\bc_*(B^n, c))$ as a 1-step complex concentrated in degree 0. |
43 Here we think of $H_0(\bc_*(B^n; c))$ as a 1-step complex concentrated in degree 0. |
44 \end{prop} |
44 \end{prop} |
45 \begin{proof} |
45 \begin{proof} |
46 By assumption $p\circ s = \id$, so all that remains is to find a degree 1 map |
46 By assumption $p\circ s = \id$, so all that remains is to find a degree 1 map |
47 $h : \bc_*(B^n; c) \to \bc_*(B^n; c)$ such that $\bd h + h\bd = \id - s \circ p$. |
47 $h : \bc_*(B^n; c) \to \bc_*(B^n; c)$ such that $\bd h + h\bd = \id - s \circ p$. |
48 For $i \ge 1$, define $h_i : \bc_i(B^n; c) \to \bc_{i+1}(B^n; c)$ by adding |
48 For $i \ge 1$, define $h_i : \bc_i(B^n; c) \to \bc_{i+1}(B^n; c)$ by adding |
65 |
65 |
66 \begin{proof} |
66 \begin{proof} |
67 This follows from Properties \ref{property:disjoint-union} and \ref{property:contractibility}. |
67 This follows from Properties \ref{property:disjoint-union} and \ref{property:contractibility}. |
68 \end{proof} |
68 \end{proof} |
69 |
69 |
70 Recall the definition of the support of a blob diagram as the union of all the |
70 %Recall the definition of the support of a blob diagram as the union of all the |
71 blobs of the diagram. |
71 %blobs of the diagram. |
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72 We define the {\it support} of a blob diagram $b$, $\supp(b) \sub X$, |
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73 to be the union of the blobs of $b$. |
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74 For $y \in \bc_*(X)$ with $y = \sum c_i b_i$ ($c_i$ a non-zero number, $b_i$ a blob diagram), |
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75 we define $\supp(y) \deq \bigcup_i \supp(b_i)$. |
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76 |
72 For future use we prove the following lemma. |
77 For future use we prove the following lemma. |
73 |
78 |
74 \begin{lemma} \label{support-shrink} |
79 \begin{lemma} \label{support-shrink} |
75 Let $L_* \sub \bc_*(X)$ be a subcomplex generated by some |
80 Let $L_* \sub \bc_*(X)$ be a subcomplex generated by some |
76 subset of the blob diagrams on $X$, and let $f: L_* \to L_*$ |
81 subset of the blob diagrams on $X$, and let $f: L_* \to L_*$ |