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67 Given a decomposition $K$ of $Y$ into $k$-balls $X_i$, let $K\times F$ denote the corresponding |
67 Given a decomposition $K$ of $Y$ into $k$-balls $X_i$, let $K\times F$ denote the corresponding |
68 decomposition of $Y\times F$ into the pieces $X_i\times F$. |
68 decomposition of $Y\times F$ into the pieces $X_i\times F$. |
69 |
69 |
70 Let $G_*\sub \bc_*(Y\times F;C)$ be the subcomplex generated by blob diagrams $a$ such that there |
70 Let $G_*\sub \bc_*(Y\times F;C)$ be the subcomplex generated by blob diagrams $a$ such that there |
71 exists a decomposition $K$ of $Y$ such that $a$ splits along $K\times F$. |
71 exists a decomposition $K$ of $Y$ such that $a$ splits along $K\times F$. |
72 It follows from Proposition \ref{thm:small-blobs} that $\bc_*(Y\times F; C)$ is homotopic to a subcomplex of $G_*$. |
72 It follows from Proposition \ref{thm:small-blobs} that $\bc_*(Y\times F; C)$ |
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73 is homotopic to a subcomplex of $G_*$. |
73 (If the blobs of $a$ are small with respect to a sufficiently fine cover then their |
74 (If the blobs of $a$ are small with respect to a sufficiently fine cover then their |
74 projections to $Y$ are contained in some disjoint union of balls.) |
75 projections to $Y$ are contained in some disjoint union of balls.) |
75 Note that the image of $\psi$ is equal to $G_*$. |
76 Note that the image of $\psi$ is equal to $G_*$. |
76 |
77 |
77 We will define $\phi: G_* \to \cl{\cC_F}(Y)$ using the method of acyclic models. |
78 We will define $\phi: G_* \to \cl{\cC_F}(Y)$ using the method of acyclic models. |
93 \begin{lemma} \label{lem:d-a-acyclic} |
94 \begin{lemma} \label{lem:d-a-acyclic} |
94 $D(a)$ is acyclic. |
95 $D(a)$ is acyclic. |
95 \end{lemma} |
96 \end{lemma} |
96 |
97 |
97 \begin{proof} |
98 \begin{proof} |
98 We will prove acyclicity in the first couple of degrees, and \nn{in this draft, at least} |
99 We will prove acyclicity in the first couple of degrees, and |
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100 %\nn{in this draft, at least} |
99 leave the general case to the reader. |
101 leave the general case to the reader. |
100 |
102 |
101 Let $K$ and $K'$ be two decompositions (0-simplices) of $Y$ compatible with $a$. |
103 Let $K$ and $K'$ be two decompositions (0-simplices) of $Y$ compatible with $a$. |
102 We want to find 1-simplices which connect $K$ and $K'$. |
104 We want to find 1-simplices which connect $K$ and $K'$. |
103 We might hope that $K$ and $K'$ have a common refinement, but this is not necessarily |
105 We might hope that $K$ and $K'$ have a common refinement, but this is not necessarily |