33 (where $X$ is a component of a permissible decomposition of $Y$) to get a blob diagram on |
33 (where $X$ is a component of a permissible decomposition of $Y$) to get a blob diagram on |
34 $Y\times F$. |
34 $Y\times F$. |
35 In filtration degrees 1 and higher we define the map to be zero. |
35 In filtration degrees 1 and higher we define the map to be zero. |
36 It is easy to check that this is a chain map. |
36 It is easy to check that this is a chain map. |
37 |
37 |
38 Next we define a map from $\bc_*^C(Y\times F)$ to $\bc_*^\cF(Y)$. |
38 Next we define a map from $\phi: \bc_*^C(Y\times F) \to \bc_*^\cF(Y)$. |
39 Actually, we will define it on the homotopy equivalent subcomplex |
39 Actually, we will define it on the homotopy equivalent subcomplex |
40 $\cS_* \sub \bc_*^C(Y\times F)$ generated by blob diagrams which are small with respect to some open cover |
40 $\cS_* \sub \bc_*^C(Y\times F)$ generated by blob diagrams which are small with |
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41 respect to some open cover |
41 of $Y\times F$. |
42 of $Y\times F$. |
42 \nn{need reference to small blob lemma} |
43 \nn{need reference to small blob lemma} |
43 We will have to show eventually that this is independent (up to homotopy) of the choice of cover. |
44 We will have to show eventually that this is independent (up to homotopy) of the choice of cover. |
44 Also, for a fixed choice of cover we will only be able to define the map for blob degree less than |
45 Also, for a fixed choice of cover we will only be able to define the map for blob degree less than |
45 some bound, but this bound goes to infinity as the cover become finer. |
46 some bound, but this bound goes to infinity as the cover become finer. |
46 |
47 |
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48 Given a decomposition $K$ of $Y$ into $k$-balls $X_i$, let $K\times F$ denote the corresponding |
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49 decomposition of $Y\times F$ into the pieces $X_i\times F$. |
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50 |
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51 %We will define $\phi$ inductively, starting at blob degree 0. |
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52 %Given a 0-blob diagram $x$ on $Y\times F$, we can choose a decomposition $K$ of $Y$ |
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53 %such that $x$ is splittable with respect to $K\times F$. |
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54 %This defines a filtration degree 0 element of $\bc_*^\cF(Y)$ |
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55 |
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56 We will define $\phi$ using a variant of the method of acyclic models. |
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57 Let $a\in S_m$ be a blob diagram on $Y\times F$. |
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58 For $m$ sufficiently small there exist decompositions of $K$ of $Y$ into $k$-balls such that the |
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59 codimension 1 cells of $K\times F$ miss the blobs of $a$, and more generally such that $a$ is splittable along $K\times F$. |
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60 Let $D(a)$ denote the subcomplex of $\bc_*^\cF(Y)$ generated by all $(a, \bar{K})$ |
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61 such that each $K_i$ has the aforementioned splittable property |
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62 (see Subsection \ref{ss:ncat_fields}). |
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63 (By $(a, \bar{K})$ we really mean $(a', \bar{K})$, where $a^\sharp$ is |
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64 $a$ split according to $K_0\times F$. |
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65 To simplify notation we will just write plain $a$ instead of $a^\sharp$.) |
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66 Roughly speaking, $D(a)$ consists of filtration degree 0 stuff which glues up to give |
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67 $a$, filtration degree 1 stuff which makes all of the filtration degree 0 stuff homologous, |
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68 filtration degree 2 stuff which kills the homology created by the |
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69 filtration degree 1 stuff, and so on. |
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70 More formally, |
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71 |
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72 \begin{lemma} |
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73 $D(a)$ is acyclic. |
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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 prove acyclicity in the first couple of degrees, and \nn{in this draft, at least} |
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78 leave the general case to the reader. |
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79 Let $K$ and $K'$ be two decompositions of $Y$ compatible with $a$. |
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80 We want to show that $(a, K)$ and $(a, K')$ are homologous |
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81 \nn{oops -- can't really ignore $\bd a$ like this} |
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82 \end{proof} |
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83 |
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84 |
47 \nn{....} |
85 \nn{....} |
48 \end{proof} |
86 \end{proof} |
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87 |
49 |
88 |
50 \nn{need to say something about dim $< n$ above} |
89 \nn{need to say something about dim $< n$ above} |
51 |
90 |
52 |
91 |
53 |
92 |