58 For $m$ sufficiently small there exist decompositions of $K$ of $Y$ into $k$-balls such that the |
58 For $m$ sufficiently small there exist decompositions of $K$ of $Y$ into $k$-balls such that the |
59 codimension 1 cells of $K\times F$ miss the blobs of $a$, and more generally such that $a$ is splittable along $K\times F$. |
59 codimension 1 cells of $K\times F$ miss the blobs of $a$, and more generally such that $a$ is splittable along $K\times F$. |
60 Let $D(a)$ denote the subcomplex of $\bc_*^\cF(Y)$ generated by all $(a, \bar{K})$ |
60 Let $D(a)$ denote the subcomplex of $\bc_*^\cF(Y)$ generated by all $(a, \bar{K})$ |
61 such that each $K_i$ has the aforementioned splittable property |
61 such that each $K_i$ has the aforementioned splittable property |
62 (see Subsection \ref{ss:ncat_fields}). |
62 (see Subsection \ref{ss:ncat_fields}). |
63 (By $(a, \bar{K})$ we really mean $(a', \bar{K})$, where $a^\sharp$ is |
63 \nn{need to define $D(a)$ more clearly; also includes $(b_j, \bar{K})$ where |
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64 $\bd(a) = \sum b_j$.} |
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65 (By $(a, \bar{K})$ we really mean $(a^\sharp, \bar{K})$, where $a^\sharp$ is |
64 $a$ split according to $K_0\times F$. |
66 $a$ split according to $K_0\times F$. |
65 To simplify notation we will just write plain $a$ instead of $a^\sharp$.) |
67 To simplify notation we will just write plain $a$ instead of $a^\sharp$.) |
66 Roughly speaking, $D(a)$ consists of filtration degree 0 stuff which glues up to give |
68 Roughly speaking, $D(a)$ consists of filtration degree 0 stuff which glues up to give |
67 $a$, filtration degree 1 stuff which makes all of the filtration degree 0 stuff homologous, |
69 $a$, filtration degree 1 stuff which makes all of the filtration degree 0 stuff homologous, |
68 filtration degree 2 stuff which kills the homology created by the |
70 filtration degree 2 stuff which kills the homology created by the |
74 \end{lemma} |
76 \end{lemma} |
75 |
77 |
76 \begin{proof} |
78 \begin{proof} |
77 We will prove acyclicity in the first couple of degrees, and \nn{in this draft, at least} |
79 We will prove acyclicity in the first couple of degrees, and \nn{in this draft, at least} |
78 leave the general case to the reader. |
80 leave the general case to the reader. |
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81 |
79 Let $K$ and $K'$ be two decompositions of $Y$ compatible with $a$. |
82 Let $K$ and $K'$ be two decompositions of $Y$ compatible with $a$. |
80 We want to show that $(a, K)$ and $(a, K')$ are homologous |
83 We want to show that $(a, K)$ and $(a, K')$ are homologous via filtration degree 1 stuff. |
81 \nn{oops -- can't really ignore $\bd a$ like this} |
84 \nn{need to say this better; these two chains don't have the same boundary.} |
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85 We might hope that $K$ and $K'$ have a common refinement, but this is not necessarily |
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86 the case. |
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87 (Consider the $x$-axis and the graph of $y = x^2\sin(1/x)$ in $\r^2$.) |
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88 However, we {\it can} find another decomposition $L$ such that $L$ shares common |
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89 refinements with both $K$ and $K'$. |
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90 Let $KL$ and $K'L$ denote these two refinements. |
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91 Then filtration degree 1 chains associated to the four anti-refinemnts |
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92 $KL\to K$, $KL\to L$, $K'L\to L$ and $K'L\to K'$ |
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93 give the desired chain connecting $(a, K)$ and $(a, K')$ |
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94 (see Figure xxxx). |
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95 |
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96 Consider a different choice of decomposition $L'$ in place of $L$ above. |
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97 This leads to a cycle consisting of filtration degree 1 stuff. |
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98 We want to show that this cycle bounds a chain of filtration degree 2 stuff. |
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99 Choose a decomposition $M$ which has common refinements with each of |
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100 $K$, $KL$, $L$, $K'L$, $K'$, $K'L'$, $L'$ and $KL'$. |
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101 Then we have a filtration degree 2 chain, as shown in Figure yyyy, which does the trick. |
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102 For example, .... |
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103 |
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104 |
82 \end{proof} |
105 \end{proof} |
83 |
106 |
84 |
107 |
85 \nn{....} |
108 \nn{....} |
86 \end{proof} |
109 \end{proof} |