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96 Consider a different choice of decomposition $L'$ in place of $L$ above. |
96 Consider a different choice of decomposition $L'$ in place of $L$ above. |
97 This leads to a cycle consisting of filtration degree 1 stuff. |
97 This leads to a cycle consisting of filtration degree 1 stuff. |
98 We want to show that this cycle bounds a chain of filtration degree 2 stuff. |
98 We want to show that this cycle bounds a chain of filtration degree 2 stuff. |
99 Choose a decomposition $M$ which has common refinements with each of |
99 Choose a decomposition $M$ which has common refinements with each of |
100 $K$, $KL$, $L$, $K'L$, $K'$, $K'L'$, $L'$ and $KL'$. |
100 $K$, $KL$, $L$, $K'L$, $K'$, $K'L'$, $L'$ and $KL'$. |
|
101 \nn{need to also require that $KLM$ antirefines to $KM$, etc.} |
101 Then we have a filtration degree 2 chain, as shown in Figure yyyy, which does the trick. |
102 Then we have a filtration degree 2 chain, as shown in Figure yyyy, which does the trick. |
102 For example, .... |
103 For example, .... |
103 |
104 |
104 |
105 |
105 \end{proof} |
106 \end{proof} |