89 refinements with both $K$ and $K'$. |
89 refinements with both $K$ and $K'$. |
90 Let $KL$ and $K'L$ denote these two refinements. |
90 Let $KL$ and $K'L$ denote these two refinements. |
91 Then filtration degree 1 chains associated to the four anti-refinemnts |
91 Then filtration degree 1 chains associated to the four anti-refinemnts |
92 $KL\to K$, $KL\to L$, $K'L\to L$ and $K'L\to K'$ |
92 $KL\to K$, $KL\to L$, $K'L\to L$ and $K'L\to K'$ |
93 give the desired chain connecting $(a, K)$ and $(a, K')$ |
93 give the desired chain connecting $(a, K)$ and $(a, K')$ |
94 (see Figure xxxx). |
94 (see Figure \ref{zzz4}). |
|
95 |
|
96 \begin{figure}[!ht] |
|
97 \begin{equation*} |
|
98 \mathfig{.63}{tempkw/zz4} |
|
99 \end{equation*} |
|
100 \caption{Connecting $K$ and $K'$ via $L$} |
|
101 \label{zzz4} |
|
102 \end{figure} |
95 |
103 |
96 Consider a different choice of decomposition $L'$ in place of $L$ above. |
104 Consider a different choice of decomposition $L'$ in place of $L$ above. |
97 This leads to a cycle consisting of filtration degree 1 stuff. |
105 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. |
106 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 |
107 Choose a decomposition $M$ which has common refinements with each of |
100 $K$, $KL$, $L$, $K'L$, $K'$, $K'L'$, $L'$ and $KL'$. |
108 $K$, $KL$, $L$, $K'L$, $K'$, $K'L'$, $L'$ and $KL'$. |
101 \nn{need to also require that $KLM$ antirefines to $KM$, etc.} |
109 \nn{need to also require that $KLM$ antirefines to $KM$, etc.} |
102 Then we have a filtration degree 2 chain, as shown in Figure yyyy, which does the trick. |
110 Then we have a filtration degree 2 chain, as shown in Figure \ref{zzz5}, which does the trick. |
|
111 (Each small triangle in Figure \ref{zzz5} can be filled with a filtration degree 2 chain.) |
103 For example, .... |
112 For example, .... |
104 |
113 |
|
114 \begin{figure}[!ht] |
|
115 \begin{equation*} |
|
116 \mathfig{1.0}{tempkw/zz5} |
|
117 \end{equation*} |
|
118 \caption{Filling in $K$-$KL$-$L$-$K'L$-$K'$-$K'L'$-$L'$-$KL'$-$K$} |
|
119 \label{zzz5} |
|
120 \end{figure} |
105 |
121 |
106 \end{proof} |
122 \end{proof} |
107 |
123 |
108 |
124 |
109 \nn{....} |
125 \nn{....} |