diff -r b0fc3660fa89 -r a044fda18400 text/a_inf_blob.tex --- a/text/a_inf_blob.tex Wed Aug 26 23:10:55 2009 +0000 +++ b/text/a_inf_blob.tex Tue Sep 15 02:55:39 2009 +0000 @@ -91,7 +91,15 @@ Then filtration degree 1 chains associated to the four anti-refinemnts $KL\to K$, $KL\to L$, $K'L\to L$ and $K'L\to K'$ give the desired chain connecting $(a, K)$ and $(a, K')$ -(see Figure xxxx). +(see Figure \ref{zzz4}). + +\begin{figure}[!ht] +\begin{equation*} +\mathfig{.63}{tempkw/zz4} +\end{equation*} +\caption{Connecting $K$ and $K'$ via $L$} +\label{zzz4} +\end{figure} Consider a different choice of decomposition $L'$ in place of $L$ above. This leads to a cycle consisting of filtration degree 1 stuff. @@ -99,9 +107,17 @@ Choose a decomposition $M$ which has common refinements with each of $K$, $KL$, $L$, $K'L$, $K'$, $K'L'$, $L'$ and $KL'$. \nn{need to also require that $KLM$ antirefines to $KM$, etc.} -Then we have a filtration degree 2 chain, as shown in Figure yyyy, which does the trick. +Then we have a filtration degree 2 chain, as shown in Figure \ref{zzz5}, which does the trick. +(Each small triangle in Figure \ref{zzz5} can be filled with a filtration degree 2 chain.) For example, .... +\begin{figure}[!ht] +\begin{equation*} +\mathfig{1.0}{tempkw/zz5} +\end{equation*} +\caption{Filling in $K$-$KL$-$L$-$K'L$-$K'$-$K'L'$-$L'$-$KL'$-$K$} +\label{zzz5} +\end{figure} \end{proof}