equal
deleted
inserted
replaced
105 leaving the $M_i$ and $N_i$ fixed. |
105 leaving the $M_i$ and $N_i$ fixed. |
106 (Keep in mind the case $R'_i = R_i$.) |
106 (Keep in mind the case $R'_i = R_i$.) |
107 (See Figure \ref{xdfig3}.) |
107 (See Figure \ref{xdfig3}.) |
108 \begin{figure}[t] |
108 \begin{figure}[t] |
109 $$\mathfig{.4}{deligne/dfig3a} \to \mathfig{.4}{deligne/dfig3b} $$ |
109 $$\mathfig{.4}{deligne/dfig3a} \to \mathfig{.4}{deligne/dfig3b} $$ |
110 \caption{Conjugating by a homeomorphism |
110 \caption{Conjugating by a homeomorphism.} |
111 \label{xdfig3} |
111 \label{xdfig3} |
112 \end{figure} |
112 \end{figure} |
113 \item If $M_i = M'_i \du M''_i$ and $N_i = N'_i \du N''_i$ (and there is a |
113 \item If $M_i = M'_i \du M''_i$ and $N_i = N'_i \du N''_i$ (and there is a |
114 compatible disjoint union of $\bd M = \bd N$), we can replace |
114 compatible disjoint union of $\bd M = \bd N$), we can replace |
115 \begin{eqnarray*} |
115 \begin{eqnarray*} |
120 (\ldots, f_{i-1}, f_i, \ldots) &\to& (\ldots, f_{i-1}, \rm{id}, f_i, \ldots) . |
120 (\ldots, f_{i-1}, f_i, \ldots) &\to& (\ldots, f_{i-1}, \rm{id}, f_i, \ldots) . |
121 \end{eqnarray*} |
121 \end{eqnarray*} |
122 (See Figure \ref{xdfig1}.) |
122 (See Figure \ref{xdfig1}.) |
123 \begin{figure}[t] |
123 \begin{figure}[t] |
124 $$\mathfig{.3}{deligne/dfig1a} \leftarrow \mathfig{.3}{deligne/dfig1b} \rightarrow \mathfig{.3}{deligne/dfig1c}$$ |
124 $$\mathfig{.3}{deligne/dfig1a} \leftarrow \mathfig{.3}{deligne/dfig1b} \rightarrow \mathfig{.3}{deligne/dfig1c}$$ |
125 \caption{Changing the order of a surgery}\label{xdfig1} |
125 \caption{Changing the order of a surgery.}\label{xdfig1} |
126 \end{figure} |
126 \end{figure} |
127 \end{itemize} |
127 \end{itemize} |
128 |
128 |
129 Note that the second equivalence increases the number of holes (or arity) by 1. |
129 Note that the second equivalence increases the number of holes (or arity) by 1. |
130 We can make a similar identification with the roles of $M'_i$ and $M''_i$ reversed. |
130 We can make a similar identification with the roles of $M'_i$ and $M''_i$ reversed. |