90 cylinders. |
90 cylinders. |
91 More specifically, we impose the following two equivalence relations: |
91 More specifically, we impose the following two equivalence relations: |
92 \begin{itemize} |
92 \begin{itemize} |
93 \item If $g:R_i\to R_i$ is a homeomorphism, we can replace |
93 \item If $g:R_i\to R_i$ is a homeomorphism, we can replace |
94 \[ |
94 \[ |
95 (\ldots, f_{i-1}, f_i, \ldots) \to (\ldots, g\circ f_{i-1}, f_i\circ g^{-1}, \ldots) . |
95 (\ldots, f_{i-1}, f_i, \ldots) \to (\ldots, g\circ f_{i-1}, f_i\circ g^{-1}, \ldots), |
96 \] |
96 \] |
|
97 leaving the $M_i$, $N_i$ and $R_i$ fixed. |
97 (See Figure xxx.) |
98 (See Figure xxx.) |
98 \item If $M_i = M'_i \du M''_i$ and $N_i = N'_i \du N''_i$ (and there is a |
99 \item If $M_i = M'_i \du M''_i$ and $N_i = N'_i \du N''_i$ (and there is a |
99 compatible disjoint union of $\bd M = \bd N$), we can replace |
100 compatible disjoint union of $\bd M = \bd N$), we can replace |
100 \begin{eqnarray*} |
101 \begin{eqnarray*} |
101 (\ldots, M_{i-1}, M_i, M_{i+1}, \ldots) &\to& (\ldots, M_{i-1}, M'_i, M''_i, M_{i+1}, \ldots) \\ |
102 (\ldots, M_{i-1}, M_i, M_{i+1}, \ldots) &\to& (\ldots, M_{i-1}, M'_i, M''_i, M_{i+1}, \ldots) \\ |