87 We regard two such surgery cylinders as the same if there is a homeomorphism between them which is the |
87 We regard two such surgery cylinders as the same if there is a homeomorphism between them which is the |
88 identity on the boundary and which preserves the 1-dimensional fibers coming from the mapping |
88 identity on the boundary and which preserves the 1-dimensional fibers coming from the mapping |
89 cylinders. |
89 cylinders. |
90 More specifically, we impose the following two equivalence relations: |
90 More specifically, we impose the following two equivalence relations: |
91 \begin{itemize} |
91 \begin{itemize} |
92 \item If $g: R_i\to R'_i$ is a homeomorphism, we can replace |
92 \item If $g: R_i\to R'_i$ is a homeomorphism which restricts to the identity on |
|
93 $\bd R_i = \bd R'_i = E_0\cup \bd M_i$, we can replace |
93 \begin{eqnarray*} |
94 \begin{eqnarray*} |
94 (\ldots, R_{i-1}, R_i, R_{i+1}, \ldots) &\to& (\ldots, R_{i-1}, R'_i, R_{i+1}, \ldots) \\ |
95 (\ldots, R_{i-1}, R_i, R_{i+1}, \ldots) &\to& (\ldots, R_{i-1}, R'_i, R_{i+1}, \ldots) \\ |
95 (\ldots, f_{i-1}, f_i, \ldots) &\to& (\ldots, g\circ f_{i-1}, f_i\circ g^{-1}, \ldots), |
96 (\ldots, f_{i-1}, f_i, \ldots) &\to& (\ldots, g\circ f_{i-1}, f_i\circ g^{-1}, \ldots), |
96 \end{eqnarray*} |
97 \end{eqnarray*} |
97 leaving the $M_i$ and $N_i$ fixed. |
98 leaving the $M_i$ and $N_i$ fixed. |