79 (By convention, $M_i = N_i = \emptyset$ if $i <1$ or $i>k$.) |
79 (By convention, $M_i = N_i = \emptyset$ if $i <1$ or $i>k$.) |
80 We call $R_0$ the outer incoming manifold and $R_{k+1}$ the outer outgoing manifold |
80 We call $R_0$ the outer incoming manifold and $R_{k+1}$ the outer outgoing manifold |
81 \item Homeomorphisms $f_i : R_i\cup N_i\to R_{i+1}\cup M_{i+1}$, $0\le i \le k$. |
81 \item Homeomorphisms $f_i : R_i\cup N_i\to R_{i+1}\cup M_{i+1}$, $0\le i \le k$. |
82 \end{itemize} |
82 \end{itemize} |
83 We can think of the above data as encoding the union of the mapping cylinders $C(f_0),\ldots,C(f_k)$, |
83 We can think of the above data as encoding the union of the mapping cylinders $C(f_0),\ldots,C(f_k)$, |
84 with $C(f_i)$ glued to $C(f_{i+1})$ along $R_{i+1}$. |
84 with $C(f_i)$ glued to $C(f_{i+1})$ along $R_{i+1}$ |
85 \nn{need figure} |
85 (see Figure xxxx). |
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86 \nn{also need to revise outer labels of older fig} |
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87 The $n$-manifolds are the ``$n$-dimensional graph" and the $I$ direction of the mapping cylinders is the ``fat" part. |
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88 We regard two such fat graphs as the same if there is a homeomorphism between them which is the |
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89 identity on the boundary and which preserves the 1-dimensional fibers coming from the mapping |
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90 cylinders. |
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91 More specifically, we impose the following two equivalence relations: |
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92 \begin{itemize} |
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93 \item If $g:R_i\to R_i$ is a homeomorphism, we can replace |
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94 \[ |
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95 (\ldots, f_{i-1}, f_i, \ldots) \to (\ldots, g\circ f_{i-1}, f_i\circ g^{-1}, \ldots) . |
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96 \] |
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97 (See Figure xxx.) |
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98 \item If $M_i = M'_i \du M''_i$ and $N_i = N'_i \du N''_i$ (and there is a |
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99 compatible disjoint union of $\bd M = \bd N$), we can replace |
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100 \begin{eqnarray*} |
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101 (\ldots, M_{i-1}, M_i, M_{i+1}, \ldots) &\to& (\ldots, M_{i-1}, M'_i, M''_i, M_{i+1}, \ldots) \\ |
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102 (\ldots, N_{i-1}, N_i, N_{i+1}, \ldots) &\to& (\ldots, N_{i-1}, N'_i, N''_i, N_{i+1}, \ldots) \\ |
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103 (\ldots, R_{i-1}, R_i, R_{i+1}, \ldots) &\to& |
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104 (\ldots, R_{i-1}, R_i\cup M''_i, R_i\cup N'_i, R_{i+1}, \ldots) \\ |
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105 (\ldots, f_{i-1}, f_i, \ldots) &\to& (\ldots, f_{i-1}, \rm{id}, f_i, \ldots) . |
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106 \end{eqnarray*} |
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107 (See Figure xxxx.) |
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108 \end{itemize} |
86 |
109 |
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110 Note that the second equivalence increases the number of holes (or arity) by 1. |
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111 We can make a similar identification with the rolls of $M'_i$ and $M''_i$ reversed. |
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112 In terms of the ``sequence of surgeries" picture, this says that if two successive surgeries |
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113 do not overlap, we can perform them in reverse order or simultaneously. |
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114 |
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115 \nn{operad structure (need to ntro mroe terminology above} |
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116 |
89 \nn{*** resume revising here} |
117 \nn{*** resume revising here} |
90 |
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91 |
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