122 Note that the second equivalence increases the number of holes (or arity) by 1. |
122 Note that the second equivalence increases the number of holes (or arity) by 1. |
123 We can make a similar identification with the roles of $M'_i$ and $M''_i$ reversed. |
123 We can make a similar identification with the roles of $M'_i$ and $M''_i$ reversed. |
124 In terms of the ``sequence of surgeries" picture, this says that if two successive surgeries |
124 In terms of the ``sequence of surgeries" picture, this says that if two successive surgeries |
125 do not overlap, we can perform them in reverse order or simultaneously. |
125 do not overlap, we can perform them in reverse order or simultaneously. |
126 |
126 |
127 There is an operad structure on $n$-dimensional surgery cylinders, given by gluing the outer boundary |
127 There is a colored operad structure on $n$-dimensional surgery cylinders, given by gluing the outer boundary |
128 of one cylinder into one of the inner boundaries of another cylinder. |
128 of one cylinder into one of the inner boundaries of another cylinder. |
129 We leave it to the reader to work out a more precise statement in terms of $M_i$'s, $f_i$'s etc. |
129 We leave it to the reader to work out a more precise statement in terms of $M_i$'s, $f_i$'s etc. |
130 |
130 |
131 For fixed $\ol{M} = (M_0,\ldots,M_k)$ and $\ol{N} = (N_0,\ldots,N_k)$, we let |
131 For fixed $\ol{M} = (M_0,\ldots,M_k)$ and $\ol{N} = (N_0,\ldots,N_k)$, we let |
132 $SC^n_{\ol{M}\ol{N}}$ denote the topological space of all $n$-dimensional surgery cylinders as above. |
132 $SC^n_{\ol{M}\ol{N}}$ denote the topological space of all $n$-dimensional surgery cylinders as above. |