150 We will denote the typical element of $FG^n_{\ol{M}\ol{N}}$ by $\ol{f} = (f_0,\ldots,f_k)$. |
150 We will denote the typical element of $FG^n_{\ol{M}\ol{N}}$ by $\ol{f} = (f_0,\ldots,f_k)$. |
151 |
151 |
152 \medskip |
152 \medskip |
153 |
153 |
154 %The little $n{+}1$-ball operad injects into the $n$-FG operad. |
154 %The little $n{+}1$-ball operad injects into the $n$-FG operad. |
155 The $n$-FG operad contains the little $n{+}1$-ball operad. |
155 The $n$-FG operad contains the little $n{+}1$-balls operad. |
156 Roughly speaking, given a configuration of $k$ little $n{+}1$-balls in the standard |
156 Roughly speaking, given a configuration of $k$ little $n{+}1$-balls in the standard |
157 $n{+}1$-ball, we fiber the complement of the balls by vertical intervals |
157 $n{+}1$-ball, we fiber the complement of the balls by vertical intervals |
158 and let $M_i$ [$N_i$] be the southern [northern] hemisphere of the $i$-th ball. |
158 and let $M_i$ [$N_i$] be the southern [northern] hemisphere of the $i$-th ball. |
159 More precisely, let $x_0,\ldots,x_n$ be the coordinates of $\r^{n+1}$. |
159 More precisely, let $x_1,\ldots,x_{n+1}$ be the coordinates of $\r^{n+1}$. |
160 Let $z$ be a point of the $k$-th space of the little $n{+}1$-ball operad, with |
160 Let $z$ be a point of the $k$-th space of the little $n{+}1$-ball operad, with |
161 little balls $D_1,\ldots,D_k$ inside the standard $n{+}1$-ball. |
161 little balls $D_1,\ldots,D_k$ inside the standard $n{+}1$-ball. |
162 We assume the $D_i$'s are ordered according to the $x_n$ coordinate of their centers. |
162 We assume the $D_i$'s are ordered according to the $x_{n+1}$ coordinate of their centers. |
163 Let $\pi:\r^{n+1}\to \r^n$ be the projection corresponding to $x_n$. |
163 Let $\pi:\r^{n+1}\to \r^n$ be the projection corresponding to $x_{n+1}$. |
164 Let $B\sub\r^n$ be the standard $n$-ball. |
164 Let $B\sub\r^n$ be the standard $n$-ball. |
165 Let $M_i$ and $N_i$ be $B$ for all $i$. |
165 Let $M_i$ and $N_i$ be $B$ for all $i$. |
166 Identify $\pi(D_i)$ with $B$ (a.k.a.\ $M_i$ or $N_i$) via translations and dilations (no rotations). |
166 Identify $\pi(D_i)$ with $B$ (a.k.a.\ $M_i$ or $N_i$) via translations and dilations (no rotations). |
167 Let $R_i = B\setmin \pi(D_i)$. |
167 Let $R_i = B\setmin \pi(D_i)$. |
168 Let $f_i = \rm{id}$ for all $i$. |
168 Let $f_i = \rm{id}$ for all $i$. |
169 We have now defined a map from the little $n{+}1$-ball operad to the $n$-FG operad, |
169 We have now defined a map from the little $n{+}1$-ball operad to the $n$-FG operad, |
170 with contractible fibers. |
170 with contractible fibers. |
171 (The fibers correspond to moving the $D_i$'s in the $x_n$ direction without changing their ordering.) |
171 (The fibers correspond to moving the $D_i$'s in the $x_{n+1}$ |
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172 direction without changing their ordering.) |
172 \nn{issue: we've described this by varying the $R_i$'s, but above we emphasize varying the $f_i$'s. |
173 \nn{issue: we've described this by varying the $R_i$'s, but above we emphasize varying the $f_i$'s. |
173 does this need more explanation?} |
174 does this need more explanation?} |
174 |
175 |
175 Another familiar subspace of the $n$-FG operad is $\Homeo(M\to N)$, which corresponds to |
176 Another familiar subspace of the $n$-FG operad is $\Homeo(M\to N)$, which corresponds to |
176 case $k=0$ (no holes). |
177 case $k=0$ (no holes). |
192 \stackrel{f_1}{\to} \bc_*(R_2\cup M_2) \stackrel{\id\ot\alpha_2}{\to} |
193 \stackrel{f_1}{\to} \bc_*(R_2\cup M_2) \stackrel{\id\ot\alpha_2}{\to} |
193 \cdots \stackrel{\id\ot\alpha_k}{\to} \bc_*(R_k\cup N_k) |
194 \cdots \stackrel{\id\ot\alpha_k}{\to} \bc_*(R_k\cup N_k) |
194 \stackrel{f_k}{\to} \bc_*(N_0) |
195 \stackrel{f_k}{\to} \bc_*(N_0) |
195 \] |
196 \] |
196 (Recall that the maps $\id\ot\alpha_i$ were defined in \nn{need ref}.) |
197 (Recall that the maps $\id\ot\alpha_i$ were defined in \nn{need ref}.) |
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198 \nn{issue: haven't we only defined $\id\ot\alpha_i$ when $\alpha_i$ is closed?} |
197 It is easy to check that the above definition is compatible with the equivalence relations |
199 It is easy to check that the above definition is compatible with the equivalence relations |
198 and also the operad structure. |
200 and also the operad structure. |
199 We can reinterpret the above as a chain map |
201 We can reinterpret the above as a chain map |
200 \[ |
202 \[ |
201 p: C_0(FG^n_{\ol{M}\ol{N}})\ot \hom(\bc_*(M_1), \bc_*(N_1))\ot\cdots\ot\hom(\bc_*(M_k), \bc_*(N_k)) |
203 p: C_0(FG^n_{\ol{M}\ol{N}})\ot \hom(\bc_*(M_1), \bc_*(N_1))\ot\cdots\ot\hom(\bc_*(M_k), \bc_*(N_k)) |