224 |
224 |
225 We will write $\cC(B)_Y$ for the image of $\gl_Y$ in $\cC(B)$. |
225 We will write $\cC(B)_Y$ for the image of $\gl_Y$ in $\cC(B)$. |
226 We will call $\cC(B)_Y$ morphisms which are splittable along $Y$ or transverse to $Y$. |
226 We will call $\cC(B)_Y$ morphisms which are splittable along $Y$ or transverse to $Y$. |
227 We have $\cC(B)_Y \sub \cC(B)_E \sub \cC(B)$. |
227 We have $\cC(B)_Y \sub \cC(B)_E \sub \cC(B)$. |
228 |
228 |
229 More generally, if $X$ is a sphere or ball subdivided \nn{...} |
229 More generally, let $\alpha$ be a subdivision of a ball (or sphere) $X$ into smaller balls. |
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230 Let $\cC(X)_\alpha \sub \cC(X)$ denote the image of the iterated gluing maps from |
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231 the smaller balls to $X$. |
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232 We will also say that $\cC(X)_\alpha$ are morphisms which are splittable along $\alpha$. |
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233 In situations where the subdivision is notationally anonymous, we will write |
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234 $\cC(X)\spl$ for the morphisms which are splittable along (a.k.a.\ transverse to) |
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235 the unnamed subdivision. |
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236 If $\beta$ is a subdivision of $\bd X$, we define $\cC(X)_\beta \deq \bd\inv(\cC(\bd X)_\beta)$; |
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237 this can also be denoted $\cC(X)\spl$ if the context contains an anonymous |
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238 subdivision of $\bd X$ and no competing subdivision of $X$. |
230 |
239 |
231 The above two composition axioms are equivalent to the following one, |
240 The above two composition axioms are equivalent to the following one, |
232 which we state in slightly vague form. |
241 which we state in slightly vague form. |
233 |
242 |
234 \xxpar{Multi-composition:} |
243 \xxpar{Multi-composition:} |
235 {Given any decomposition $B = B_1\cup\cdots\cup B_m$ of a $k$-ball |
244 {Given any decomposition $B = B_1\cup\cdots\cup B_m$ of a $k$-ball |
236 into small $k$-balls, there is a |
245 into small $k$-balls, there is a |
237 map from an appropriate subset (like a fibered product) |
246 map from an appropriate subset (like a fibered product) |
238 of $\cC(B_1)\times\cdots\times\cC(B_m)$ to $\cC(B)$, |
247 of $\cC(B_1)\spl\times\cdots\times\cC(B_m)\spl$ to $\cC(B)\spl$, |
239 and these various $m$-fold composition maps satisfy an |
248 and these various $m$-fold composition maps satisfy an |
240 operad-type strict associativity condition (Figure \ref{blah7}).} |
249 operad-type strict associativity condition (Figure \ref{blah7}).} |
241 |
250 |
242 \begin{figure}[!ht] |
251 \begin{figure}[!ht] |
243 $$\mathfig{.8}{tempkw/blah7}$$ |
252 $$\mathfig{.8}{tempkw/blah7}$$ |