211 |
211 |
212 \begin{figure}[!ht] |
212 \begin{figure}[!ht] |
213 $$\mathfig{.65}{tempkw/blah6}$$ |
213 $$\mathfig{.65}{tempkw/blah6}$$ |
214 \caption{An example of strict associativity}\label{blah6}\end{figure} |
214 \caption{An example of strict associativity}\label{blah6}\end{figure} |
215 |
215 |
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216 \nn{figure \ref{blah6} (blah6) needs a dotted line in the south split ball} |
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217 |
216 Notation: $a\bullet b \deq \gl_Y(a, b)$ and/or $a\cup b \deq \gl_Y(a, b)$. |
218 Notation: $a\bullet b \deq \gl_Y(a, b)$ and/or $a\cup b \deq \gl_Y(a, b)$. |
217 In the other direction, we will call the projection from $\cC(B)_E$ to $\cC(B_i)_E$ |
219 In the other direction, we will call the projection from $\cC(B)_E$ to $\cC(B_i)_E$ |
218 a {\it restriction} map and write $\res_{B_i}(a)$ for $a\in \cC(B)_E$. |
220 a {\it restriction} map and write $\res_{B_i}(a)$ for $a\in \cC(B)_E$. |
219 Compositions of boundary and restriction maps will also be called restriction maps. |
221 Compositions of boundary and restriction maps will also be called restriction maps. |
220 For example, if $B$ is a $k$-ball and $Y\sub \bd B$ is a $k{-}1$-ball, there is a |
222 For example, if $B$ is a $k$-ball and $Y\sub \bd B$ is a $k{-}1$-ball, there is a |
221 restriction map from $\cC(B)_{\bd Y}$ to $\cC(Y)$. |
223 restriction map from $\cC(B)_{\bd Y}$ to $\cC(Y)$. |
222 |
224 |
223 %More notation and terminology: |
225 We will write $\cC(B)_Y$ for the image of $\gl_Y$ in $\cC(B)$. |
224 %We will call the projection from $\cC(B)_E$ to $\cC(B_i)_E$ a {\it restriction} |
226 We will call $\cC(B)_Y$ morphisms which are splittable along $Y$ or transverse to $Y$. |
225 %map |
227 We have $\cC(B)_Y \sub \cC(B)_E \sub \cC(B)$. |
226 |
228 |
227 The above two axioms are equivalent to the following axiom, |
229 More generally, if $X$ is a sphere or ball subdivided \nn{...} |
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230 |
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231 The above two composition axioms are equivalent to the following one, |
228 which we state in slightly vague form. |
232 which we state in slightly vague form. |
229 |
233 |
230 \xxpar{Multi-composition:} |
234 \xxpar{Multi-composition:} |
231 {Given any decomposition $B = B_1\cup\cdots\cup B_m$ of a $k$-ball |
235 {Given any decomposition $B = B_1\cup\cdots\cup B_m$ of a $k$-ball |
232 into small $k$-balls, there is a |
236 into small $k$-balls, there is a |