141 That is, given compatible domain and range, we should be able to combine them into |
141 That is, given compatible domain and range, we should be able to combine them into |
142 the full boundary of a morphism: |
142 the full boundary of a morphism: |
143 |
143 |
144 \xxpar{Domain $+$ range $\to$ boundary:} |
144 \xxpar{Domain $+$ range $\to$ boundary:} |
145 {Let $S = B_1 \cup_E B_2$, where $S$ is a $k$-sphere ($0\le k\le n-1$), |
145 {Let $S = B_1 \cup_E B_2$, where $S$ is a $k$-sphere ($0\le k\le n-1$), |
146 $B_i$ is a $k$-ball, and $E = B_1\cap B_2$ is a $k{-}1$-sphere. |
146 $B_i$ is a $k$-ball, and $E = B_1\cap B_2$ is a $k{-}1$-sphere (Figure \ref{blah3}). |
147 Let $\cC(B_1) \times_{\cC(E)} \cC(B_2)$ denote the fibered product of the |
147 Let $\cC(B_1) \times_{\cC(E)} \cC(B_2)$ denote the fibered product of the |
148 two maps $\bd: \cC(B_i)\to \cC(E)$. |
148 two maps $\bd: \cC(B_i)\to \cC(E)$. |
149 Then (axiom) we have an injective map |
149 Then (axiom) we have an injective map |
150 \[ |
150 \[ |
151 \gl_E : \cC(B_1) \times_{\cC(E)} \cC(B_2) \to \cC(S) |
151 \gl_E : \cC(B_1) \times_{\cC(E)} \cC(B_2) \to \cC(S) |
152 \] |
152 \] |
153 which is natural with respect to the actions of homeomorphisms.} |
153 which is natural with respect to the actions of homeomorphisms.} |
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154 |
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155 \begin{figure}[!ht] |
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156 $$\mathfig{.4}{tempkw/blah3}$$ |
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157 \caption{Combining two balls to get a full boundary}\label{blah3}\end{figure} |
154 |
158 |
155 Note that we insist on injectivity above. |
159 Note that we insist on injectivity above. |
156 |
160 |
157 Let $\cC(S)_E$ denote the image of $\gl_E$. |
161 Let $\cC(S)_E$ denote the image of $\gl_E$. |
158 We will refer to elements of $\cC(S)_E$ as ``splittable along $E$" or ``transverse to $E$". |
162 We will refer to elements of $\cC(S)_E$ as ``splittable along $E$" or ``transverse to $E$". |
173 In the presence of strong duality, these $k$ distinct compositions are subsumed into |
177 In the presence of strong duality, these $k$ distinct compositions are subsumed into |
174 one general type of composition which can be in any ``direction". |
178 one general type of composition which can be in any ``direction". |
175 |
179 |
176 \xxpar{Composition:} |
180 \xxpar{Composition:} |
177 {Let $B = B_1 \cup_Y B_2$, where $B$, $B_1$ and $B_2$ are $k$-balls ($0\le k\le n$) |
181 {Let $B = B_1 \cup_Y B_2$, where $B$, $B_1$ and $B_2$ are $k$-balls ($0\le k\le n$) |
178 and $Y = B_1\cap B_2$ is a $k{-}1$-ball. |
182 and $Y = B_1\cap B_2$ is a $k{-}1$-ball (Figure \ref{blah5}). |
179 Let $E = \bd Y$, which is a $k{-}2$-sphere. |
183 Let $E = \bd Y$, which is a $k{-}2$-sphere. |
180 Note that each of $B$, $B_1$ and $B_2$ has its boundary split into two $k{-}1$-balls by $E$. |
184 Note that each of $B$, $B_1$ and $B_2$ has its boundary split into two $k{-}1$-balls by $E$. |
181 We have restriction (domain or range) maps $\cC(B_i)_E \to \cC(Y)$. |
185 We have restriction (domain or range) maps $\cC(B_i)_E \to \cC(Y)$. |
182 Let $\cC(B_1)_E \times_{\cC(Y)} \cC(B_2)_E$ denote the fibered product of these two maps. |
186 Let $\cC(B_1)_E \times_{\cC(Y)} \cC(B_2)_E$ denote the fibered product of these two maps. |
183 Then (axiom) we have a map |
187 Then (axiom) we have a map |
187 which is natural with respect to the actions of homeomorphisms, and also compatible with restrictions |
191 which is natural with respect to the actions of homeomorphisms, and also compatible with restrictions |
188 to the intersection of the boundaries of $B$ and $B_i$. |
192 to the intersection of the boundaries of $B$ and $B_i$. |
189 If $k < n$ we require that $\gl_Y$ is injective. |
193 If $k < n$ we require that $\gl_Y$ is injective. |
190 (For $k=n$, see below.)} |
194 (For $k=n$, see below.)} |
191 |
195 |
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196 \begin{figure}[!ht] |
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197 $$\mathfig{.4}{tempkw/blah5}$$ |
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198 \caption{From two balls to one ball}\label{blah5}\end{figure} |
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199 |
192 \xxpar{Strict associativity:} |
200 \xxpar{Strict associativity:} |
193 {The composition (gluing) maps above are strictly associative.} |
201 {The composition (gluing) maps above are strictly associative.} |
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202 |
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203 \begin{figure}[!ht] |
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204 $$\mathfig{.65}{tempkw/blah6}$$ |
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205 \caption{An example of strict associativity}\label{blah6}\end{figure} |
194 |
206 |
195 Notation: $a\bullet b \deq \gl_Y(a, b)$ and/or $a\cup b \deq \gl_Y(a, b)$. |
207 Notation: $a\bullet b \deq \gl_Y(a, b)$ and/or $a\cup b \deq \gl_Y(a, b)$. |
196 In the other direction, we will call the projection from $\cC(B)_E$ to $\cC(B_i)_E$ |
208 In the other direction, we will call the projection from $\cC(B)_E$ to $\cC(B_i)_E$ |
197 a {\it restriction} map and write $\res_{B_i}(a)$ for $a\in \cC(B)_E$. |
209 a {\it restriction} map and write $\res_{B_i}(a)$ for $a\in \cC(B)_E$. |
198 Compositions of boundary and restriction maps will also be called restriction maps. |
210 Compositions of boundary and restriction maps will also be called restriction maps. |
210 {Given any decomposition $B = B_1\cup\cdots\cup B_m$ of a $k$-ball |
222 {Given any decomposition $B = B_1\cup\cdots\cup B_m$ of a $k$-ball |
211 into small $k$-balls, there is a |
223 into small $k$-balls, there is a |
212 map from an appropriate subset (like a fibered product) |
224 map from an appropriate subset (like a fibered product) |
213 of $\cC(B_1)\times\cdots\times\cC(B_m)$ to $\cC(B)$, |
225 of $\cC(B_1)\times\cdots\times\cC(B_m)$ to $\cC(B)$, |
214 and these various $m$-fold composition maps satisfy an |
226 and these various $m$-fold composition maps satisfy an |
215 operad-type strict associativity condition.} |
227 operad-type strict associativity condition (Figure \ref{blah7}).} |
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228 |
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229 \begin{figure}[!ht] |
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230 $$\mathfig{.8}{tempkw/blah7}$$ |
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231 \caption{Operadish composition and associativity}\label{blah7}\end{figure} |
216 |
232 |
217 The next axiom is related to identity morphisms, though that might not be immediately obvious. |
233 The next axiom is related to identity morphisms, though that might not be immediately obvious. |
218 |
234 |
219 \xxpar{Product (identity) morphisms:} |
235 \xxpar{Product (identity) morphisms:} |
220 {Let $X$ be a $k$-ball and $D$ be an $m$-ball, with $k+m \le n$. |
236 {Let $X$ be a $k$-ball and $D$ be an $m$-ball, with $k+m \le n$. |