212 in the case that $X = X_1 \du X_2$, with $x_i \in \cC(X_i)$. |
212 in the case that $X = X_1 \du X_2$, with $x_i \in \cC(X_i)$. |
213 |
213 |
214 \medskip |
214 \medskip |
215 |
215 |
216 |
216 |
217 Using the functoriality and product field properties above, together |
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218 with boundary collar homeomorphisms of manifolds, we can define |
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219 {\it collar maps} $\cC(M)\to \cC(M)$. |
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220 Let $M$ be an $n$-manifold and $Y \subset \bd M$ be a codimension zero submanifold |
217 Let $M$ be an $n$-manifold and $Y \subset \bd M$ be a codimension zero submanifold |
221 of $\bd M$. |
218 of $\bd M$. |
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219 Let $M \cup (Y\times I)$ denote $M$ glued to $Y\times I$ along $Y$. |
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220 Extend the product structure on $Y\times I$ to a bicollar neighborhood of |
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221 $Y$ inside $M \cup (Y\times I)$. |
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222 We call a homeomorphism |
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223 \[ |
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224 f: M \cup (Y\times I) \to M |
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225 \] |
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226 a {\it collaring homeomorphism} if $f$ is the identity outside of the bicollar |
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227 and $f$ preserves the fibers of the bicollar. |
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228 |
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229 Using the functoriality and product field properties above, together |
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230 with collaring homeomorphisms, we can define |
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231 {\it collar maps} $\cC(M)\to \cC(M)$. |
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232 Let $M$ and $Y \sub \bd M$ be as above. |
222 Let $x \in \cC(M)$ be a field on $M$ and such that $\bd x$ is splittable along $\bd Y$. |
233 Let $x \in \cC(M)$ be a field on $M$ and such that $\bd x$ is splittable along $\bd Y$. |
223 Let $c$ be $x$ restricted to $Y$. |
234 Let $c$ be $x$ restricted to $Y$. |
224 Let $M \cup (Y\times I)$ denote $M$ glued to $Y\times I$ along $Y$. |
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225 Then we have the glued field $x \bullet (c\times I)$ on $M \cup (Y\times I)$. |
235 Then we have the glued field $x \bullet (c\times I)$ on $M \cup (Y\times I)$. |
226 Let $f: M \cup (Y\times I) \to M$ be a collaring homeomorphism. |
236 Let $f: M \cup (Y\times I) \to M$ be a collaring homeomorphism. |
227 Then we call the map $x \mapsto f(x \bullet (c\times I))$ a {\it collar map}. |
237 Then we call the map $x \mapsto f(x \bullet (c\times I))$ a {\it collar map}. |
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238 |
228 We call the equivalence relation generated by collar maps and |
239 We call the equivalence relation generated by collar maps and |
229 homeomorphisms isotopic to the identity {\it extended isotopy}, since the collar maps |
240 homeomorphisms isotopic to the identity {\it extended isotopy}, since the collar maps |
230 can be thought of (informally) as the limit of homeomorphisms |
241 can be thought of (informally) as the limit of homeomorphisms |
231 which expand an infinitesimally thin collar neighborhood of $Y$ to a thicker |
242 which expand an infinitesimally thin collar neighborhood of $Y$ to a thicker |
232 collar neighborhood. |
243 collar neighborhood. |