190 Furthermore, up to homeomorphisms of $X\sgl$ isotopic to the identity |
190 Furthermore, up to homeomorphisms of $X\sgl$ isotopic to the identity |
191 and collaring maps, |
191 and collaring maps, |
192 the gluing map is surjective. |
192 the gluing map is surjective. |
193 We say that fields in the image of the gluing map |
193 We say that fields in the image of the gluing map |
194 are transverse to $Y$ or splittable along $Y$. |
194 are transverse to $Y$ or splittable along $Y$. |
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195 \item Splittings. |
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196 Let $c\in \cC_k(X)$ and let $Y\sub X$ be a codimension 1 properly embedded submanifold of $X$. |
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197 Then for most small perturbations of $Y$ (i.e.\ for an open dense |
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198 subset of such perturbations) $c$ splits along $Y$. |
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199 (In Example \ref{ex:maps-to-a-space(fields)}, $c$ splits along all such $Y$. |
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200 In Example \ref{ex:traditional-n-categories(fields)}, $c$ splits along $Y$ so long as $Y$ |
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201 is in general position with respect to the cell decomposition |
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202 associated to $c$.) |
195 \item Product fields. |
203 \item Product fields. |
196 There are maps $\cC_{k-1}(Y) \to \cC_k(Y \times I)$, denoted |
204 There are maps $\cC_{k-1}(Y) \to \cC_k(Y \times I)$, denoted |
197 $c \mapsto c\times I$. |
205 $c \mapsto c\times I$. |
198 These maps comprise a natural transformation of functors, and commute appropriately |
206 These maps comprise a natural transformation of functors, and commute appropriately |
199 with all the structure maps above (disjoint union, boundary restriction, etc.). |
207 with all the structure maps above (disjoint union, boundary restriction, etc.). |