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194 the gluing map is surjective. |
194 the gluing map is surjective. |
195 We say that fields in the image of the gluing map |
195 We say that fields in the image of the gluing map |
196 are transverse to $Y$ or splittable along $Y$. |
196 are transverse to $Y$ or splittable along $Y$. |
197 \item Splittings. |
197 \item Splittings. |
198 Let $c\in \cC_k(X)$ and let $Y\sub X$ be a codimension 1 properly embedded submanifold of $X$. |
198 Let $c\in \cC_k(X)$ and let $Y\sub X$ be a codimension 1 properly embedded submanifold of $X$. |
199 Then for most small perturbations of $Y$ (i.e.\ for an open dense |
199 Then for most small perturbations of $Y$ (e.g.\ for an open dense |
200 subset of such perturbations) $c$ splits along $Y$. |
200 subset of such perturbations, or for all perturbations satisfying |
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201 a transversality condition) $c$ splits along $Y$. |
201 (In Example \ref{ex:maps-to-a-space(fields)}, $c$ splits along all such $Y$. |
202 (In Example \ref{ex:maps-to-a-space(fields)}, $c$ splits along all such $Y$. |
202 In Example \ref{ex:traditional-n-categories(fields)}, $c$ splits along $Y$ so long as $Y$ |
203 In Example \ref{ex:traditional-n-categories(fields)}, $c$ splits along $Y$ so long as $Y$ |
203 is in general position with respect to the cell decomposition |
204 is in general position with respect to the cell decomposition |
204 associated to $c$.) |
205 associated to $c$.) |
205 \item Product fields. |
206 \item Product fields. |