166 \begin{proof} |
166 \begin{proof} |
167 Apply Theorem \ref{product_thm} with the fiber $F$ equal to a point. |
167 Apply Theorem \ref{product_thm} with the fiber $F$ equal to a point. |
168 \end{proof} |
168 \end{proof} |
169 |
169 |
170 \medskip |
170 \medskip |
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171 |
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172 Next we prove a gluing theorem. |
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173 Let $X$ be a closed $k$-manifold with a splitting $X = X'_1\cup_Y X'_2$. |
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174 We will need an explicit collar on $Y$, so rewrite this as |
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175 $X = X_1\cup (Y\times J) \cup X_2$. |
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176 \nn{need figure} |
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177 Given this data we have: \nn{need refs to above for these} |
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178 \begin{itemize} |
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179 \item An $A_\infty$ $n{-}k$-category $\bc(X)$, which assigns to an $m$-ball |
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180 $D$ fields on $D\times X$ (for $m+k < n$) or the blob complex $\bc_*(D\times X; c)$ |
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181 (for $m+k = n$). \nn{need to explain $c$}. |
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182 \item An $A_\infty$ $n{-}k{+}1$-category $\bc(Y)$, defined similarly. |
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183 \item Two $\bc(Y)$ modules $\bc(X_1)$ and $\bc(X_2)$, which assign to a marked |
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184 $m$-ball $(D, H)$ either fields on $(D\times Y) \cup (H\times X_i)$ (if $m+k < n$) |
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185 or the blob complex $\bc_*((D\times Y) \cup (H\times X_i))$ (if $m+k = n$). |
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186 \end{itemize} |
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187 |
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188 \begin{thm} |
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189 $\bc(X) \cong \bc(X_1) \otimes_{\bc(Y), J} \bc(X_2)$. |
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190 \end{thm} |
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191 |
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192 \begin{proof} |
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193 The proof is similar to that of Theorem \ref{product_thm}. |
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194 \nn{need to say something about dimensions less than $n$, |
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195 but for now concentrate on top dimension.} |
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196 |
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197 Let $\cT$ denote the $n{-}k$-category $\bc(X_1) \otimes_{\bc(Y), J} \bc(X_2)$. |
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198 Let $D$ be an $n{-}k$-ball. |
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199 There is an obvious map from $\cT(D)$ to $\bc_*(D\times X)$. |
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200 To get a map in the other direction, we replace $\bc_*(D\times X)$ with a subcomplex |
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201 $\cS_*$ which is adapted to a fine open cover of $D\times X$. |
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202 For sufficiently small $j$ (depending on the cover), we can find, for each $j$-blob diagram $b$ |
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203 on $D\times X$, a decomposition of $J$ such that $b$ splits on the corresponding |
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204 decomposition of $D\times X$. |
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205 The proof that these two maps are inverse to each other is the same as in |
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206 Theorem \ref{product_thm}. |
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207 \end{proof} |
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208 |
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209 |
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210 \medskip |
171 \hrule |
211 \hrule |
172 \medskip |
212 \medskip |
173 |
213 |
174 \nn{to be continued...} |
214 \nn{to be continued...} |
175 \medskip |
215 \medskip |
176 |
216 \nn{still to do: fiber bundles, general maps} |
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217 |