13 When we need to distinguish between the new and old definitions, we will refer to the |
13 When we need to distinguish between the new and old definitions, we will refer to the |
14 new-fangled and old-fashioned blob complex. |
14 new-fangled and old-fashioned blob complex. |
15 |
15 |
16 \medskip |
16 \medskip |
17 |
17 |
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18 \subsection{The small blob complex} |
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19 |
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20 \input{text/smallblobs} |
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21 |
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22 \subsection{A product formula} |
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23 |
18 Let $M^n = Y^k\times F^{n-k}$. |
24 Let $M^n = Y^k\times F^{n-k}$. |
19 Let $C$ be a plain $n$-category. |
25 Let $C$ be a plain $n$-category. |
20 Let $\cF$ be the $A_\infty$ $k$-category which assigns to a $k$-ball |
26 Let $\cF$ be the $A_\infty$ $k$-category which assigns to a $k$-ball |
21 $X$ the old-fashioned blob complex $\bc_*(X\times F)$. |
27 $X$ the old-fashioned blob complex $\bc_*(X\times F)$. |
22 |
28 |
23 \begin{thm} \label{product_thm} |
29 \begin{thm} \label{product_thm} |
24 The old-fashioned blob complex $\bc_*^C(Y\times F)$ is homotopy equivalent to the |
30 The old-fashioned blob complex $\bc_*^C(Y\times F)$ is homotopy equivalent to the |
25 new-fangled blob complex $\bc_*^\cF(Y)$. |
31 new-fangled blob complex $\bc_*^\cF(Y)$. |
26 \end{thm} |
32 \end{thm} |
27 |
33 |
28 \input{text/smallblobs} |
34 |
29 |
35 |
30 \begin{proof}[Proof of Theorem \ref{product_thm}] |
36 \begin{proof}[Proof of Theorem \ref{product_thm}] |
31 We will use the concrete description of the colimit from Subsection \ref{ss:ncat_fields}. |
37 We will use the concrete description of the colimit from Subsection \ref{ss:ncat_fields}. |
32 |
38 |
33 First we define a map |
39 First we define a map |
211 Apply Theorem \ref{product_thm} with the fiber $F$ equal to a point. |
217 Apply Theorem \ref{product_thm} with the fiber $F$ equal to a point. |
212 \end{proof} |
218 \end{proof} |
213 |
219 |
214 \medskip |
220 \medskip |
215 |
221 |
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222 \subsection{A gluing theorem} |
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223 \label{sec:gluing} |
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224 |
216 Next we prove a gluing theorem. |
225 Next we prove a gluing theorem. |
217 Let $X$ be a closed $k$-manifold with a splitting $X = X'_1\cup_Y X'_2$. |
226 Let $X$ be a closed $k$-manifold with a splitting $X = X'_1\cup_Y X'_2$. |
218 We will need an explicit collar on $Y$, so rewrite this as |
227 We will need an explicit collar on $Y$, so rewrite this as |
219 $X = X_1\cup (Y\times J) \cup X_2$. |
228 $X = X_1\cup (Y\times J) \cup X_2$. |
220 \nn{need figure} |
229 \nn{need figure} |
228 $m$-ball $(D, H)$ either fields on $(D\times Y) \cup (H\times X_i)$ (if $m+k < n$) |
237 $m$-ball $(D, H)$ either fields on $(D\times Y) \cup (H\times X_i)$ (if $m+k < n$) |
229 or the blob complex $\bc_*((D\times Y) \cup (H\times X_i))$ (if $m+k = n$). |
238 or the blob complex $\bc_*((D\times Y) \cup (H\times X_i))$ (if $m+k = n$). |
230 \end{itemize} |
239 \end{itemize} |
231 |
240 |
232 \begin{thm} |
241 \begin{thm} |
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242 \label{thm:gluing} |
233 $\bc(X) \cong \bc(X_1) \otimes_{\bc(Y), J} \bc(X_2)$. |
243 $\bc(X) \cong \bc(X_1) \otimes_{\bc(Y), J} \bc(X_2)$. |
234 \end{thm} |
244 \end{thm} |
235 |
245 |
236 \begin{proof} |
246 \begin{proof} |
237 The proof is similar to that of Theorem \ref{product_thm}. |
247 The proof is similar to that of Theorem \ref{product_thm}. |
251 \end{proof} |
261 \end{proof} |
252 |
262 |
253 This establishes Property \ref{property:gluing}. |
263 This establishes Property \ref{property:gluing}. |
254 |
264 |
255 \medskip |
265 \medskip |
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266 |
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267 \subsection{Reconstructing mapping spaces} |
256 |
268 |
257 The next theorem shows how to reconstruct a mapping space from local data. |
269 The next theorem shows how to reconstruct a mapping space from local data. |
258 Let $T$ be a topological space, let $M$ be an $n$-manifold, |
270 Let $T$ be a topological space, let $M$ be an $n$-manifold, |
259 and recall the $A_\infty$ $n$-category $\pi^\infty_{\leq n}(T)$ |
271 and recall the $A_\infty$ $n$-category $\pi^\infty_{\leq n}(T)$ |
260 of Example \ref{ex:chains-of-maps-to-a-space}. |
272 of Example \ref{ex:chains-of-maps-to-a-space}. |