23 \begin{thm} \label{product_thm} |
23 \begin{thm} \label{product_thm} |
24 The old-fashioned blob complex $\bc_*^C(Y\times F)$ is homotopy equivalent to the |
24 The old-fashioned blob complex $\bc_*^C(Y\times F)$ is homotopy equivalent to the |
25 new-fangled blob complex $\bc_*^\cF(Y)$. |
25 new-fangled blob complex $\bc_*^\cF(Y)$. |
26 \end{thm} |
26 \end{thm} |
27 |
27 |
28 \begin{proof} |
28 \input{text/smallblobs} |
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29 |
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30 \begin{proof}[Proof of Theorem \ref{product_thm}] |
29 We will use the concrete description of the colimit from Subsection \ref{ss:ncat_fields}. |
31 We will use the concrete description of the colimit from Subsection \ref{ss:ncat_fields}. |
30 |
32 |
31 First we define a map |
33 First we define a map |
32 \[ |
34 \[ |
33 \psi: \bc_*^\cF(Y) \to \bc_*^C(Y\times F) . |
35 \psi: \bc_*^\cF(Y) \to \bc_*^C(Y\times F) . |
213 |
215 |
214 \nn{to be continued...} |
216 \nn{to be continued...} |
215 \medskip |
217 \medskip |
216 \nn{still to do: fiber bundles, general maps} |
218 \nn{still to do: fiber bundles, general maps} |
217 |
219 |
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220 \todo{} |
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221 Various citations we might want to make: |
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222 \begin{itemize} |
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223 \item \cite{MR2061854} McClure and Smith's review article |
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224 \item \cite{MR0420610} May, (inter alia, definition of $E_\infty$ operad) |
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225 \item \cite{MR0236922,MR0420609} Boardman and Vogt |
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226 \item \cite{MR1256989} definition of framed little-discs operad |
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227 \end{itemize} |
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228 |
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229 We now turn to establishing the gluing formula for blob homology, restated from Property \ref{property:gluing} in the Introduction |
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230 \begin{itemize} |
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231 %\mbox{}% <-- gets the indenting right |
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232 \item For any $(n-1)$-manifold $Y$, the blob homology of $Y \times I$ is |
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233 naturally an $A_\infty$ category. % We'll write $\bc_*(Y)$ for $\bc_*(Y \times I)$ below. |
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234 |
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235 \item For any $n$-manifold $X$, with $Y$ a codimension $0$-submanifold of its boundary, the blob homology of $X$ is naturally an |
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236 $A_\infty$ module for $\bc_*(Y \times I)$. |
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237 |
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238 \item For any $n$-manifold $X$, with $Y \cup Y^{\text{op}}$ a codimension |
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239 $0$-submanifold of its boundary, the blob homology of $X'$, obtained from |
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240 $X$ by gluing along $Y$, is the $A_\infty$ self-tensor product of |
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241 $\bc_*(X)$ as an $\bc_*(Y \times I)$-bimodule. |
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242 \begin{equation*} |
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243 \bc_*(X') \iso \bc_*(X) \Tensor^{A_\infty}_{\mathclap{\bc_*(Y \times I)}} \!\!\!\!\!\!\xymatrix{ \ar@(ru,rd)@<-1ex>[]} |
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244 \end{equation*} |
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245 \end{itemize} |
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246 |