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224 \end{axiom} |
224 \end{axiom} |
225 |
225 |
226 Note that the functoriality in the above axiom allows us to operate via |
226 Note that the functoriality in the above axiom allows us to operate via |
227 homeomorphisms which are not the identity on the boundary of the $k$-ball. |
227 homeomorphisms which are not the identity on the boundary of the $k$-ball. |
228 The action of these homeomorphisms gives the ``strong duality" structure. |
228 The action of these homeomorphisms gives the ``strong duality" structure. |
229 |
229 As such, we don't subdivide the boundary of a morphism |
230 Next we consider domains and ranges of $k$-morphisms. |
230 into domain and range --- the duality operations can convert between domain and range. |
231 Because we assume strong duality, it doesn't make much sense to subdivide the boundary of a morphism |
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232 into domain and range --- the duality operations can convert domain to range and vice-versa. |
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233 Instead, we will use a unified domain/range, which we will call a ``boundary". |
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234 |
231 |
235 Later \todo{} we inductively define an extension of the functors $\cC_k$ to functors $\cl{\cC}_k$ from arbitrary manifolds to sets. We need the restriction of these functors to $k$-spheres, for $k<n$, for the next axiom. |
232 Later \todo{} we inductively define an extension of the functors $\cC_k$ to functors $\cl{\cC}_k$ from arbitrary manifolds to sets. We need the restriction of these functors to $k$-spheres, for $k<n$, for the next axiom. |
236 |
233 |
237 \begin{axiom}[Boundaries]\label{nca-boundary} |
234 \begin{axiom}[Boundaries]\label{nca-boundary} |
238 For each $k$-ball $X$, we have a map of sets $\bd: \cC_k(X)\to \cl{\cC}_{k-1}(\bd X)$. |
235 For each $k$-ball $X$, we have a map of sets $\bd: \cC_k(X)\to \cl{\cC}_{k-1}(\bd X)$. |
240 \end{axiom} |
237 \end{axiom} |
241 |
238 |
242 For $c\in \cl{\cC}_{k-1}(\bd X)$ we define $\cC_k(X; c) = \bd^{-1}(c)$. |
239 For $c\in \cl{\cC}_{k-1}(\bd X)$ we define $\cC_k(X; c) = \bd^{-1}(c)$. |
243 |
240 |
244 Many of the examples we are interested in are enriched in some auxiliary category $\cS$ |
241 Many of the examples we are interested in are enriched in some auxiliary category $\cS$ |
245 (e.g. $\cS$ is vector spaces or rings, or, in the $A_\infty$ case, chain complex or topological spaces). |
242 (e.g. vector spaces or rings, or, in the $A_\infty$ case, chain complex or topological spaces). |
246 This means (by definition) that in the top dimension $k=n$ the sets $\cC_n(X; c)$ have the structure |
243 This means that in the top dimension $k=n$ the sets $\cC_n(X; c)$ have the structure |
247 of an object of $\cS$, and all of the structure maps of the category (above and below) are |
244 of an object of $\cS$, and all of the structure maps of the category (above and below) are |
248 compatible with the $\cS$ structure on $\cC_n(X; c)$. |
245 compatible with the $\cS$ structure on $\cC_n(X; c)$. |
249 |
246 |
250 |
247 |
251 Given two hemispheres (a `domain' and `range') that agree on the equator, we need to be able to assemble them into a boundary value of the entire sphere. |
248 Given two hemispheres (a `domain' and `range') that agree on the equator, we need to be able to assemble them into a boundary value of the entire sphere. |
296 For the next axiom, a \emph{pinched product} is a map locally modeled on a degeneracy map between simplices. |
293 For the next axiom, a \emph{pinched product} is a map locally modeled on a degeneracy map between simplices. |
297 \begin{axiom}[Product (identity) morphisms] |
294 \begin{axiom}[Product (identity) morphisms] |
298 \label{axiom:product} |
295 \label{axiom:product} |
299 For each pinched product $\pi:E\to X$, with $X$ a $k$-ball and $E$ a $k{+}m$-ball ($m\ge 1$), |
296 For each pinched product $\pi:E\to X$, with $X$ a $k$-ball and $E$ a $k{+}m$-ball ($m\ge 1$), |
300 there is a map $\pi^*:\cC(X)\to \cC(E)$. |
297 there is a map $\pi^*:\cC(X)\to \cC(E)$. |
298 These maps must be |
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299 \begin{enumerate} |
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300 \item natural with respect to maps of pinched products, |
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301 \item functorial with respect to composition of pinched products, |
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302 \item compatible with gluing and restriction of pinched products. |
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303 \end{enumerate} |
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304 |
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305 %%% begin noop %%% |
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306 % this was the original list of conditions, which I've replaced with the much terser list above -S |
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307 \noop{ |
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301 These maps must satisfy the following conditions. |
308 These maps must satisfy the following conditions. |
302 \begin{enumerate} |
309 \begin{enumerate} |
303 \item |
310 \item |
304 If $\pi:E\to X$ and $\pi':E'\to X'$ are pinched products, and |
311 If $\pi:E\to X$ and $\pi':E'\to X'$ are pinched products, and |
305 if $f:X\to X'$ and $\tilde{f}:E \to E'$ are maps such that the diagram |
312 if $f:X\to X'$ and $\tilde{f}:E \to E'$ are maps such that the diagram |
336 such that $\rho$ and $\pi$ are pinched products, then |
343 such that $\rho$ and $\pi$ are pinched products, then |
337 \[ |
344 \[ |
338 \res_D\circ\pi^* = \rho^*\circ\res_Y . |
345 \res_D\circ\pi^* = \rho^*\circ\res_Y . |
339 \] |
346 \] |
340 \end{enumerate} |
347 \end{enumerate} |
348 } %%% end \noop %%% |
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341 \end{axiom} |
349 \end{axiom} |
342 \begin{axiom}[\textup{\textbf{[plain version]}} Extended isotopy invariance in dimension $n$.] |
350 \begin{axiom}[\textup{\textbf{[plain version]}} Extended isotopy invariance in dimension $n$.] |
343 \label{axiom:extended-isotopies} |
351 \label{axiom:extended-isotopies} |
344 Let $X$ be an $n$-ball and $f: X\to X$ be a homeomorphism which restricts |
352 Let $X$ be an $n$-ball and $f: X\to X$ be a homeomorphism which restricts |
345 to the identity on $\bd X$ and isotopic (rel boundary) to the identity. |
353 to the identity on $\bd X$ and isotopic (rel boundary) to the identity. |
674 |
682 |
675 An $n$-dimensional surgery cylinder is a sequence of mapping cylinders and surgeries (Figure \ref{delfig2}), modulo changing the order of distant surgeries, and conjugating a submanifold not modified in a surgery by a homeomorphism. Surgery cylinders form an operad, by gluing the outer boundary of one cylinder into an inner boundary of another. |
683 An $n$-dimensional surgery cylinder is a sequence of mapping cylinders and surgeries (Figure \ref{delfig2}), modulo changing the order of distant surgeries, and conjugating a submanifold not modified in a surgery by a homeomorphism. Surgery cylinders form an operad, by gluing the outer boundary of one cylinder into an inner boundary of another. |
676 |
684 |
677 By the `blob cochains' of a manifold $X$, we mean the $A_\infty$ maps of $\bc_*(X)$ as a $\bc_*(\bdy X)$ $A_\infty$-module. |
685 By the `blob cochains' of a manifold $X$, we mean the $A_\infty$ maps of $\bc_*(X)$ as a $\bc_*(\bdy X)$ $A_\infty$-module. |
678 |
686 |
679 \todo{Sketch proof} |
687 \begin{proof} |
688 We have already defined the action of mapping cylinders, in Theorem \ref{thm:evaluation}, and the action of surgeries is just composition of maps of $A_\infty$-modules. We only need to check that the relations of the $n$-SC operad are satisfied. This follows immediately from the locality of the action of $\CH{-}$ (i.e., that it is compatible with gluing) and functoriality. |
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689 \end{proof} |
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680 |
690 |
681 The little disks operad $LD$ is homotopy equivalent to the $n=1$ case of the $n$-SC operad. The blob complex $\bc_*(I, \cC)$ is a bimodule over itself, and the $A_\infty$-bimodule intertwiners are homotopy equivalent to the Hochschild cohains $Hoch^*(C, C)$. The usual Deligne conjecture (proved variously in \cite{hep-th/9403055, MR1805894, MR2064592, MR1805923}) gives a map |
691 The little disks operad $LD$ is homotopy equivalent to the $n=1$ case of the $n$-SC operad. The blob complex $\bc_*(I, \cC)$ is a bimodule over itself, and the $A_\infty$-bimodule intertwiners are homotopy equivalent to the Hochschild cohains $Hoch^*(C, C)$. The usual Deligne conjecture (proved variously in \cite{hep-th/9403055, MR1805894, MR2064592, MR1805923}) gives a map |
682 \[ |
692 \[ |
683 C_*(LD_k)\otimes \overbrace{Hoch^*(C, C)\otimes\cdots\otimes Hoch^*(C, C)}^{\text{$k$ copies}} |
693 C_*(LD_k)\otimes \overbrace{Hoch^*(C, C)\otimes\cdots\otimes Hoch^*(C, C)}^{\text{$k$ copies}} |
684 \to Hoch^*(C, C), |
694 \to Hoch^*(C, C), |
701 %% \appendix Appendix text... |
711 %% \appendix Appendix text... |
702 %% or, for appendix with title, use square brackets: |
712 %% or, for appendix with title, use square brackets: |
703 %% \appendix[Appendix Title] |
713 %% \appendix[Appendix Title] |
704 |
714 |
705 \begin{acknowledgments} |
715 \begin{acknowledgments} |
706 -- text of acknowledgments here, including grant info -- |
716 \nn{say something here} |
707 \end{acknowledgments} |
717 \end{acknowledgments} |
708 |
718 |
709 %% PNAS does not support submission of supporting .tex files such as BibTeX. |
719 %% PNAS does not support submission of supporting .tex files such as BibTeX. |
710 %% Instead all references must be included in the article .tex document. |
720 %% Instead all references must be included in the article .tex document. |
711 %% If you currently use BibTeX, your bibliography is formed because the |
721 %% If you currently use BibTeX, your bibliography is formed because the |