159 %% \section{} |
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164 \input{../text/intro} |
163 \section{} |
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164 |
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165 \nn{ |
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166 background: TQFTs are important, historically, semisimple categories well-understood. |
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167 Many new examples arising recently which do not fit this framework, e.g. SW and OS theory. |
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168 These have more complicated gluing formulas (\cite{1003.0598,1005.1248}, etc); |
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169 it would be nice to give generalized TQFT axioms that encompass these. |
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170 Triangulated categories are important; often calculations are via exact sequences, |
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171 and the standard TQFT constructions are quotients, which destroy exactness. |
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172 A first attempt to deal with this might be to replace all the tensor products in gluing formulas |
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173 with derived tensor products (cite Kh?). |
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174 However, in this approach it's probably difficult to prove invariance of constructions, |
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175 because they depend on explicit presentations of the manifold. |
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176 We'll give a manifestly invariant construction, |
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177 and deduce gluing formulas based on derived (actually, $A_\infty$) tensor products.} |
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178 |
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179 \section{Definitions} |
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180 \subsection{$n$-categories} |
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181 \nn{ |
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182 Axioms for $n$-categories, examples (maps, string diagrams) |
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183 } |
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184 \nn{ |
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185 Decide if we need a friendlier, skein-module version. |
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186 } |
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187 \subsection{The blob complex} |
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188 \subsubsection{Decompositions of manifolds} |
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189 \nn{Mention that the axioms for $n$-categories can be stated in terms of decompositions of balls} |
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190 \subsubsection{Homotopy colimits} |
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191 \nn{How can we extend an $n$-category from balls to arbitrary manifolds?} |
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192 |
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193 \nn{In practice, this gives the old definition} |
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194 \subsubsection{} |
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195 \section{Properties of the blob complex} |
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196 \subsection{Formal properties} |
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197 \label{sec:properties} |
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198 The blob complex enjoys the following list of formal properties. |
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199 |
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200 \begin{property}[Functoriality] |
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201 \label{property:functoriality}% |
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202 The blob complex is functorial with respect to homeomorphisms. |
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203 That is, |
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204 for a fixed $n$-dimensional system of fields $\cF$, the association |
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205 \begin{equation*} |
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206 X \mapsto \bc_*(X; \cF) |
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207 \end{equation*} |
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208 is a functor from $n$-manifolds and homeomorphisms between them to chain |
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209 complexes and isomorphisms between them. |
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210 \end{property} |
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211 As a consequence, there is an action of $\Homeo(X)$ on the chain complex $\bc_*(X; \cF)$; |
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212 this action is extended to all of $C_*(\Homeo(X))$ in Theorem \ref{thm:CH} below. |
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213 |
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214 \begin{property}[Disjoint union] |
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215 \label{property:disjoint-union} |
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216 The blob complex of a disjoint union is naturally isomorphic to the tensor product of the blob complexes. |
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217 \begin{equation*} |
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218 \bc_*(X_1 \du X_2) \iso \bc_*(X_1) \tensor \bc_*(X_2) |
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219 \end{equation*} |
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220 \end{property} |
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221 |
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222 If an $n$-manifold $X$ contains $Y \sqcup Y^\text{op}$ as a codimension $0$ submanifold of its boundary, |
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223 write $X_\text{gl} = X \bigcup_{Y}\selfarrow$ for the manifold obtained by gluing together $Y$ and $Y^\text{op}$. |
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224 Note that this includes the case of gluing two disjoint manifolds together. |
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225 \begin{property}[Gluing map] |
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226 \label{property:gluing-map}% |
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227 %If $X_1$ and $X_2$ are $n$-manifolds, with $Y$ a codimension $0$-submanifold of $\bdy X_1$, and $Y^{\text{op}}$ a codimension $0$-submanifold of $\bdy X_2$, there is a chain map |
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228 %\begin{equation*} |
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229 %\gl_Y: \bc_*(X_1) \tensor \bc_*(X_2) \to \bc_*(X_1 \cup_Y X_2). |
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230 %\end{equation*} |
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231 Given a gluing $X \to X_\mathrm{gl}$, there is |
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232 a natural map |
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233 \[ |
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234 \bc_*(X) \to \bc_*(X_\mathrm{gl}) |
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235 \] |
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236 (natural with respect to homeomorphisms, and also associative with respect to iterated gluings). |
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237 \end{property} |
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238 |
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239 \begin{property}[Contractibility] |
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240 \label{property:contractibility}% |
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241 With field coefficients, the blob complex on an $n$-ball is contractible in the sense |
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242 that it is homotopic to its $0$-th homology. |
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243 Moreover, the $0$-th homology of balls can be canonically identified with the vector spaces |
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244 associated by the system of fields $\cF$ to balls. |
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245 \begin{equation*} |
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246 \xymatrix{\bc_*(B^n;\cF) \ar[r]^(0.4){\iso}_(0.4){\text{qi}} & H_0(\bc_*(B^n;\cF)) \ar[r]^(0.6)\iso & A_\cF(B^n)} |
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247 \end{equation*} |
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248 \end{property} |
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249 |
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250 \nn{Properties \ref{property:functoriality} will be immediate from the definition given in |
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251 \S \ref{sec:blob-definition}, and we'll recall it at the appropriate point there. |
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252 Properties \ref{property:disjoint-union}, \ref{property:gluing-map} and |
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253 \ref{property:contractibility} are established in \S \ref{sec:basic-properties}.} |
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254 |
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255 \subsection{Specializations} |
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256 \label{sec:specializations} |
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257 |
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258 The blob complex is a simultaneous generalization of the TQFT skein module construction and of Hochschild homology. |
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259 |
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260 \begin{thm}[Skein modules] |
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261 \label{thm:skein-modules} |
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262 The $0$-th blob homology of $X$ is the usual |
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263 (dual) TQFT Hilbert space (a.k.a.\ skein module) associated to $X$ |
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264 by $\cF$. |
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265 \begin{equation*} |
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266 H_0(\bc_*(X;\cF)) \iso A_{\cF}(X) |
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267 \end{equation*} |
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268 \end{thm} |
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269 |
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270 \begin{thm}[Hochschild homology when $X=S^1$] |
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271 \label{thm:hochschild} |
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272 The blob complex for a $1$-category $\cC$ on the circle is |
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273 quasi-isomorphic to the Hochschild complex. |
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274 \begin{equation*} |
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275 \xymatrix{\bc_*(S^1;\cC) \ar[r]^(0.47){\iso}_(0.47){\text{qi}} & \HC_*(\cC).} |
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276 \end{equation*} |
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277 \end{thm} |
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278 |
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279 Proposition \ref{thm:skein-modules} is immediate from the definition, and |
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280 Theorem \ref{thm:hochschild} is established by extending the statement to bimodules as well as categories, then verifying that the universal properties of Hochschild homology also hold for $\bc_*(S^1; -)$. |
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281 |
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282 |
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283 \subsection{Structure of the blob complex} |
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284 \label{sec:structure} |
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285 |
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286 In the following $\CH{X} = C_*(\Homeo(X))$ is the singular chain complex of the space of homeomorphisms of $X$, fixed on $\bdy X$. |
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287 |
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288 \begin{thm}[$C_*(\Homeo(-))$ action] |
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289 \label{thm:CH}\label{thm:evaluation} |
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290 There is a chain map |
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291 \begin{equation*} |
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292 e_X: \CH{X} \tensor \bc_*(X) \to \bc_*(X). |
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293 \end{equation*} |
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294 such that |
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295 \begin{enumerate} |
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296 \item Restricted to $CH_0(X)$ this is the action of homeomorphisms described in Property \ref{property:functoriality}. |
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297 |
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298 \item For |
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299 any codimension $0$-submanifold $Y \sqcup Y^\text{op} \subset \bdy X$ the following diagram |
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300 (using the gluing maps described in Property \ref{property:gluing-map}) commutes (up to homotopy). |
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301 \begin{equation*} |
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302 \xymatrix@C+0.3cm{ |
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303 \CH{X} \otimes \bc_*(X) |
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304 \ar[r]_{e_{X}} \ar[d]^{\gl^{\Homeo}_Y \otimes \gl_Y} & |
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305 \bc_*(X) \ar[d]_{\gl_Y} \\ |
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306 \CH{X \bigcup_Y \selfarrow} \otimes \bc_*(X \bigcup_Y \selfarrow) \ar[r]_<<<<<<<{e_{(X \bigcup_Y \scalebox{0.5}{\selfarrow})}} & \bc_*(X \bigcup_Y \selfarrow) |
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307 } |
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308 \end{equation*} |
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309 \end{enumerate} |
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310 |
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311 Futher, this map is associative, in the sense that the following diagram commutes (up to homotopy). |
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312 \begin{equation*} |
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313 \xymatrix{ |
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314 \CH{X} \tensor \CH{X} \tensor \bc_*(X) \ar[r]^<<<<<{\id \tensor e_X} \ar[d]^{\compose \tensor \id} & \CH{X} \tensor \bc_*(X) \ar[d]^{e_X} \\ |
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315 \CH{X} \tensor \bc_*(X) \ar[r]^{e_X} & \bc_*(X) |
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316 } |
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317 \end{equation*} |
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318 \end{thm} |
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319 |
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320 Since the blob complex is functorial in the manifold $X$, this is equivalent to having chain maps |
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321 $$ev_{X \to Y} : \CH{X \to Y} \tensor \bc_*(X) \to \bc_*(Y)$$ |
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322 for any homeomorphic pair $X$ and $Y$, |
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323 satisfying corresponding conditions. |
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324 |
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325 \begin{thm}[Blob complexes of products with balls form an $A_\infty$ $n$-category] |
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326 \label{thm:blobs-ainfty} |
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327 Let $\cC$ be a topological $n$-category. |
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328 Let $Y$ be an $n{-}k$-manifold. |
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329 There is an $A_\infty$ $k$-category $\bc_*(Y;\cC)$, defined on each $m$-ball $D$, for $0 \leq m < k$, |
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330 to be the set $$\bc_*(Y;\cC)(D) = \cC(Y \times D)$$ and on $k$-balls $D$ to be the set |
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331 $$\bc_*(Y;\cC)(D) = \bc_*(Y \times D; \cC).$$ |
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332 (When $m=k$ the subsets with fixed boundary conditions form a chain complex.) |
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333 These sets have the structure of an $A_\infty$ $k$-category, with compositions coming from the gluing map in |
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334 Property \ref{property:gluing-map} and with the action of families of homeomorphisms given in Theorem \ref{thm:evaluation}. |
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335 \end{thm} |
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336 \begin{rem} |
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337 Perhaps the most interesting case is when $Y$ is just a point; then we have a way of building an $A_\infty$ $n$-category from a topological $n$-category. |
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338 We think of this $A_\infty$ $n$-category as a free resolution. |
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339 \end{rem} |
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340 This result is described in more detail as Example 6.2.8 of \cite{1009.5025} |
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341 |
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342 The definition is in fact simpler, almost tautological, and we use a different notation, $\cl{\cC}(M)$. The next theorem describes the blob complex for product manifolds, in terms of the $A_\infty$ blob complex of the $A_\infty$ $n$-categories constructed as in the previous example. |
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343 %The notation is intended to reflect the close parallel with the definition of the TQFT skein module via a colimit. |
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344 |
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345 \newtheorem*{thm:product}{Theorem \ref{thm:product}} |
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346 |
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347 \begin{thm}[Product formula] |
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348 \label{thm:product} |
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349 Let $W$ be a $k$-manifold and $Y$ be an $n-k$ manifold. |
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350 Let $\cC$ be an $n$-category. |
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351 Let $\bc_*(Y;\cC)$ be the $A_\infty$ $k$-category associated to $Y$ via blob homology (see Example \ref{ex:blob-complexes-of-balls}). |
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352 Then |
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353 \[ |
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354 \bc_*(Y\times W; \cC) \simeq \cl{\bc_*(Y;\cC)}(W). |
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355 \] |
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356 \end{thm} |
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357 The statement can be generalized to arbitrary fibre bundles, and indeed to arbitrary maps |
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358 (see \cite[\S7.1]{1009.5025}). |
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359 |
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360 Fix a topological $n$-category $\cC$, which we'll omit from the notation. |
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361 Recall that for any $(n-1)$-manifold $Y$, the blob complex $\bc_*(Y)$ is naturally an $A_\infty$ category. |
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362 |
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363 \begin{thm}[Gluing formula] |
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364 \label{thm:gluing} |
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365 \mbox{}% <-- gets the indenting right |
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366 \begin{itemize} |
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367 \item For any $n$-manifold $X$, with $Y$ a codimension $0$-submanifold of its boundary, the blob complex of $X$ is naturally an |
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368 $A_\infty$ module for $\bc_*(Y)$. |
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369 |
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370 \item For any $n$-manifold $X_\text{gl} = X\bigcup_Y \selfarrow$, the blob complex $\bc_*(X_\text{gl})$ is the $A_\infty$ self-tensor product of |
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371 $\bc_*(X)$ as an $\bc_*(Y)$-bimodule: |
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372 \begin{equation*} |
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373 \bc_*(X_\text{gl}) \simeq \bc_*(X) \Tensor^{A_\infty}_{\mathclap{\bc_*(Y)}} \selfarrow |
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374 \end{equation*} |
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375 \end{itemize} |
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376 \end{thm} |
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377 |
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378 \nn{Theorem \ref{thm:product} is proved in \S \ref{ss:product-formula}, and Theorem \ref{thm:gluing} in \S \ref{sec:gluing}.} |
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379 |
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380 \section{Applications} |
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381 \label{sec:applications} |
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382 Finally, we give two applications of the above machinery. |
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383 |
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384 \begin{thm}[Mapping spaces] |
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385 \label{thm:map-recon} |
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386 Let $\pi^\infty_{\le n}(T)$ denote the $A_\infty$ $n$-category based on maps |
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387 $B^n \to T$. |
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388 (The case $n=1$ is the usual $A_\infty$-category of paths in $T$.) |
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389 Then |
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390 $$\bc_*(X; \pi^\infty_{\le n}(T)) \simeq \CM{X}{T}.$$ |
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391 \end{thm} |
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392 |
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393 This says that we can recover (up to homotopy) the space of maps to $T$ via blob homology from local data. |
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394 Note that there is no restriction on the connectivity of $T$ as in \cite[Theorem 3.8.6]{0911.0018}. |
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395 \nn{The proof appears in \S \ref{sec:map-recon}.} |
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396 |
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397 |
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398 \begin{thm}[Higher dimensional Deligne conjecture] |
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399 \label{thm:deligne} |
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400 The singular chains of the $n$-dimensional surgery cylinder operad act on blob cochains. |
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401 Since the little $n{+}1$-balls operad is a suboperad of the $n$-dimensional surgery cylinder operad, |
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402 this implies that the little $n{+}1$-balls operad acts on blob cochains of the $n$-ball. |
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403 \end{thm} |
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404 \nn{See \S \ref{sec:deligne} for a full explanation of the statement, and the proof.} |
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405 |
165 |
406 |
166 |
407 |
167 %% == end of paper: |
408 %% == end of paper: |
168 |
409 |
169 %% Optional Materials and Methods Section |
410 %% Optional Materials and Methods Section |