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207 \label{property:functoriality}% |
207 \label{property:functoriality}% |
208 The blob complex is functorial with respect to homeomorphisms. |
208 The blob complex is functorial with respect to homeomorphisms. |
209 That is, |
209 That is, |
210 for a fixed $n$-dimensional system of fields $\cC$, the association |
210 for a fixed $n$-dimensional system of fields $\cC$, the association |
211 \begin{equation*} |
211 \begin{equation*} |
212 X \mapsto \bc_*^{\cC}(X) |
212 X \mapsto \bc_*(X; \cC) |
213 \end{equation*} |
213 \end{equation*} |
214 is a functor from $n$-manifolds and homeomorphisms between them to chain |
214 is a functor from $n$-manifolds and homeomorphisms between them to chain |
215 complexes and isomorphisms between them. |
215 complexes and isomorphisms between them. |
216 \end{property} |
216 \end{property} |
217 As a consequence, there is an action of $\Homeo(X)$ on the chain complex $\bc_*^\cC(X)$; |
217 As a consequence, there is an action of $\Homeo(X)$ on the chain complex $\bc_*(X; \cC)$; |
218 this action is extended to all of $C_*(\Homeo(X))$ in Theorem \ref{thm:evaluation} below. |
218 this action is extended to all of $C_*(\Homeo(X))$ in Theorem \ref{thm:evaluation} below. |
219 |
219 |
220 The blob complex is also functorial (indeed, exact) with respect to $\cC$, |
220 The blob complex is also functorial (indeed, exact) with respect to $\cC$, |
221 although we will not address this in detail here. |
221 although we will not address this in detail here. |
222 |
222 |
248 \begin{property}[Contractibility] |
248 \begin{property}[Contractibility] |
249 \label{property:contractibility}% |
249 \label{property:contractibility}% |
250 With field coefficients, the blob complex on an $n$-ball is contractible in the sense that it is homotopic to its $0$-th homology. |
250 With field coefficients, the blob complex on an $n$-ball is contractible in the sense that it is homotopic to its $0$-th homology. |
251 Moreover, the $0$-th homology of balls can be canonically identified with the vector spaces associated by the system of fields $\cC$ to balls. |
251 Moreover, the $0$-th homology of balls can be canonically identified with the vector spaces associated by the system of fields $\cC$ to balls. |
252 \begin{equation*} |
252 \begin{equation*} |
253 \xymatrix{\bc_*^{\cC}(B^n) \ar[r]^(0.4){\iso}_(0.4){\text{qi}} & H_0(\bc_*^{\cC}(B^n)) \ar[r]^(0.6)\iso & \cC(B^n)} |
253 \xymatrix{\bc_*(B^n;\cC) \ar[r]^(0.4){\iso}_(0.4){\text{qi}} & H_0(\bc_*(B^n;\cC)) \ar[r]^(0.6)\iso & \cC(B^n)} |
254 \end{equation*} |
254 \end{equation*} |
255 \end{property} |
255 \end{property} |
256 |
256 |
257 Properties \ref{property:functoriality} will be immediate from the definition given in |
257 Properties \ref{property:functoriality} will be immediate from the definition given in |
258 \S \ref{sec:blob-definition}, and we'll recall it at the appropriate point there. |
258 \S \ref{sec:blob-definition}, and we'll recall it at the appropriate point there. |
269 The $0$-th blob homology of $X$ is the usual |
269 The $0$-th blob homology of $X$ is the usual |
270 (dual) TQFT Hilbert space (a.k.a.\ skein module) associated to $X$ |
270 (dual) TQFT Hilbert space (a.k.a.\ skein module) associated to $X$ |
271 by $\cC$. |
271 by $\cC$. |
272 (See \S \ref{sec:local-relations}.) |
272 (See \S \ref{sec:local-relations}.) |
273 \begin{equation*} |
273 \begin{equation*} |
274 H_0(\bc_*^{\cC}(X)) \iso A^{\cC}(X) |
274 H_0(\bc_*(X;\cC)) \iso A_{\cC}(X) |
275 \end{equation*} |
275 \end{equation*} |
276 \end{thm:skein-modules} |
276 \end{thm:skein-modules} |
277 |
277 |
278 \newtheorem*{thm:hochschild}{Theorem \ref{thm:hochschild}} |
278 \newtheorem*{thm:hochschild}{Theorem \ref{thm:hochschild}} |
279 |
279 |
280 \begin{thm:hochschild}[Hochschild homology when $X=S^1$] |
280 \begin{thm:hochschild}[Hochschild homology when $X=S^1$] |
281 The blob complex for a $1$-category $\cC$ on the circle is |
281 The blob complex for a $1$-category $\cC$ on the circle is |
282 quasi-isomorphic to the Hochschild complex. |
282 quasi-isomorphic to the Hochschild complex. |
283 \begin{equation*} |
283 \begin{equation*} |
284 \xymatrix{\bc_*^{\cC}(S^1) \ar[r]^(0.4){\iso}_(0.4){\text{qi}} & \HC_*(\cC).} |
284 \xymatrix{\bc_*(S^1;\cC) \ar[r]^(0.4){\iso}_(0.4){\text{qi}} & \HC_*(\cC).} |
285 \end{equation*} |
285 \end{equation*} |
286 \end{thm:hochschild} |
286 \end{thm:hochschild} |
287 |
287 |
288 Theorem \ref{thm:skein-modules} is immediate from the definition, and |
288 Theorem \ref{thm:skein-modules} is immediate from the definition, and |
289 Theorem \ref{thm:hochschild} is established in \S \ref{sec:hochschild}. |
289 Theorem \ref{thm:hochschild} is established in \S \ref{sec:hochschild}. |
295 |
295 |
296 In the following $\CH{X}$ is the singular chain complex of the space of homeomorphisms of $X$, fixed on $\bdy X$. |
296 In the following $\CH{X}$ is the singular chain complex of the space of homeomorphisms of $X$, fixed on $\bdy X$. |
297 |
297 |
298 \newtheorem*{thm:CH}{Theorem \ref{thm:CH}} |
298 \newtheorem*{thm:CH}{Theorem \ref{thm:CH}} |
299 |
299 |
300 \begin{thm:CH}[$C_*(\Homeo(-))$ action]\mbox{}\\ |
300 \begin{thm:CH}[$C_*(\Homeo(-))$ action] |
301 \vspace{-0.5cm} |
|
302 \label{thm:evaluation}% |
301 \label{thm:evaluation}% |
303 There is a chain map |
302 There is a chain map |
304 \begin{equation*} |
303 \begin{equation*} |
305 \ev_X: \CH{X} \tensor \bc_*(X) \to \bc_*(X). |
304 \ev_X: \CH{X} \tensor \bc_*(X) \to \bc_*(X). |
306 \end{equation*} |
305 \end{equation*} |
311 \item For |
310 \item For |
312 any codimension $0$-submanifold $Y \sqcup Y^\text{op} \subset \bdy X$ the following diagram |
311 any codimension $0$-submanifold $Y \sqcup Y^\text{op} \subset \bdy X$ the following diagram |
313 (using the gluing maps described in Property \ref{property:gluing-map}) commutes (up to homotopy). |
312 (using the gluing maps described in Property \ref{property:gluing-map}) commutes (up to homotopy). |
314 \begin{equation*} |
313 \begin{equation*} |
315 \xymatrix@C+2cm{ |
314 \xymatrix@C+2cm{ |
316 \CH{X \bigcup_Y \selfarrow} \otimes \bc_*(X \bigcup_Y \selfarrow) \ar[r]^<<<<<<<<<<<<{\ev_{(X \bigcup_Y \scalebox{0.5}{\selfarrow})}} & \bc_*(X \bigcup_Y \selfarrow) \\ |
|
317 \CH{X} \otimes \bc_*(X) |
315 \CH{X} \otimes \bc_*(X) |
318 \ar[r]_{\ev_{X}} \ar[u]^{\gl^{\Homeo}_Y \otimes \gl_Y} & |
316 \ar[r]_{\ev_{X}} \ar[d]^{\gl^{\Homeo}_Y \otimes \gl_Y} & |
319 \bc_*(X) \ar[u]_{\gl_Y} |
317 \bc_*(X) \ar[d]_{\gl_Y} \\ |
|
318 \CH{X \bigcup_Y \selfarrow} \otimes \bc_*(X \bigcup_Y \selfarrow) \ar[r]^<<<<<<<<<<<<{\ev_{(X \bigcup_Y \scalebox{0.5}{\selfarrow})}} & \bc_*(X \bigcup_Y \selfarrow) |
320 } |
319 } |
321 \end{equation*} |
320 \end{equation*} |
322 \end{enumerate} |
321 \end{enumerate} |
323 Moreover any such chain map is unique, up to an iterated homotopy. |
322 Moreover any such chain map is unique, up to an iterated homotopy. |
324 (That is, any pair of homotopies have a homotopy between them, and so on.) |
323 (That is, any pair of homotopies have a homotopy between them, and so on.) |
327 \newtheorem*{thm:CH-associativity}{Theorem \ref{thm:CH-associativity}} |
326 \newtheorem*{thm:CH-associativity}{Theorem \ref{thm:CH-associativity}} |
328 |
327 |
329 |
328 |
330 Further, |
329 Further, |
331 \begin{thm:CH-associativity} |
330 \begin{thm:CH-associativity} |
332 \item The chain map of Theorem \ref{thm:CH} is associative, in the sense that the following diagram commutes (up to homotopy). |
331 The chain map of Theorem \ref{thm:CH} is associative, in the sense that the following diagram commutes (up to homotopy). |
333 \begin{equation*} |
332 \begin{equation*} |
334 \xymatrix{ |
333 \xymatrix{ |
335 \CH{X} \tensor \CH{X} \tensor \bc_*(X) \ar[r]^<<<<<{\id \tensor \ev_X} \ar[d]^{\compose \tensor \id} & \CH{X} \tensor \bc_*(X) \ar[d]^{\ev_X} \\ |
334 \CH{X} \tensor \CH{X} \tensor \bc_*(X) \ar[r]^<<<<<{\id \tensor \ev_X} \ar[d]^{\compose \tensor \id} & \CH{X} \tensor \bc_*(X) \ar[d]^{\ev_X} \\ |
336 \CH{X} \tensor \bc_*(X) \ar[r]^{\ev_X} & \bc_*(X) |
335 \CH{X} \tensor \bc_*(X) \ar[r]^{\ev_X} & \bc_*(X) |
337 } |
336 } |