57 homeomorphisms extends to a suitably defined action of the complex of singular chains of homeomorphisms. |
57 homeomorphisms extends to a suitably defined action of the complex of singular chains of homeomorphisms. |
58 The axioms for an $A_\infty$ $n$-category are designed to capture two main examples: the blob complexes of $n$-balls labelled by a |
58 The axioms for an $A_\infty$ $n$-category are designed to capture two main examples: the blob complexes of $n$-balls labelled by a |
59 topological $n$-category, and the complex $\CM{-}{T}$ of maps to a fixed target space $T$. |
59 topological $n$-category, and the complex $\CM{-}{T}$ of maps to a fixed target space $T$. |
60 |
60 |
61 In \S \ref{ss:ncat_fields} we explain how to construct a system of fields from a topological $n$-category |
61 In \S \ref{ss:ncat_fields} we explain how to construct a system of fields from a topological $n$-category |
62 (using a colimit along cellulations of a manifold), and in \S \ref{sec:ainfblob} give an alternative definition |
62 (using a colimit along certain decompositions of a manifold into balls). With this in hand, we freely write $\bc_*(M; \cC)$ to indicate the blob complex of a manifold $M$ with the system of fields constructed from the $n$-category $\cC$. In \S \ref{sec:ainfblob} we give an alternative definition |
63 of the blob complex for an $A_\infty$ $n$-category on an $n$-manifold (analogously, using a homotopy colimit). |
63 of the blob complex for an $A_\infty$ $n$-category on an $n$-manifold (analogously, using a homotopy colimit). |
64 Using these definitions, we show how to use the blob complex to ``resolve" any topological $n$-category as an |
64 Using these definitions, we show how to use the blob complex to ``resolve" any topological $n$-category as an |
65 $A_\infty$ $n$-category, and relate the first and second definitions of the blob complex. |
65 $A_\infty$ $n$-category, and relate the first and second definitions of the blob complex. |
66 We use the blob complex for $A_\infty$ $n$-categories to establish important properties of the blob complex (in both variants), |
66 We use the blob complex for $A_\infty$ $n$-categories to establish important properties of the blob complex (in both variants), |
67 in particular the ``gluing formula" of Theorem \ref{thm:gluing} below. |
67 in particular the ``gluing formula" of Theorem \ref{thm:gluing} below. |
205 |
205 |
206 \begin{property}[Functoriality] |
206 \begin{property}[Functoriality] |
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 $\cF$, the association |
211 \begin{equation*} |
211 \begin{equation*} |
212 X \mapsto \bc_*(X; \cC) |
212 X \mapsto \bc_*(X; \cF) |
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 $\bc_*(X; \cC)$; |
217 As a consequence, there is an action of $\Homeo(X)$ on the chain complex $\bc_*(X; \cF)$; |
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 $\cF$, |
221 although we will not address this in detail here. |
221 although we will not address this in detail here. |
222 |
222 |
223 \begin{property}[Disjoint union] |
223 \begin{property}[Disjoint union] |
224 \label{property:disjoint-union} |
224 \label{property:disjoint-union} |
225 The blob complex of a disjoint union is naturally the tensor product of the blob complexes. |
225 The blob complex of a disjoint union is naturally the tensor product of the blob complexes. |
246 \end{property} |
246 \end{property} |
247 |
247 |
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 $\cF$ to balls. |
252 \begin{equation*} |
252 \begin{equation*} |
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)} |
253 \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 & \cF(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. |
266 \newtheorem*{thm:skein-modules}{Theorem \ref{thm:skein-modules}} |
266 \newtheorem*{thm:skein-modules}{Theorem \ref{thm:skein-modules}} |
267 |
267 |
268 \begin{thm:skein-modules}[Skein modules] |
268 \begin{thm:skein-modules}[Skein modules] |
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 $\cF$. |
272 (See \S \ref{sec:local-relations}.) |
272 (See \S \ref{sec:local-relations}.) |
273 \begin{equation*} |
273 \begin{equation*} |
274 H_0(\bc_*(X;\cC)) \iso A_{\cC}(X) |
274 H_0(\bc_*(X;\cF)) \iso A_{\cF}(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 |