2 |
2 |
3 \section{Introduction} |
3 \section{Introduction} |
4 |
4 |
5 We construct the ``blob complex'' $\bc_*(M; \cC)$ associated to an $n$-manifold $M$ and a ``linear $n$-category with strong duality'' $\cC$. This blob complex provides a simultaneous generalisation of several well-understood constructions: |
5 We construct the ``blob complex'' $\bc_*(M; \cC)$ associated to an $n$-manifold $M$ and a ``linear $n$-category with strong duality'' $\cC$. This blob complex provides a simultaneous generalisation of several well-understood constructions: |
6 \begin{itemize} |
6 \begin{itemize} |
7 \item The vector space $H_0(\bc_*(M; \cC))$ is isomorphic to the usual topological quantum field theory invariant of $M$ associated to $\cC$. (See \S \ref{sec:fields} \nn{more specific}.) |
7 \item The vector space $H_0(\bc_*(M; \cC))$ is isomorphic to the usual topological quantum field theory invariant of $M$ associated to $\cC$. (See Property \ref{property:skein-modules} later in the introduction and \S \ref{sec:constructing-a-tqft}.) |
8 \item When $n=1$, $\cC$ is just a 1-category (e.g.\ an associative algebra), and $\bc_*(S^1; \cC)$ is quasi-isomorphic to the Hochschild complex $\HC_*(\cC)$. (See \S \ref{sec:hochschild}.) |
8 \item When $n=1$ and $\cC$ is just a 1-category (e.g.\ an associative algebra), the blob complex $\bc_*(S^1; \cC)$ is quasi-isomorphic to the Hochschild complex $\HC_*(\cC)$. (See Property \ref{property:hochschild} and \S \ref{sec:hochschild}.) |
9 \item When $\cC$ is the polynomial algebra $k[t]$, thought of as an n-category (see \S \ref{sec:comm_alg}), we have |
9 \item When $\cC$ is the polynomial algebra $k[t]$, thought of as an n-category (see \S \ref{sec:comm_alg}), we have |
10 that $\bc_*(M; k[t])$ is homotopy equivalent to $C_*(\Sigma^\infty(M), k)$, the singular chains |
10 that $\bc_*(M; k[t])$ is homotopy equivalent to $C_*(\Sigma^\infty(M), k)$, the singular chains |
11 on the configurations space of unlabeled points in $M$. |
11 on the configurations space of unlabeled points in $M$. |
12 %$$H_*(\bc_*(M; k[t])) = H^{\text{sing}}_*(\Delta^\infty(M), k).$$ |
12 %$$H_*(\bc_*(M; k[t])) = H^{\text{sing}}_*(\Delta^\infty(M), k).$$ |
13 \end{itemize} |
13 \end{itemize} |
14 The blob complex definition is motivated by the desire for a derived analogue of the usual TQFT Hilbert space (replacing quotient of fields by local relations with some sort of resolution), |
14 The blob complex definition is motivated by the desire for a derived analogue of the usual TQFT Hilbert space (replacing quotient of fields by local relations with some sort of resolution), |
15 and for a generalization of Hochschild homology to higher $n$-categories. We would also like to be able to talk about $\CM{M}{T}$ when $T$ is an $n$-category rather than a manifold. The blob complex allows us to do all of these! More detailed motivations are described in \S \ref{sec:motivations}. |
15 and for a generalization of Hochschild homology to higher $n$-categories. We would also like to be able to talk about $\CM{M}{T}$ when $T$ is an $n$-category rather than a manifold. The blob complex gives us all of these! More detailed motivations are described in \S \ref{sec:motivations}. |
16 |
16 |
17 The blob complex has good formal properties, summarized in \S \ref{sec:properties}. These include an action of $\CH{M}$, |
17 The blob complex has good formal properties, summarized in \S \ref{sec:properties}. These include an action of $\CH{M}$, |
18 extending the usual $\Homeo(M)$ action on the TQFT space $H_0$ (see Property \ref{property:evaluation}) and a gluing formula allowing calculations by cutting manifolds into smaller parts (see Property \ref{property:gluing}). |
18 extending the usual $\Homeo(M)$ action on the TQFT space $H_0$ (see Property \ref{property:evaluation}) and a gluing formula allowing calculations by cutting manifolds into smaller parts (see Property \ref{property:gluing}). |
19 |
19 |
20 We expect applications of the blob complex to contact topology and Khovanov homology but do not address these in this paper. See \S \ref{sec:future} for slightly more detail. |
20 We expect applications of the blob complex to contact topology and Khovanov homology but do not address these in this paper. See \S \ref{sec:future} for slightly more detail. |
126 \begin{equation*} |
126 \begin{equation*} |
127 X \mapsto \bc_*^{\cC}(X) |
127 X \mapsto \bc_*^{\cC}(X) |
128 \end{equation*} |
128 \end{equation*} |
129 is a functor from $n$-manifolds and homeomorphisms between them to chain complexes and isomorphisms between them. |
129 is a functor from $n$-manifolds and homeomorphisms between them to chain complexes and isomorphisms between them. |
130 \end{property} |
130 \end{property} |
131 |
131 As a consequence, there is an action of $\Homeo(X)$ on the chain complex $\bc_*^\cC(X)$; this action is extended to all of $C_*(\Homeo(X))$ in Property \ref{property:evaluation} below. |
132 The blob complex is also functorial with respect to $\cC$, although we will not address this in detail here. |
132 |
|
133 The blob complex is also functorial with respect to $\cC$, although we will not address this in detail here. \todo{exact w.r.t $\cC$?} |
133 |
134 |
134 \begin{property}[Disjoint union] |
135 \begin{property}[Disjoint union] |
135 \label{property:disjoint-union} |
136 \label{property:disjoint-union} |
136 The blob complex of a disjoint union is naturally the tensor product of the blob complexes. |
137 The blob complex of a disjoint union is naturally the tensor product of the blob complexes. |
137 \begin{equation*} |
138 \begin{equation*} |
138 \bc_*(X_1 \du X_2) \iso \bc_*(X_1) \tensor \bc_*(X_2) |
139 \bc_*(X_1 \du X_2) \iso \bc_*(X_1) \tensor \bc_*(X_2) |
139 \end{equation*} |
140 \end{equation*} |
140 \end{property} |
141 \end{property} |
141 |
142 |
142 If an $n$-manifold $X_\text{cut}$ contains $Y \sqcup Y^\text{op}$ as a codimension $0$-submanifold of its boundary, write $X_\text{glued} = X_\text{cut} \bigcup_{Y}\selfarrow$ for the manifold obtained by gluing together $Y$ and $Y^\text{op}$. Note that this includes the case of gluing two disjoint manifolds together. |
143 If an $n$-manifold $X_\text{cut}$ contains $Y \sqcup Y^\text{op}$ as a codimension $0$ submanifold of its boundary, write $X_\text{glued} = X_\text{cut} \bigcup_{Y}\selfarrow$ for the manifold obtained by gluing together $Y$ and $Y^\text{op}$. Note that this includes the case of gluing two disjoint manifolds together. |
143 \begin{property}[Gluing map] |
144 \begin{property}[Gluing map] |
144 \label{property:gluing-map}% |
145 \label{property:gluing-map}% |
145 %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 |
146 %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 |
146 %\begin{equation*} |
147 %\begin{equation*} |
147 %\gl_Y: \bc_*(X_1) \tensor \bc_*(X_2) \to \bc_*(X_1 \cup_Y X_2). |
148 %\gl_Y: \bc_*(X_1) \tensor \bc_*(X_2) \to \bc_*(X_1 \cup_Y X_2). |
154 (natural with respect to homeomorphisms, and also associative with respect to iterated gluings). |
155 (natural with respect to homeomorphisms, and also associative with respect to iterated gluings). |
155 \end{property} |
156 \end{property} |
156 |
157 |
157 \begin{property}[Contractibility] |
158 \begin{property}[Contractibility] |
158 \label{property:contractibility}% |
159 \label{property:contractibility}% |
159 \nn{this holds with field coefficients, or more generally when |
160 With field coefficients, the blob complex on an $n$-ball is contractible in the sense that it is homotopic to its $0$-th homology. Moreover, the $0$-th homology of balls can be canonically identified with the vector spaces associated by the system of fields $\cC$ to balls. |
160 the map to 0-th homology has a splitting; need to fix statement} |
|
161 The blob complex on an $n$-ball is contractible in the sense that it is quasi-isomorphic to its $0$-th homology. Moreover, the $0$-th homology of balls can be canonically identified with the original $n$-category $\cC$. |
|
162 \begin{equation} |
161 \begin{equation} |
163 \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)} |
162 \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)} |
164 \end{equation} |
163 \end{equation} |
165 \end{property} |
164 \end{property} |
166 |
165 |
211 |
210 |
212 Since the blob complex is functorial in the manifold $X$, we can use this to build a chain map |
211 Since the blob complex is functorial in the manifold $X$, we can use this to build a chain map |
213 $$ev_{X \to Y} : \CH{X \to Y} \tensor \bc_*(X) \to \bc_*(Y)$$ |
212 $$ev_{X \to Y} : \CH{X \to Y} \tensor \bc_*(X) \to \bc_*(Y)$$ |
214 satisfying corresponding conditions. |
213 satisfying corresponding conditions. |
215 |
214 |
216 In \S \ref{sec:ncats} we introduce the notion of topological $n$-categories, from which we can construct systems of fields, as well as the notion of an $A_\infty$ $n$-category. |
215 In \S \ref{sec:ncats} we introduce the notion of topological $n$-categories, from which we can construct systems of fields. Below, we talk about the blob complex associated to a topological $n$-category, implicitly passing to the system of fields. Further, in \S \ref{sec:ncats} we also have the notion of an $A_\infty$ $n$-category. |
217 |
216 |
218 \begin{property}[Blob complexes of (products with) balls form an $A_\infty$ $n$-category] |
217 \begin{property}[Blob complexes of (products with) balls form an $A_\infty$ $n$-category] |
219 \label{property:blobs-ainfty} |
218 \label{property:blobs-ainfty} |
220 Let $\cC$ be a topological $n$-category. Let $Y$ be an $n{-}k$-manifold. |
219 Let $\cC$ be a topological $n$-category. Let $Y$ be an $n{-}k$-manifold. |
221 Define $A_*(Y, \cC)$ on each $m$-ball $D$, for $0 \leq m < k$, to be the set $$A_*(Y,\cC)(D) = A^\cC(Y \times D)$$ and on $k$-balls $D$ to be the set $$A_*(Y, \cC)(D) = \bc_*(Y \times D, \cC).$$ (When $m=k$ the subsets with fixed boundary conditions form a chain complex.) These sets have the structure of an $A_\infty$ $k$-category. |
220 There is an $A_\infty$ $k$-category $A_*(Y, \cC)$, defined on each $m$-ball $D$, for $0 \leq m < k$, to be the set $$A_*(Y,\cC)(D) = A^\cC(Y \times D)$$ and on $k$-balls $D$ to be the set $$A_*(Y, \cC)(D) = \bc_*(Y \times D, \cC).$$ (When $m=k$ the subsets with fixed boundary conditions form a chain complex.) These sets have the structure of an $A_\infty$ $k$-category, with compositions coming from the gluing map in Property \ref{property:gluing-map} and with the action of families of homeomorphisms given in Property \ref{property:evaluation}. |
222 \end{property} |
221 \end{property} |
223 \begin{rem} |
222 \begin{rem} |
224 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. |
223 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. We think of this $A_\infty$ $n$-category as a free resolution. |
225 \end{rem} |
224 \end{rem} |
226 |
225 |
227 There is a version of the blob complex for $\cC$ an $A_\infty$ $n$-category |
226 There is a version of the blob complex for $\cC$ an $A_\infty$ $n$-category |
228 instead of a garden variety $n$-category; this is described in \S \ref{sec:ainfblob}. |
227 instead of a garden variety $n$-category; this is described in \S \ref{sec:ainfblob}. |
229 |
228 |
230 \begin{property}[Product formula] |
229 \begin{property}[Product formula] |
|
230 \label{property:product} |
231 Let $W$ be a $k$-manifold and $Y$ be an $n-k$ manifold. Let $\cC$ be an $n$-category. |
231 Let $W$ be a $k$-manifold and $Y$ be an $n-k$ manifold. Let $\cC$ be an $n$-category. |
232 Let $A_*(Y,\cC)$ be the $A_\infty$ $k$-category associated to $Y$ via blob homology (see Property \ref{property:blobs-ainfty}). |
232 Let $A_*(Y,\cC)$ be the $A_\infty$ $k$-category associated to $Y$ via blob homology (see Property \ref{property:blobs-ainfty}). |
233 Then |
233 Then |
234 \[ |
234 \[ |
235 \bc_*(Y\times W, \cC) \simeq \bc_*(W, A_*(Y,\cC)) . |
235 \bc_*(Y\times W, \cC) \simeq \bc_*(W, A_*(Y,\cC)) . |
240 |
240 |
241 \begin{property}[Gluing formula] |
241 \begin{property}[Gluing formula] |
242 \label{property:gluing}% |
242 \label{property:gluing}% |
243 \mbox{}% <-- gets the indenting right |
243 \mbox{}% <-- gets the indenting right |
244 \begin{itemize} |
244 \begin{itemize} |
245 \item For any $(n-1)$-manifold $Y$, the blob homology of $Y \times I$ is |
245 \item For any $(n-1)$-manifold $Y$, the blob complex of $Y \times I$ is |
246 naturally an $A_\infty$ category. % We'll write $\bc_*(Y)$ for $\bc_*(Y \times I)$ below. |
246 naturally an $A_\infty$ category. % We'll write $\bc_*(Y)$ for $\bc_*(Y \times I)$ below. |
247 |
247 |
248 \item For any $n$-manifold $X$, with $Y$ a codimension $0$-submanifold of its boundary, the blob homology of $X$ is naturally an |
248 \item For any $n$-manifold $X$, with $Y$ a codimension $0$-submanifold of its boundary, the blob complex of $X$ is naturally an |
249 $A_\infty$ module for $\bc_*(Y \times I)$. |
249 $A_\infty$ module for $\bc_*(Y \times I)$. |
250 |
250 |
251 \item For any $n$-manifold $X_\text{glued} = X_\text{cut} \bigcup_Y \selfarrow$, the blob complex $\bc_*(X_\text{glued})$ is the $A_\infty$ self-tensor product of |
251 \item For any $n$-manifold $X_\text{glued} = X_\text{cut} \bigcup_Y \selfarrow$, the blob complex $\bc_*(X_\text{glued})$ is the $A_\infty$ self-tensor product of |
252 $\bc_*(X_\text{cut})$ as an $\bc_*(Y \times I)$-bimodule. |
252 $\bc_*(X_\text{cut})$ as an $\bc_*(Y \times I)$-bimodule: |
253 \begin{equation*} |
253 \begin{equation*} |
254 \bc_*(X_\text{glued}) \simeq \bc_*(X_\text{cut}) \Tensor^{A_\infty}_{\mathclap{\bc_*(Y \times I)}} \selfarrow |
254 \bc_*(X_\text{glued}) \simeq \bc_*(X_\text{cut}) \Tensor^{A_\infty}_{\mathclap{\bc_*(Y \times I)}} \selfarrow |
255 \end{equation*} |
255 \end{equation*} |
256 \end{itemize} |
256 \end{itemize} |
257 \end{property} |
257 \end{property} |
259 Finally, we state two more properties, which we will not prove in this paper. |
259 Finally, we state two more properties, which we will not prove in this paper. |
260 |
260 |
261 \begin{property}[Mapping spaces] |
261 \begin{property}[Mapping spaces] |
262 Let $\pi^\infty_{\le n}(T)$ denote the $A_\infty$ $n$-category based on maps |
262 Let $\pi^\infty_{\le n}(T)$ denote the $A_\infty$ $n$-category based on maps |
263 $B^n \to T$. |
263 $B^n \to T$. |
264 (The case $n=1$ is the usual $A_\infty$ category of paths in $T$.) |
264 (The case $n=1$ is the usual $A_\infty$-category of paths in $T$.) |
265 Then |
265 Then |
266 $$\bc_*(X, \pi^\infty_{\le n}(T)) \simeq \CM{X}{T}.$$ |
266 $$\bc_*(X, \pi^\infty_{\le n}(T)) \simeq \CM{X}{T}.$$ |
267 \end{property} |
267 \end{property} |
268 |
268 |
269 This says that we can recover the (homotopic) space of maps to $T$ via blob homology from local data. |
269 This says that we can recover the (homotopic) space of maps to $T$ via blob homology from local data. |
275 See \S \ref{sec:deligne} for an explanation of the terms appearing here. The proof will appear elsewhere. |
275 See \S \ref{sec:deligne} for an explanation of the terms appearing here. The proof will appear elsewhere. |
276 |
276 |
277 Properties \ref{property:functoriality}, \ref{property:gluing-map} and \ref{property:skein-modules} will be immediate from the definition given in |
277 Properties \ref{property:functoriality}, \ref{property:gluing-map} and \ref{property:skein-modules} will be immediate from the definition given in |
278 \S \ref{sec:blob-definition}, and we'll recall them at the appropriate points there. \todo{Make sure this gets done.} |
278 \S \ref{sec:blob-definition}, and we'll recall them at the appropriate points there. \todo{Make sure this gets done.} |
279 Properties \ref{property:disjoint-union} and \ref{property:contractibility} are established in \S \ref{sec:basic-properties}. |
279 Properties \ref{property:disjoint-union} and \ref{property:contractibility} are established in \S \ref{sec:basic-properties}. |
280 Property \ref{property:hochschild} is established in \S \ref{sec:hochschild}, Property \ref{property:evaluation} in \S \ref{sec:evaluation}, |
280 Property \ref{property:hochschild} is established in \S \ref{sec:hochschild}, Property \ref{property:evaluation} in \S \ref{sec:evaluation}, Property \ref{property:blobs-ainfty} as Example \ref{ex:blob-complexes-of-balls} in \S \ref{sec:ncats}, |
281 and Property \ref{property:gluing} in \S \ref{sec:gluing}. |
281 and Properties \ref{property:product} and \ref{property:gluing} in \S \ref{sec:ainfblob} as consequences of Theorem \ref{product_thm}. |
282 \nn{need to say where the remaining properties are proved.} |
|
283 |
282 |
284 \subsection{Future directions} |
283 \subsection{Future directions} |
285 \label{sec:future} |
284 \label{sec:future} |
286 Throughout, we have resisted the temptation to work in the greatest generality possible (don't worry, it wasn't that hard). |
285 Throughout, we have resisted the temptation to work in the greatest generality possible (don't worry, it wasn't that hard). |
287 In most of the places where we say ``set" or ``vector space", any symmetric monoidal category would do. |
286 In most of the places where we say ``set" or ``vector space", any symmetric monoidal category would do. We could presumably also replace many of our chain complexes with topological spaces (or indeed, work at the generality of model categories), and likely it will prove useful to think about the connections between what we do here and $(\infty,k)$-categories. |
288 \nn{maybe make similar remark about chain complexes and $(\infty, 0)$-categories} |
|
289 More could be said about finite characteristic (there appears in be $2$-torsion in $\bc_1(S^2, \cC)$ for any spherical $2$-category $\cC$, for example). Much more could be said about other types of manifolds, in particular oriented, $\operatorname{Spin}$ and $\operatorname{Pin}^{\pm}$ manifolds, where boundary issues become more complicated. (We'd recommend thinking about boundaries as germs, rather than just codimension $1$ manifolds.) We've also take the path of least resistance by considering $\operatorname{PL}$ manifolds; there may be some differences for topological manifolds and smooth manifolds. |
287 More could be said about finite characteristic (there appears in be $2$-torsion in $\bc_1(S^2, \cC)$ for any spherical $2$-category $\cC$, for example). Much more could be said about other types of manifolds, in particular oriented, $\operatorname{Spin}$ and $\operatorname{Pin}^{\pm}$ manifolds, where boundary issues become more complicated. (We'd recommend thinking about boundaries as germs, rather than just codimension $1$ manifolds.) We've also take the path of least resistance by considering $\operatorname{PL}$ manifolds; there may be some differences for topological manifolds and smooth manifolds. |
290 |
288 |
291 Many results in Hochschild homology can be understood `topologically' via the blob complex. For example, we expect that the shuffle product on the Hochschild homology of a commutative algebra $A$ simply corresponds to the gluing operation on $\bc_*(S^1 \times [0,1], A)$, but haven't investigated the details. |
289 Many results in Hochschild homology can be understood `topologically' via the blob complex. For example, we expect that the shuffle product on the Hochschild homology of a commutative algebra $A$ simply corresponds to the gluing operation on $\bc_*(S^1 \times [0,1], A)$, but haven't investigated the details. |
292 |
290 |
293 Most importantly, however, \nn{applications!} \nn{$n=2$ cases, contact, Kh} |
291 Most importantly, however, \nn{applications!} \nn{cyclic homology, $n=2$ cases, contact, Kh} |
294 |
292 |
295 |
293 |
296 \subsection{Thanks and acknowledgements} |
294 \subsection{Thanks and acknowledgements} |
297 We'd like to thank David Ben-Zvi, Michael Freedman, Vaughan Jones, Justin Roberts, Chris Schommer-Pries, Peter Teichner \nn{probably lots more} for many interesting and useful conversations. During this work, Kevin Walker has been at Microsoft Station Q, and Scott Morrison has been at Microsoft Station Q and the Miller Institute for Basic Research at UC Berkeley. |
295 We'd like to thank David Ben-Zvi, Michael Freedman, Vaughan Jones, Justin Roberts, Chris Schommer-Pries, Peter Teichner \nn{probably lots more} for many interesting and useful conversations. During this work, Kevin Walker has been at Microsoft Station Q, and Scott Morrison has been at Microsoft Station Q and the Miller Institute for Basic Research at UC Berkeley. |
298 |
296 |