6 associated to an $n$-manifold $M$ and a linear $n$-category $\cC$ with strong duality. |
6 associated to an $n$-manifold $M$ and a linear $n$-category $\cC$ with strong duality. |
7 This blob complex provides a simultaneous generalization of several well known constructions: |
7 This blob complex provides a simultaneous generalization of several well known constructions: |
8 \begin{itemize} |
8 \begin{itemize} |
9 \item The 0-th homology $H_0(\bc_*(M; \cC))$ is isomorphic to the usual |
9 \item The 0-th homology $H_0(\bc_*(M; \cC))$ is isomorphic to the usual |
10 topological quantum field theory invariant of $M$ associated to $\cC$. |
10 topological quantum field theory invariant of $M$ associated to $\cC$. |
11 (See Theorem \ref{thm:skein-modules} later in the introduction and \S \ref{sec:constructing-a-tqft}.) |
11 (See Proposition \ref{thm:skein-modules} later in the introduction and \S \ref{sec:constructing-a-tqft}.) |
12 \item When $n=1$ and $\cC$ is just a 1-category (e.g.\ an associative algebra), |
12 \item When $n=1$ and $\cC$ is just a 1-category (e.g.\ an associative algebra), |
13 the blob complex $\bc_*(S^1; \cC)$ is quasi-isomorphic to the Hochschild complex $\HC_*(\cC)$. |
13 the blob complex $\bc_*(S^1; \cC)$ is quasi-isomorphic to the Hochschild complex $\HC_*(\cC)$. |
14 (See Theorem \ref{thm:hochschild} and \S \ref{sec:hochschild}.) |
14 (See Theorem \ref{thm:hochschild} and \S \ref{sec:hochschild}.) |
15 \item When $\cC$ is the polynomial algebra $k[t]$, thought of as an n-category, we have |
15 \item When $\cC$ is the polynomial algebra $k[t]$, thought of as an n-category, we have |
16 that $\bc_*(M; k[t])$ is homotopy equivalent to $C_*(\Sigma^\infty(M), k)$, the singular chains |
16 that $\bc_*(M; k[t])$ is homotopy equivalent to $C_*(\Sigma^\infty(M), k)$, the singular chains |
122 \S \ref{ss:ncat_fields} |
122 \S \ref{ss:ncat_fields} |
123 %$\displaystyle \cF(M) = \DirectSum_{c \in\cell(M)} \cC(c)$ \\ $\displaystyle U(B) = \DirectSum_{c \in \cell(B)} \ker \ev: \cC(c) \to \cC(B)$ |
123 %$\displaystyle \cF(M) = \DirectSum_{c \in\cell(M)} \cC(c)$ \\ $\displaystyle U(B) = \DirectSum_{c \in \cell(B)} \ker \ev: \cC(c) \to \cC(B)$ |
124 } (FU.100); |
124 } (FU.100); |
125 \draw[->] (C) -- node[above left=3pt] {restrict to \\ standard balls} (tC); |
125 \draw[->] (C) -- node[above left=3pt] {restrict to \\ standard balls} (tC); |
126 \draw[->] (FU.80) -- node[right] {restrict \\ to balls} (C.-80); |
126 \draw[->] (FU.80) -- node[right] {restrict \\ to balls} (C.-80); |
127 \draw[->] (BC) -- node[right] {$H_0$ \\ c.f. Theorem \ref{thm:skein-modules}} (A); |
127 \draw[->] (BC) -- node[right] {$H_0$ \\ c.f. Proposition \ref{thm:skein-modules}} (A); |
128 |
128 |
129 \draw[->] (FU) -- node[left] {blob complex \\ for balls} (Cs); |
129 \draw[->] (FU) -- node[left] {blob complex \\ for balls} (Cs); |
130 \draw[<->] (BC) -- node[right] {$\iso$ by \\ Corollary \ref{cor:new-old}} (BCs); |
130 \draw[<->] (BC) -- node[right] {$\iso$ by \\ Corollary \ref{cor:new-old}} (BCs); |
131 \end{tikzpicture} |
131 \end{tikzpicture} |
132 |
132 |
284 \subsection{Specializations} |
284 \subsection{Specializations} |
285 \label{sec:specializations} |
285 \label{sec:specializations} |
286 |
286 |
287 The blob complex is a simultaneous generalization of the TQFT skein module construction and of Hochschild homology. |
287 The blob complex is a simultaneous generalization of the TQFT skein module construction and of Hochschild homology. |
288 |
288 |
289 \newtheorem*{thm:skein-modules}{Theorem \ref{thm:skein-modules}} |
289 \newtheorem*{thm:skein-modules}{Proposition \ref{thm:skein-modules}} |
290 |
290 |
291 \begin{thm:skein-modules}[Skein modules] |
291 \begin{thm:skein-modules}[Skein modules] |
292 The $0$-th blob homology of $X$ is the usual |
292 The $0$-th blob homology of $X$ is the usual |
293 (dual) TQFT Hilbert space (a.k.a.\ skein module) associated to $X$ |
293 (dual) TQFT Hilbert space (a.k.a.\ skein module) associated to $X$ |
294 by $\cF$. |
294 by $\cF$. |
306 \begin{equation*} |
306 \begin{equation*} |
307 \xymatrix{\bc_*(S^1;\cC) \ar[r]^(0.47){\iso}_(0.47){\text{qi}} & \HC_*(\cC).} |
307 \xymatrix{\bc_*(S^1;\cC) \ar[r]^(0.47){\iso}_(0.47){\text{qi}} & \HC_*(\cC).} |
308 \end{equation*} |
308 \end{equation*} |
309 \end{thm:hochschild} |
309 \end{thm:hochschild} |
310 |
310 |
311 Theorem \ref{thm:skein-modules} is immediate from the definition, and |
311 Proposition \ref{thm:skein-modules} is immediate from the definition, and |
312 Theorem \ref{thm:hochschild} is established in \S \ref{sec:hochschild}. |
312 Theorem \ref{thm:hochschild} is established in \S \ref{sec:hochschild}. |
313 We also note \S \ref{sec:comm_alg} which describes the blob complex when $\cC$ is a one of |
313 We also note \S \ref{sec:comm_alg} which describes the blob complex when $\cC$ is a one of |
314 certain commutative algebras thought of as $n$-categories. |
314 certain commutative algebras thought of as $n$-categories. |
315 |
315 |
316 |
316 |