author | Scott Morrison <scott@tqft.net> |
Sun, 31 Oct 2010 20:41:53 +0900 | |
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\input{preamble} |
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\input{../text/kw_macros} |
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\contributor{Submitted to Proceedings |
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of the National Academy of Sciences of the United States of America} |
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%\url{www.pnas.org/cgi/doi/10.1073/pnas.0709640104} |
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\author{Scott Morrison\affil{1}{Miller Institute for Basic Research, UC Berkeley, CA 94704, USA} \and Kevin Walker\affil{2}{Microsoft Station Q, 2243 CNSI Building, UC Santa Barbara, CA 93106, USA}} |
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of the United States of America} |
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\nn{ |
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background: TQFTs are important, historically, semisimple categories well-understood. |
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Many new examples arising recently which do not fit this framework, e.g. SW and OS theory. |
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These have more complicated gluing formulas (\cite{1003.0598,1005.1248}, etc); |
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it would be nice to give generalized TQFT axioms that encompass these. |
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Triangulated categories are important; often calculations are via exact sequences, |
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and the standard TQFT constructions are quotients, which destroy exactness. |
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A first attempt to deal with this might be to replace all the tensor products in gluing formulas |
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with derived tensor products (cite Kh?). |
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However, in this approach it's probably difficult to prove invariance of constructions, |
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because they depend on explicit presentations of the manifold. |
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We'll give a manifestly invariant construction, |
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and deduce gluing formulas based on derived (actually, $A_\infty$) tensor products.} |
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\section{Definitions} |
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\subsection{$n$-categories} \mbox{} |
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\todo{This is just a copy and paste of the statements of the axioms. We need to rewrite this into something that's both compact and comprehensible! The first few at least aren't that terrifying, but we definitely don't want to derail the reader with the actual product axiom, for example.} |
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\begin{axiom}[Morphisms] |
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\label{axiom:morphisms} |
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For each $0 \le k \le n$, we have a functor $\cC_k$ from |
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the category of $k$-balls and |
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homeomorphisms to the category of sets and bijections. |
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\end{axiom} |
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\begin{lem} |
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\label{lem:spheres} |
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For each $1 \le k \le n$, we have a functor $\cl{\cC}_{k-1}$ from |
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the category of $k{-}1$-spheres and |
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homeomorphisms to the category of sets and bijections. |
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\end{lem} |
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\begin{axiom}[Boundaries]\label{nca-boundary} |
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For each $k$-ball $X$, we have a map of sets $\bd: \cC_k(X)\to \cl{\cC}_{k-1}(\bd X)$. |
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These maps, for various $X$, comprise a natural transformation of functors. |
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\end{axiom} |
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\begin{lem}[Boundary from domain and range] |
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\label{lem:domain-and-range} |
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Let $S = B_1 \cup_E B_2$, where $S$ is a $k{-}1$-sphere $(1\le k\le n)$, |
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$B_i$ is a $k{-}1$-ball, and $E = B_1\cap B_2$ is a $k{-}2$-sphere (Figure \ref{blah3}). |
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Let $\cC(B_1) \times_{\cl{\cC}(E)} \cC(B_2)$ denote the fibered product of the |
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two maps $\bd: \cC(B_i)\to \cl{\cC}(E)$. |
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Then we have an injective map |
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\[ |
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\gl_E : \cC(B_1) \times_{\cl{\cC}(E)} \cC(B_2) \into \cl{\cC}(S) |
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\] |
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which is natural with respect to the actions of homeomorphisms. |
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(When $k=1$ we stipulate that $\cl{\cC}(E)$ is a point, so that the above fibered product |
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becomes a normal product.) |
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\end{lem} |
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\begin{axiom}[Composition] |
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\label{axiom:composition} |
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Let $B = B_1 \cup_Y B_2$, where $B$, $B_1$ and $B_2$ are $k$-balls ($0\le k\le n$) |
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and $Y = B_1\cap B_2$ is a $k{-}1$-ball (Figure \ref{blah5}). |
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Let $E = \bd Y$, which is a $k{-}2$-sphere. |
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Note that each of $B$, $B_1$ and $B_2$ has its boundary split into two $k{-}1$-balls by $E$. |
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We have restriction (domain or range) maps $\cC(B_i)_E \to \cC(Y)$. |
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Let $\cC(B_1)_E \times_{\cC(Y)} \cC(B_2)_E$ denote the fibered product of these two maps. |
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We have a map |
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\[ |
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\gl_Y : \cC(B_1)_E \times_{\cC(Y)} \cC(B_2)_E \to \cC(B)_E |
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\] |
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which is natural with respect to the actions of homeomorphisms, and also compatible with restrictions |
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to the intersection of the boundaries of $B$ and $B_i$. |
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If $k < n$, |
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or if $k=n$ and we are in the $A_\infty$ case, |
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we require that $\gl_Y$ is injective. |
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(For $k=n$ in the plain (non-$A_\infty$) case, see below.) |
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\end{axiom} |
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\begin{axiom}[Strict associativity] \label{nca-assoc} |
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The composition (gluing) maps above are strictly associative. |
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Given any splitting of a ball $B$ into smaller balls |
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$$\bigsqcup B_i \to B,$$ |
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any sequence of gluings (in the sense of Definition \ref{defn:gluing-decomposition}, where all the intermediate steps are also disjoint unions of balls) yields the same result. |
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\end{axiom} |
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\begin{axiom}[Product (identity) morphisms] |
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\label{axiom:product} |
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For each pinched product $\pi:E\to X$, with $X$ a $k$-ball and $E$ a $k{+}m$-ball ($m\ge 1$), |
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there is a map $\pi^*:\cC(X)\to \cC(E)$. |
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These maps must satisfy the following conditions. |
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\begin{enumerate} |
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\item |
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If $\pi:E\to X$ and $\pi':E'\to X'$ are pinched products, and |
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if $f:X\to X'$ and $\tilde{f}:E \to E'$ are maps such that the diagram |
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\[ \xymatrix{ |
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E \ar[r]^{\tilde{f}} \ar[d]_{\pi} & E' \ar[d]^{\pi'} \\ |
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X \ar[r]^{f} & X' |
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} \] |
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commutes, then we have |
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\[ |
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\pi'^*\circ f = \tilde{f}\circ \pi^*. |
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\] |
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\item |
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Product morphisms are compatible with gluing (composition). |
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Let $\pi:E\to X$, $\pi_1:E_1\to X_1$, and $\pi_2:E_2\to X_2$ |
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be pinched products with $E = E_1\cup E_2$. |
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Let $a\in \cC(X)$, and let $a_i$ denote the restriction of $a$ to $X_i\sub X$. |
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Then |
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\[ |
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\pi^*(a) = \pi_1^*(a_1)\bullet \pi_2^*(a_2) . |
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\] |
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\item |
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Product morphisms are associative. |
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If $\pi:E\to X$ and $\rho:D\to E$ are pinched products then |
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\[ |
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\rho^*\circ\pi^* = (\pi\circ\rho)^* . |
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\] |
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\item |
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Product morphisms are compatible with restriction. |
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If we have a commutative diagram |
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\[ \xymatrix{ |
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D \ar@{^(->}[r] \ar[d]_{\rho} & E \ar[d]^{\pi} \\ |
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Y \ar@{^(->}[r] & X |
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} \] |
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such that $\rho$ and $\pi$ are pinched products, then |
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\[ |
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\res_D\circ\pi^* = \rho^*\circ\res_Y . |
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\] |
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\end{enumerate} |
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\end{axiom} |
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\begin{axiom}[\textup{\textbf{[plain version]}} Extended isotopy invariance in dimension $n$.] |
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\label{axiom:extended-isotopies} |
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Let $X$ be an $n$-ball and $f: X\to X$ be a homeomorphism which restricts |
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to the identity on $\bd X$ and isotopic (rel boundary) to the identity. |
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Then $f$ acts trivially on $\cC(X)$. |
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In addition, collar maps act trivially on $\cC(X)$. |
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\end{axiom} |
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\smallskip |
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For $A_\infty$ $n$-categories, we replace |
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isotopy invariance with the requirement that families of homeomorphisms act. |
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For the moment, assume that our $n$-morphisms are enriched over chain complexes. |
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Let $\Homeo_\bd(X)$ denote homeomorphisms of $X$ which fix $\bd X$ and |
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$C_*(\Homeo_\bd(X))$ denote the singular chains on this space. |
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\begin{axiom}[\textup{\textbf{[$A_\infty$ version]}} Families of homeomorphisms act in dimension $n$.] |
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\label{axiom:families} |
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For each $n$-ball $X$ and each $c\in \cl{\cC}(\bd X)$ we have a map of chain complexes |
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\[ |
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C_*(\Homeo_\bd(X))\ot \cC(X; c) \to \cC(X; c) . |
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\] |
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These action maps are required to be associative up to homotopy, |
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and also compatible with composition (gluing) in the sense that |
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a diagram like the one in Theorem \ref{thm:CH} commutes. |
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\end{axiom} |
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\todo{ |
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Decide if we need a friendlier, skein-module version. |
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} |
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\subsubsection{Examples} |
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\todo{maps to a space, string diagrams} |
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\subsection{The blob complex} |
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\subsubsection{Decompositions of manifolds} |
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A \emph{ball decomposition} of $W$ is a |
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sequence of gluings $M_0\to M_1\to\cdots\to M_m = W$ such that $M_0$ is a disjoint union of balls |
574 | 322 |
$\du_a X_a$ and each $M_i$ is a manifold. |
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If $X_a$ is some component of $M_0$, its image in $W$ need not be a ball; $\bd X_a$ may have been glued to itself. |
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A {\it permissible decomposition} of $W$ is a map |
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\[ |
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\coprod_a X_a \to W, |
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\] |
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which can be completed to a ball decomposition $\du_a X_a = M_0\to\cdots\to M_m = W$. |
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A permissible decomposition is weaker than a ball decomposition; we forget the order in which the balls |
330 |
are glued up to yield $W$, and just require that there is some non-pathological way to do this. |
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Given permissible decompositions $x = \{X_a\}$ and $y = \{Y_b\}$ of $W$, we say that $x$ is a refinement |
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of $y$, or write $x \le y$, if there is a ball decomposition $\du_a X_a = M_0\to\cdots\to M_m = W$ |
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with $\du_b Y_b = M_i$ for some $i$. |
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\begin{defn} |
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The poset $\cell(W)$ has objects the permissible decompositions of $W$, |
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and a unique morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$. |
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See Figure \ref{partofJfig} for an example. |
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\end{defn} |
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An $n$-category $\cC$ determines |
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a functor $\psi_{\cC;W}$ from $\cell(W)$ to the category of sets |
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(possibly with additional structure if $k=n$). |
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Each $k$-ball $X$ of a decomposition $y$ of $W$ has its boundary decomposed into $k{-}1$-balls, |
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and there is a subset $\cC(X)\spl \sub \cC(X)$ of morphisms whose boundaries |
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are splittable along this decomposition. |
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\begin{defn} |
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Define the functor $\psi_{\cC;W} : \cell(W) \to \Set$ as follows. |
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For a decomposition $x = \bigsqcup_a X_a$ in $\cell(W)$, $\psi_{\cC;W}(x)$ is the subset |
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\begin{equation*} |
354 |
%\label{eq:psi-C} |
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\psi_{\cC;W}(x) \sub \prod_a \cC(X_a)\spl |
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\end{equation*} |
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where the restrictions to the various pieces of shared boundaries amongst the cells |
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$X_a$ all agree (this is a fibered product of all the labels of $n$-cells over the labels of $n-1$-cells). When $k=n$, the `subset' and `product' in the above formula should be interpreted in the appropriate enriching category. |
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If $x$ is a refinement of $y$, the map $\psi_{\cC;W}(x) \to \psi_{\cC;W}(y)$ is given by the composition maps of $\cC$. |
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\end{defn} |
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We will use the term `field on $W$' to refer to \nn{a point} of this functor, |
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that is, a permissible decomposition $x$ of $W$ together with an element of $\psi_{\cC;W}(x)$. |
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\todo{Mention that the axioms for $n$-categories can be stated in terms of decompositions of balls?} |
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\subsubsection{Homotopy colimits} |
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\nn{Motivation: How can we extend an $n$-category from balls to arbitrary manifolds?} |
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We now define the blob complex $\bc_*(W; \cC)$ of an $n$-manifold $W$ |
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with coefficients in the $n$-category $\cC$ to be the homotopy colimit |
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of the functor $\psi_{\cC; W}$ described above. |
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When $\cC$ is a topological $n$-category, |
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the flexibility available in the construction of a homotopy colimit allows |
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us to give a much more explicit description of the blob complex. |
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We say a collection of balls $\{B_i\}$ in a manifold $W$ is \emph{permissible} |
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if there exists a permissible decomposition $M_0\to\cdots\to M_m = W$ such that |
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each $B_i$ appears as a connected component of one of the $M_j$. Note that this allows the balls to be pairwise either disjoint or nested. Such a collection of balls cuts $W$ into pieces, the connected components of $W \setminus \bigcup \bdy B_i$. These pieces need not be manifolds, but they do automatically have permissible decompositions. |
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The $k$-blob group $\bc_k(W; \cC)$ is generated by the $k$-blob diagrams. A $k$-blob diagram consists of |
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\begin{itemize} |
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\item a permissible collection of $k$ embedded balls, |
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\item an ordering of the balls, and |
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\item for each resulting piece of $W$, a field, |
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\end{itemize} |
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such that for any innermost blob $B$, the field on $B$ goes to zero under the composition map from $\cC$. We call such a field a `null field on $B$'. |
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The differential acts on a $k$-blob diagram by summing over ways to forget one of the $k$ blobs, with signs given by the ordering. |
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\todo{Say why this really is the homotopy colimit} |
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We now spell this out for some small values of $k$. For $k=0$, the $0$-blob group is simply fields on $W$. For $k=1$, a generator consists of a field on $W$ and a ball, such that the restriction of the field that that ball is a null field. The differential simply forgets the ball. Thus we see that $H_0$ of the blob complex is the quotient of fields by null fields. |
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For $k=2$, we have a two types of generators; they each consists of a field $f$ on $W$, and two balls $B_1$ and $B_2$. In the first case, the balls are disjoint, and $f$ restricted to either of the $B_i$ is a null field. In the second case, the balls are properly nested, say $B_1 \subset B_2$, and $f$ restricted to $B_1$ is null. Note that this implies that $f$ restricted to $B_2$ is also null, by the associativity of the gluing operation. This ensures that the differential is well-defined. |
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\section{Properties of the blob complex} |
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\subsection{Formal properties} |
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\label{sec:properties} |
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The blob complex enjoys the following list of formal properties. |
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\begin{property}[Functoriality] |
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\label{property:functoriality}% |
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The blob complex is functorial with respect to homeomorphisms. |
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That is, |
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for a fixed $n$-category $\cC$, the association |
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\begin{equation*} |
574 | 409 |
X \mapsto \bc_*(X; \cC) |
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\end{equation*} |
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is a functor from $n$-manifolds and homeomorphisms between them to chain |
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complexes and isomorphisms between them. |
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\end{property} |
574 | 414 |
As a consequence, there is an action of $\Homeo(X)$ on the chain complex $\bc_*(X; \cC)$; |
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this action is extended to all of $C_*(\Homeo(X))$ in Theorem \ref{thm:CH} below. |
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\begin{property}[Disjoint union] |
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\label{property:disjoint-union} |
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The blob complex of a disjoint union is naturally isomorphic to the tensor product of the blob complexes. |
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\begin{equation*} |
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\bc_*(X_1 \du X_2) \iso \bc_*(X_1) \tensor \bc_*(X_2) |
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\end{equation*} |
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\end{property} |
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If an $n$-manifold $X$ contains $Y \sqcup Y^\text{op}$ (we allow $Y = \eset$) as a codimension $0$ submanifold of its boundary, |
426 |
write $X \bigcup_{Y}\selfarrow$ for the manifold obtained by gluing together $Y$ and $Y^\text{op}$. |
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\begin{property}[Gluing map] |
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\label{property:gluing-map}% |
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%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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%\begin{equation*} |
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%\gl_Y: \bc_*(X_1) \tensor \bc_*(X_2) \to \bc_*(X_1 \cup_Y X_2). |
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%\end{equation*} |
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Given a gluing $X \to X_\mathrm{gl}$, there is |
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a map |
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\[ |
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\bc_*(X) \to \bc_*(X \bigcup_{Y}\selfarrow), |
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\] |
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natural with respect to homeomorphisms, and associative with respect to iterated gluings. |
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\end{property} |
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\begin{property}[Contractibility] |
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\label{property:contractibility}% |
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With field coefficients, the blob complex on an $n$-ball is contractible in the sense |
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that it is homotopic to its $0$-th homology. |
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Moreover, the $0$-th homology of balls can be canonically identified with the vector spaces |
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associated by the system of fields $\cF$ to balls. |
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\begin{equation*} |
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\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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\end{equation*} |
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\end{property} |
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\nn{Properties \ref{property:functoriality} will be immediate from the definition given in |
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\S \ref{sec:blob-definition}, and we'll recall it at the appropriate point there. |
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Properties \ref{property:disjoint-union}, \ref{property:gluing-map} and |
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\ref{property:contractibility} are established in \S \ref{sec:basic-properties}.} |
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\subsection{Specializations} |
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\label{sec:specializations} |
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The blob complex has two important special cases. |
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\begin{thm}[Skein modules] |
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\label{thm:skein-modules} |
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The $0$-th blob homology of $X$ is the usual |
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(dual) TQFT Hilbert space (a.k.a.\ skein module) associated to $X$ |
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by $\cF$. |
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\begin{equation*} |
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H_0(\bc_*(X;\cF)) \iso A_{\cF}(X) |
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\end{equation*} |
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\end{thm} |
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\begin{thm}[Hochschild homology when $X=S^1$] |
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\label{thm:hochschild} |
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The blob complex for a $1$-category $\cC$ on the circle is |
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quasi-isomorphic to the Hochschild complex. |
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\begin{equation*} |
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\xymatrix{\bc_*(S^1;\cC) \ar[r]^(0.47){\iso}_(0.47){\text{qi}} & \HC_*(\cC).} |
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\end{equation*} |
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\end{thm} |
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Theorem \ref{thm:skein-modules} is immediate from the definition, and |
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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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|
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\subsection{Structure of the blob complex} |
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\label{sec:structure} |
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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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\begin{thm}[$C_*(\Homeo(-))$ action] |
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\label{thm:CH}\label{thm:evaluation} |
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There is a chain map |
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\begin{equation*} |
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e_X: \CH{X} \tensor \bc_*(X) \to \bc_*(X). |
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\end{equation*} |
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such that |
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\begin{enumerate} |
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\item Restricted to $CH_0(X)$ this is the action of homeomorphisms described in Property \ref{property:functoriality}. |
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|
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\item For |
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any codimension $0$-submanifold $Y \sqcup Y^\text{op} \subset \bdy X$ the following diagram |
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(using the gluing maps described in Property \ref{property:gluing-map}) commutes (up to homotopy). |
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\begin{equation*} |
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\xymatrix@C+0.3cm{ |
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\CH{X} \otimes \bc_*(X) |
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\ar[r]_{e_{X}} \ar[d]^{\gl^{\Homeo}_Y \otimes \gl_Y} & |
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\bc_*(X) \ar[d]_{\gl_Y} \\ |
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\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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} |
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\end{equation*} |
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\end{enumerate} |
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Futher, this map is associative, in the sense that the following diagram commutes (up to homotopy). |
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514 |
\begin{equation*} |
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515 |
\xymatrix{ |
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\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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\CH{X} \tensor \bc_*(X) \ar[r]^{e_X} & \bc_*(X) |
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518 |
} |
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519 |
\end{equation*} |
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520 |
\end{thm} |
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521 |
|
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522 |
Since the blob complex is functorial in the manifold $X$, this is equivalent to having chain maps |
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$$ev_{X \to Y} : \CH{X \to Y} \tensor \bc_*(X) \to \bc_*(Y)$$ |
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524 |
for any homeomorphic pair $X$ and $Y$, |
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525 |
satisfying corresponding conditions. |
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526 |
|
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|
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528 |
|
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529 |
\begin{thm}[Blob complexes of products with balls form an $A_\infty$ $n$-category] |
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530 |
\label{thm:blobs-ainfty} |
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531 |
Let $\cC$ be a topological $n$-category. |
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532 |
Let $Y$ be an $n{-}k$-manifold. |
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533 |
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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534 |
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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535 |
$$\bc_*(Y;\cC)(D) = \bc_*(Y \times D; \cC).$$ |
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536 |
(When $m=k$ the subsets with fixed boundary conditions form a chain complex.) |
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537 |
These sets have the structure of an $A_\infty$ $k$-category, with compositions coming from the gluing map in |
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538 |
Property \ref{property:gluing-map} and with the action of families of homeomorphisms given in Theorem \ref{thm:evaluation}. |
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539 |
\end{thm} |
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540 |
\begin{rem} |
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541 |
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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542 |
We think of this $A_\infty$ $n$-category as a free resolution. |
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543 |
\end{rem} |
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544 |
This result is described in more detail as Example 6.2.8 of \cite{1009.5025} |
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545 |
|
574 | 546 |
We next describe the blob complex for product manifolds, in terms of the $A_\infty$ blob complex of the $A_\infty$ $n$-categories constructed as above. |
572
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547 |
|
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548 |
\begin{thm}[Product formula] |
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549 |
\label{thm:product} |
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550 |
Let $W$ be a $k$-manifold and $Y$ be an $n-k$ manifold. |
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551 |
Let $\cC$ be an $n$-category. |
574 | 552 |
Let $\bc_*(Y;\cC)$ be the $A_\infty$ $k$-category associated to $Y$ via blob homology. |
572
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553 |
Then |
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|
554 |
\[ |
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555 |
\bc_*(Y\times W; \cC) \simeq \cl{\bc_*(Y;\cC)}(W). |
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556 |
\] |
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|
557 |
\end{thm} |
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558 |
The statement can be generalized to arbitrary fibre bundles, and indeed to arbitrary maps |
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559 |
(see \cite[\S7.1]{1009.5025}). |
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|
560 |
|
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561 |
Fix a topological $n$-category $\cC$, which we'll omit from the notation. |
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562 |
Recall that for any $(n-1)$-manifold $Y$, the blob complex $\bc_*(Y)$ is naturally an $A_\infty$ category. |
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563 |
|
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|
564 |
\begin{thm}[Gluing formula] |
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565 |
\label{thm:gluing} |
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|
566 |
\mbox{}% <-- gets the indenting right |
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|
567 |
\begin{itemize} |
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568 |
\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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|
569 |
$A_\infty$ module for $\bc_*(Y)$. |
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|
570 |
|
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571 |
\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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572 |
$\bc_*(X)$ as an $\bc_*(Y)$-bimodule: |
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|
573 |
\begin{equation*} |
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|
574 |
\bc_*(X_\text{gl}) \simeq \bc_*(X) \Tensor^{A_\infty}_{\mathclap{\bc_*(Y)}} \selfarrow |
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|
575 |
\end{equation*} |
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|
576 |
\end{itemize} |
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|
577 |
\end{thm} |
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578 |
|
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579 |
\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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580 |
|
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|
581 |
\section{Applications} |
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|
582 |
\label{sec:applications} |
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583 |
Finally, we give two applications of the above machinery. |
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|
584 |
|
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585 |
\begin{thm}[Mapping spaces] |
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|
586 |
\label{thm:map-recon} |
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587 |
Let $\pi^\infty_{\le n}(T)$ denote the $A_\infty$ $n$-category based on maps |
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588 |
$B^n \to T$. |
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|
589 |
(The case $n=1$ is the usual $A_\infty$-category of paths in $T$.) |
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|
590 |
Then |
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|
591 |
$$\bc_*(X; \pi^\infty_{\le n}(T)) \simeq \CM{X}{T}.$$ |
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|
592 |
\end{thm} |
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|
593 |
|
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|
594 |
This says that we can recover (up to homotopy) the space of maps to $T$ via blob homology from local data. |
574 | 595 |
Note that there is no restriction on the connectivity of $T$ as there is for the corresponding result in topological chiral homology \cite[Theorem 3.8.6]{0911.0018}. |
580
99611dfed1f3
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|
596 |
\todo{sketch proof} |
572
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|
597 |
|
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|
598 |
\begin{thm}[Higher dimensional Deligne conjecture] |
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|
599 |
\label{thm:deligne} |
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|
600 |
The singular chains of the $n$-dimensional surgery cylinder operad act on blob cochains. |
577
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|
601 |
Since the little $n{+}1$-balls operad is a suboperad of the $n$-SC operad, |
572
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|
602 |
this implies that the little $n{+}1$-balls operad acts on blob cochains of the $n$-ball. |
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|
603 |
\end{thm} |
577
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|
604 |
|
580
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|
605 |
An $n$-dimensional surgery cylinder is a sequence of mapping cylinders and surgeries (Figure \ref{delfig2}), modulo changing the order of distant surgeries, and conjugating a submanifold not modified in a surgery by a homeomorphism. Surgery cylinders form an operad, by gluing the outer boundary of one cylinder into an inner boundary of another. |
579 | 606 |
|
580
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|
607 |
By the `blob cochains' of a manifold $X$, we mean the $A_\infty$ maps of $\bc_*(X)$ as a $\bc_*(\bdy X)$ $A_\infty$-module. |
577
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|
608 |
|
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|
609 |
\todo{Sketch proof} |
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|
610 |
|
580
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|
611 |
The little disks operad $LD$ is homotopy equivalent to the $n=1$ case of the $n$-SC operad. The blob complex $\bc_*(I, \cC)$ is a bimodule over itself, and the $A_\infty$-bimodule intertwiners are homotopy equivalent to the Hochschild cohains $Hoch^*(C, C)$. The usual Deligne conjecture (proved variously in \cite{hep-th/9403055, MR1805894, MR2064592, MR1805923}) gives a map |
577
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|
612 |
\[ |
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|
613 |
C_*(LD_k)\otimes \overbrace{Hoch^*(C, C)\otimes\cdots\otimes Hoch^*(C, C)}^{\text{$k$ copies}} |
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|
614 |
\to Hoch^*(C, C), |
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|
615 |
\] |
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|
616 |
which we now see to be a specialization of Theorem \ref{thm:deligne}. |
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|
617 |
|
566 | 618 |
|
619 |
%% == end of paper: |
|
620 |
||
621 |
%% Optional Materials and Methods Section |
|
622 |
%% The Materials and Methods section header will be added automatically. |
|
623 |
||
624 |
%% Enter any subheads and the Materials and Methods text below. |
|
625 |
%\begin{materials} |
|
626 |
% Materials text |
|
627 |
%\end{materials} |
|
628 |
||
629 |
||
630 |
%% Optional Appendix or Appendices |
|
631 |
%% \appendix Appendix text... |
|
632 |
%% or, for appendix with title, use square brackets: |
|
633 |
%% \appendix[Appendix Title] |
|
634 |
||
635 |
\begin{acknowledgments} |
|
636 |
-- text of acknowledgments here, including grant info -- |
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%% bibliography environment described above. |
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% A short C-terminal sequence is necessary and sufficient for the |
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% {\it Proc Natl Acad Sci USA} 88:10362-10366. |
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\end{article} |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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\begin{equation*} |
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\end{equation*} |
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\caption{A small part of $\cell(W)$} |
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\label{partofJfig} |
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\end{figure} |
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\begin{figure} |
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$$\mathfig{.4}{deligne/manifolds}$$ |
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\caption{An $n$-dimensional surgery cylinder}\label{delfig2} |
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\end{figure} |
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%% keep that much space between every line in the table. |
|
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||
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%% \begin{table} |
|
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%% \caption{Repeat length of longer allele by age of onset class} |
|
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%% \begin{tabular}{@{\vrule height 10.5pt depth4pt width0pt}lrcccc} |
|
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|
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%% \end{tabular} |
|
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%% For two column figures and tables, use the following: |
|
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||
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|
719 |
%% \caption{Almost Sharp Front}\label{afoto} |
|
720 |
%% \end{figure*} |
|
721 |
||
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%% \begin{table*} |
|
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|
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|
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%% \end{table*} |
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||
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\end{document} |
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