author | Kevin Walker <kevin@canyon23.net> |
Tue, 02 Nov 2010 06:38:40 -0700 | |
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\input{preamble} |
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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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\contributor{Submitted to Proceedings of the National Academy of Sciences |
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of the United States of America} |
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\keywords{n-categories | topological quantum field theory | hochschild homology} |
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%\dropcap{I}n this article we study the evolution of ''almost-sharp'' fronts |
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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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\nn{rough draft of n-cat stuff...} |
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\nn{maybe say something about goals: well-suited to TQFTs; avoid proliferation of coherency axioms; |
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non-recursive (n-cats not defined n terms of (n-1)-cats; easy to show that the motivating |
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examples satisfy the axioms; strong duality; both plain and infty case; |
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(?) easy to see that axioms are correct, in the sense of nothing missing (need |
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to say this better if we keep it)} |
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\nn{maybe: the typical n-cat definition tries to do two things at once: (1) give a list of basic properties |
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which are weak enough to include the basic examples and strong enough to support the proofs |
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of the main theorems; and (2) specify a minimal set of generators and/or axioms. |
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We separate these two tasks, and address only the first, which becomes much easier when not burdened by the second. |
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More specifically, life is easier when working with maximal, rather than minimal, collections of axioms.} |
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\nn{say something about defining plain and infty cases simultaneously} |
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There are five basic ingredients of an $n$-category definition: |
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$k$-morphisms (for $0\le k \le n$), domain and range, composition, |
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identity morphisms, and special behavior in dimension $n$ (e.g. enrichment |
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in some auxiliary category, or strict associativity instead of weak associativity). |
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We will treat each of these in turn. |
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To motivate our morphism axiom, consider the venerable notion of the Moore loop space |
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\nn{need citation -- \S 2.2 of Adams' ``Infinite Loop Spaces''?}. |
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In the standard definition of a loop space, loops are always parameterized by the unit interval $I = [0,1]$, |
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so composition of loops requires a reparameterization $I\cup I \cong I$, and this leads to a proliferation |
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of higher associativity relations. |
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While this proliferation is manageable for 1-categories (and indeed leads to an elegant theory |
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of Stasheff polyhedra and $A_\infty$ categories), it becomes undesirably complex for higher categories. |
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In a Moore loop space, we have a separate space $\Omega_r$ for each interval $[0,r]$, and a |
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{\it strictly associative} composition $\Omega_r\times \Omega_s\to \Omega_{r+s}$. |
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Thus we can have the simplicity of strict associativity in exchange for more morphisms. |
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We wish to imitate this strategy in higher categories. |
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Because we are mainly interested in the case of strong duality, we replace the intervals $[0,r]$ not with |
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a product of $k$ intervals \nn{cf xxxx} but rather with any $k$-ball, that is, any $k$-manifold which is homeomorphic |
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to the standard $k$-ball $B^k$. |
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\nn{maybe add that in addition we want functoriality} |
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In fact, the axioms here may easily be varied by considering balls with structure (e.g. $m$ independent vector fields, a map to some target space, etc.). Such variations are useful for axiomatizing categories with less duality, and also as technical tools in proofs. |
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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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Note that the functoriality in the above axiom allows us to operate via |
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homeomorphisms which are not the identity on the boundary of the $k$-ball. |
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The action of these homeomorphisms gives the ``strong duality" structure. |
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Next we consider domains and ranges of $k$-morphisms. |
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Because we assume strong duality, it doesn't make much sense to subdivide the boundary of a morphism |
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into domain and range --- the duality operations can convert domain to range and vice-versa. |
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Instead, we will use a unified domain/range, which we will call a ``boundary". |
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Later \todo{} we inductively define an extension of the functors $\cC_k$ to functors $\cl{\cC}_k$ from arbitrary manifolds to sets. We need the restriction of these functors to $k$-spheres, for $k<n$, for the next axiom. |
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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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For $c\in \cl{\cC}_{k-1}(\bd X)$ we let $\cC_k(X; c)$ denote the preimage $\bd^{-1}(c)$. |
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Many of the examples we are interested in are enriched in some auxiliary category $\cS$ |
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(e.g. $\cS$ is vector spaces or rings, or, in the $A_\infty$ case, chain complex or topological spaces). |
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This means (by definition) that in the top dimension $k=n$ the sets $\cC_n(X; c)$ have the structure |
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of an object of $\cS$, and all of the structure maps of the category (above and below) are |
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compatible with the $\cS$ structure on $\cC_n(X; c)$. |
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Given two hemispheres (a `domain' and `range') that agree on the equator, we need to be able to assemble them into a boundary value of the entire sphere. |
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\begin{lem} |
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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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258 |
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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266 |
\end{lem} |
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If $\bdy B = S$, we denote $\bdy^{-1}(\im(\gl_E))$ by $\cC(B)_E$. |
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\begin{axiom}[Gluing] |
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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 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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278 |
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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282 |
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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284 |
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 gluing maps above are strictly associative. |
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Given any decomposition of a ball $B$ into smaller balls |
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$$\bigsqcup B_i \to B,$$ |
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any sequence of gluings (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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For the next axiom, a \emph{pinched product} is a map locally modeled on a degeneracy map between simplices. |
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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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306 |
\[ \xymatrix{ |
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E \ar[r]^{\tilde{f}} \ar[d]_{\pi} & E' \ar[d]^{\pi'} \\ |
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X \ar[r]^{f} & X' |
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} \] |
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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. |
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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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\[ |
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321 |
\pi^*(a) = \pi_1^*(a_1)\bullet \pi_2^*(a_2) . |
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\] |
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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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349 |
|
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350 |
\smallskip |
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351 |
|
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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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357 |
|
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358 |
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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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362 |
\[ |
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C_*(\Homeo_\bd(X))\ot \cC(X; c) \to \cC(X; c) . |
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364 |
\] |
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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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368 |
\end{axiom} |
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369 |
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370 |
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371 |
\todo{ |
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Decide if we need a friendlier, skein-module version. |
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} |
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374 |
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375 |
\subsubsection{Examples} |
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\todo{maps to a space, string diagrams} |
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377 |
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\subsection{The blob complex} |
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\subsubsection{Decompositions of manifolds} |
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|
583 | 381 |
\nn{KW: I'm inclined to suppress all discussion of the subtleties of decompositions. |
382 |
Maybe just a single remark that we are omitting some details which appear in our |
|
383 |
longer paper.} |
|
584 | 384 |
\nn{SM: for now I disagree: the space expense is pretty minor, and it always us to be "in principle" complete. Let's see how we go for length.} |
587
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\nn{KW: It's not the length I'm worried about --- I was worried about distracting the reader |
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with an arcane technical issue. But we can decide later.} |
583 | 387 |
|
574 | 388 |
A \emph{ball decomposition} of $W$ is a |
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389 |
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 | 390 |
$\du_a X_a$ and each $M_i$ is a manifold. |
391 |
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. |
|
392 |
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 $\du_a X_a = M_0\to\cdots\to M_m = W$. |
574 | 397 |
A permissible decomposition is weaker than a ball decomposition; we forget the order in which the balls |
398 |
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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403 |
|
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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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409 |
|
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410 |
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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, |
574 | 415 |
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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|
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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 |
574 | 421 |
\begin{equation*} |
422 |
%\label{eq:psi-C} |
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\psi_{\cC;W}(x) \sub \prod_a \cC(X_a)\spl |
574 | 424 |
\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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429 |
|
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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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432 |
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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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435 |
\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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438 |
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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441 |
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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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445 |
|
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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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448 |
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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449 |
|
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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} |
580
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\item a permissible collection of $k$ embedded balls, |
575
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453 |
\item an ordering of the balls, and |
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454 |
\item for each resulting piece of $W$, a field, |
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455 |
\end{itemize} |
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456 |
such that for any innermost blob $B$, the field on $B$ goes to zero under the gluing map from $\cC$. We call such a field a `null field on $B$'. |
575
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457 |
|
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458 |
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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459 |
|
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460 |
\todo{Say why this really is the homotopy colimit} |
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461 |
|
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462 |
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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465 |
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466 |
\section{Properties of the blob complex} |
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467 |
\subsection{Formal properties} |
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\label{sec:properties} |
584 | 469 |
The blob complex enjoys the following list of formal properties. The first three properties are immediate from the definitions. |
583 | 470 |
|
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\begin{property}[Functoriality] |
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472 |
\label{property:functoriality}% |
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The blob complex is functorial with respect to homeomorphisms. |
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474 |
That is, |
574 | 475 |
for a fixed $n$-category $\cC$, the association |
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\begin{equation*} |
574 | 477 |
X \mapsto \bc_*(X; \cC) |
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478 |
\end{equation*} |
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479 |
is a functor from $n$-manifolds and homeomorphisms between them to chain |
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480 |
complexes and isomorphisms between them. |
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481 |
\end{property} |
574 | 482 |
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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484 |
|
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\begin{property}[Disjoint union] |
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486 |
\label{property:disjoint-union} |
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487 |
The blob complex of a disjoint union is naturally isomorphic to the tensor product of the blob complexes. |
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488 |
\begin{equation*} |
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\bc_*(X_1 \du X_2) \iso \bc_*(X_1) \tensor \bc_*(X_2) |
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490 |
\end{equation*} |
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491 |
\end{property} |
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492 |
|
574 | 493 |
If an $n$-manifold $X$ contains $Y \sqcup Y^\text{op}$ (we allow $Y = \eset$) as a codimension $0$ submanifold of its boundary, |
494 |
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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496 |
\label{property:gluing-map}% |
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497 |
%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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498 |
%\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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500 |
%\end{equation*} |
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Given a gluing $X \to X_\mathrm{gl}$, there is |
574 | 502 |
a map |
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\[ |
574 | 504 |
\bc_*(X) \to \bc_*(X \bigcup_{Y}\selfarrow), |
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505 |
\] |
574 | 506 |
natural with respect to homeomorphisms, and associative with respect to iterated gluings. |
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507 |
\end{property} |
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508 |
|
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509 |
\begin{property}[Contractibility] |
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510 |
\label{property:contractibility}% |
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511 |
With field coefficients, the blob complex on an $n$-ball is contractible in the sense |
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512 |
that it is homotopic to its $0$-th homology. |
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513 |
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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515 |
\begin{equation*} |
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516 |
\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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517 |
\end{equation*} |
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518 |
\end{property} |
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519 |
|
583 | 520 |
\begin{proof}(Sketch) |
521 |
For $k\ge 1$, the contracting homotopy sends a $k$-blob diagram to the $(k{+}1)$-blob diagram |
|
522 |
obtained by adding an outer $(k{+}1)$-st blob consisting of all $B^n$. |
|
523 |
For $k=0$ we choose a splitting $s: H_0(\bc_*(B^n)) \to \bc_0(B^n)$ and send |
|
524 |
$x\in \bc_0(B^n)$ to $x - s([x])$, where $[x]$ denotes the image of $x$ in $H_0(\bc_*(B^n))$. |
|
525 |
\end{proof} |
|
526 |
||
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528 |
\subsection{Specializations} |
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529 |
\label{sec:specializations} |
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530 |
|
574 | 531 |
The blob complex has two important special cases. |
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532 |
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\begin{thm}[Skein modules] |
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534 |
\label{thm:skein-modules} |
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535 |
The $0$-th blob homology of $X$ is the usual |
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536 |
(dual) TQFT Hilbert space (a.k.a.\ skein module) associated to $X$ |
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by $\cF$. |
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538 |
\begin{equation*} |
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539 |
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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542 |
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\begin{thm}[Hochschild homology when $X=S^1$] |
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\label{thm:hochschild} |
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545 |
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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|
574 | 552 |
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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557 |
\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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560 |
|
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|
561 |
\begin{thm} |
572
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|
562 |
\label{thm:CH}\label{thm:evaluation} |
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|
563 |
There is a chain map |
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|
564 |
\begin{equation*} |
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|
565 |
e_X: \CH{X} \tensor \bc_*(X) \to \bc_*(X) |
572
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|
566 |
\end{equation*} |
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|
567 |
such that |
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|
568 |
\begin{enumerate} |
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569 |
\item Restricted to $CH_0(X)$ this is the action of homeomorphisms described in Property \ref{property:functoriality}. |
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|
570 |
|
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|
571 |
\item For |
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|
572 |
any codimension $0$-submanifold $Y \sqcup Y^\text{op} \subset \bdy X$ the following diagram |
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|
573 |
(using the gluing maps described in Property \ref{property:gluing-map}) commutes (up to homotopy). |
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|
574 |
\begin{equation*} |
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|
575 |
\xymatrix@C+0.3cm{ |
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|
576 |
\CH{X} \otimes \bc_*(X) |
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|
577 |
\ar[r]_{e_{X}} \ar[d]^{\gl^{\Homeo}_Y \otimes \gl_Y} & |
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578 |
\bc_*(X) \ar[d]_{\gl_Y} \\ |
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579 |
\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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|
580 |
} |
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|
581 |
\end{equation*} |
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|
582 |
\end{enumerate} |
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|
583 |
|
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|
584 |
Futher, this map is associative, in the sense that the following diagram commutes (up to homotopy). |
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|
585 |
\begin{equation*} |
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|
586 |
\xymatrix{ |
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|
587 |
\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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588 |
\CH{X} \tensor \bc_*(X) \ar[r]^{e_X} & \bc_*(X) |
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|
589 |
} |
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|
590 |
\end{equation*} |
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|
591 |
\end{thm} |
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|
592 |
|
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|
593 |
Since the blob complex is functorial in the manifold $X$, this is equivalent to having chain maps |
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|
594 |
$$ev_{X \to Y} : \CH{X \to Y} \tensor \bc_*(X) \to \bc_*(Y)$$ |
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|
595 |
for any homeomorphic pair $X$ and $Y$, |
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|
596 |
satisfying corresponding conditions. |
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|
597 |
|
575
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|
598 |
|
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|
599 |
|
585
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|
600 |
\begin{thm} |
572
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|
601 |
\label{thm:blobs-ainfty} |
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|
602 |
Let $\cC$ be a topological $n$-category. |
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|
603 |
Let $Y$ be an $n{-}k$-manifold. |
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|
604 |
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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|
605 |
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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|
606 |
$$\bc_*(Y;\cC)(D) = \bc_*(Y \times D; \cC).$$ |
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|
607 |
(When $m=k$ the subsets with fixed boundary conditions form a chain complex.) |
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|
608 |
These sets have the structure of an $A_\infty$ $k$-category, with compositions coming from the gluing map in |
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|
609 |
Property \ref{property:gluing-map} and with the action of families of homeomorphisms given in Theorem \ref{thm:evaluation}. |
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|
610 |
\end{thm} |
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|
611 |
\begin{rem} |
585
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|
612 |
When $Y$ is a point this gives $A_\infty$ $n$-category from a topological $n$-category, which can be thought of as a free resolution. |
572
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|
613 |
\end{rem} |
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|
614 |
This result is described in more detail as Example 6.2.8 of \cite{1009.5025} |
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|
615 |
|
574 | 616 |
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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|
617 |
|
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|
618 |
\begin{thm}[Product formula] |
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|
619 |
\label{thm:product} |
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|
620 |
Let $W$ be a $k$-manifold and $Y$ be an $n-k$ manifold. |
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|
621 |
Let $\cC$ be an $n$-category. |
574 | 622 |
Let $\bc_*(Y;\cC)$ be the $A_\infty$ $k$-category associated to $Y$ via blob homology. |
572
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|
623 |
Then |
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|
624 |
\[ |
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|
625 |
\bc_*(Y\times W; \cC) \simeq \cl{\bc_*(Y;\cC)}(W). |
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|
626 |
\] |
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|
627 |
\end{thm} |
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|
628 |
The statement can be generalized to arbitrary fibre bundles, and indeed to arbitrary maps |
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|
629 |
(see \cite[\S7.1]{1009.5025}). |
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|
630 |
|
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|
631 |
Fix a topological $n$-category $\cC$, which we'll omit from the notation. |
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|
632 |
Recall that for any $(n-1)$-manifold $Y$, the blob complex $\bc_*(Y)$ is naturally an $A_\infty$ category. |
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|
633 |
|
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|
634 |
\begin{thm}[Gluing formula] |
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|
635 |
\label{thm:gluing} |
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|
636 |
\mbox{}% <-- gets the indenting right |
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|
637 |
\begin{itemize} |
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|
638 |
\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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|
639 |
$A_\infty$ module for $\bc_*(Y)$. |
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|
640 |
|
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|
641 |
\item The blob complex of a glued manifold $X\bigcup_Y \selfarrow$ is the $A_\infty$ self-tensor product of |
572
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|
642 |
$\bc_*(X)$ as an $\bc_*(Y)$-bimodule: |
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|
643 |
\begin{equation*} |
585
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|
644 |
\bc_*(X\bigcup_Y \selfarrow) \simeq \bc_*(X) \Tensor^{A_\infty}_{\mathclap{\bc_*(Y)}} \selfarrow |
572
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|
645 |
\end{equation*} |
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|
646 |
\end{itemize} |
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|
647 |
\end{thm} |
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|
648 |
|
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|
649 |
\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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|
650 |
|
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|
651 |
\section{Applications} |
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|
652 |
\label{sec:applications} |
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|
653 |
Finally, we give two applications of the above machinery. |
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|
654 |
|
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|
655 |
\begin{thm}[Mapping spaces] |
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|
656 |
\label{thm:map-recon} |
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|
657 |
Let $\pi^\infty_{\le n}(T)$ denote the $A_\infty$ $n$-category based on maps |
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|
658 |
$B^n \to T$. |
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|
659 |
(The case $n=1$ is the usual $A_\infty$-category of paths in $T$.) |
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|
660 |
Then |
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|
661 |
$$\bc_*(X; \pi^\infty_{\le n}(T)) \simeq \CM{X}{T}.$$ |
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|
662 |
\end{thm} |
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|
663 |
|
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|
664 |
This says that we can recover (up to homotopy) the space of maps to $T$ via blob homology from local data. |
574 | 665 |
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
k-blobs for small k, and blob cochains
Scott Morrison <scott@tqft.net>
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579
diff
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|
666 |
\todo{sketch proof} |
572
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|
667 |
|
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|
668 |
\begin{thm}[Higher dimensional Deligne conjecture] |
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|
669 |
\label{thm:deligne} |
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|
670 |
The singular chains of the $n$-dimensional surgery cylinder operad act on blob cochains. |
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|
671 |
Since the little $n{+}1$-balls operad is a suboperad of the $n$-SC operad, |
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|
672 |
this implies that the little $n{+}1$-balls operad acts on blob cochains of the $n$-ball. |
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|
673 |
\end{thm} |
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|
674 |
|
580
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|
675 |
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 | 676 |
|
580
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|
677 |
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. |
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|
678 |
|
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|
679 |
\todo{Sketch proof} |
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|
680 |
|
580
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|
681 |
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 |
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|
682 |
\[ |
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|
683 |
C_*(LD_k)\otimes \overbrace{Hoch^*(C, C)\otimes\cdots\otimes Hoch^*(C, C)}^{\text{$k$ copies}} |
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|
684 |
\to Hoch^*(C, C), |
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|
685 |
\] |
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|
686 |
which we now see to be a specialization of Theorem \ref{thm:deligne}. |
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687 |
|
566 | 688 |
|
689 |
%% == end of paper: |
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|
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%\begin{materials} |
|
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% Materials text |
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%\end{materials} |
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%% Optional Appendix or Appendices |
|
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%% \appendix Appendix text... |
|
702 |
%% or, for appendix with title, use square brackets: |
|
703 |
%% \appendix[Appendix Title] |
|
704 |
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\begin{acknowledgments} |
|
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-- text of acknowledgments here, including grant info -- |
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\end{acknowledgments} |
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%% PNAS does not support submission of supporting .tex files such as BibTeX. |
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%% Instead all references must be included in the article .tex document. |
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%% If you currently use BibTeX, your bibliography is formed because the |
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%% command \verb+\bibliography{}+ brings the <filename>.bbl file into your |
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%% .tex document. To conform to PNAS requirements, copy the reference listings |
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%% from your .bbl file and add them to the article .tex file, using the |
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%% bibliography environment described above. |
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%% Contact pnas@nas.edu if you need assistance with your |
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%% bibliography. |
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% \bibitem{Neuhaus} Neuhaus J-M, Sitcher L, Meins F, Jr, Boller T (1991) |
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% A short C-terminal sequence is necessary and sufficient for the |
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% targeting of chitinases to the plant vacuole. |
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% {\it Proc Natl Acad Sci USA} 88:10362-10366. |
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%% following \begin{thebibliography} |
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%%%% BIBTEX |
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\bibliographystyle{alpha} |
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\bibliography{../bibliography/bibliography} |
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%%%% non-BIBTEX |
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739 |
%\begin{thebibliography}{} |
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740 |
% |
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%\end{thebibliography} |
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566 | 743 |
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\end{article} |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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747 |
%% Adding Figure and Table References |
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%% Be sure to add figures and tables after \end{article} |
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749 |
%% and before \end{document} |
|
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||
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%% For figures, put the caption below the illustration. |
|
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%% |
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%% \begin{figure} |
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%% \caption{Almost Sharp Front}\label{afoto} |
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%% \end{figure} |
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||
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|
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\begin{figure} |
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\begin{equation*} |
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760 |
\mathfig{.23}{ncat/zz2} |
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761 |
\end{equation*} |
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762 |
\caption{A small part of $\cell(W)$} |
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763 |
\label{partofJfig} |
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|
764 |
\end{figure} |
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765 |
|
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766 |
\begin{figure} |
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|
767 |
$$\mathfig{.4}{deligne/manifolds}$$ |
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|
768 |
\caption{An $n$-dimensional surgery cylinder}\label{delfig2} |
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769 |
\end{figure} |
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770 |
|
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|
566 | 772 |
%% For Tables, put caption above table |
773 |
%% |
|
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%% Table caption should start with a capital letter, continue with lower case |
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%% and not have a period at the end |
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%% Using @{\vrule height ?? depth ?? width0pt} in the tabular preamble will |
|
777 |
%% keep that much space between every line in the table. |
|
778 |
||
779 |
%% \begin{table} |
|
780 |
%% \caption{Repeat length of longer allele by age of onset class} |
|
781 |
%% \begin{tabular}{@{\vrule height 10.5pt depth4pt width0pt}lrcccc} |
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782 |
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783 |
%% \end{tabular} |
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784 |
%% \end{table} |
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786 |
%% For two column figures and tables, use the following: |
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787 |
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788 |
%% \begin{figure*} |
|
789 |
%% \caption{Almost Sharp Front}\label{afoto} |
|
790 |
%% \end{figure*} |
|
791 |
||
792 |
%% \begin{table*} |
|
793 |
%% \caption{Repeat length of longer allele by age of onset class} |
|
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%% \begin{tabular}{ccc} |
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795 |
%% table text |
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%% \end{tabular} |
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%% \end{table*} |
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\end{document} |
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800 |