author | kevin@6e1638ff-ae45-0410-89bd-df963105f760 |
Fri, 05 Jun 2009 00:38:41 +0000 | |
changeset 69 | d363611b1f59 |
parent 67 | 1df2e5b38eb2 |
child 70 | 5ab0e6f0d89e |
permissions | -rw-r--r-- |
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\documentclass[11pt,leqno]{amsart} |
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\def\op{^\mathrm{op}} |
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\def\deq{\stackrel{\mathrm{def}}{=}} |
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\def\pd#1#2{\frac{\partial #1}{\partial #2}} |
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\def\lf{\overline{\cC}} |
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\def\nn#1{{{\it \small [#1]}}} |
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\applytolist{declaremathop}{pr}{im}{gl}{ev}{coinv}{tr}{rot}{Eq}{obj}{mor}{ob}{Rep}{Tet}{cat}{Maps}{Diff}{sign}{supp}; |
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%%%%%% end excerpt |
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\title{Blob Homology} |
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\begin{document} |
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\textbf{Draft version, do not distribute.} |
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\versioninfo |
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\section*{Todo} |
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\subsection*{What else?...} |
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\begin{itemize} |
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\item higher priority |
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\begin{itemize} |
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\item K\&S: learn the state of the art in A-inf categories |
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(tensor products, Kadeishvili result, ...) |
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\item K: so-called evaluation map stuff |
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\item K: topological fields |
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\item section describing intended applications |
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\item say something about starting with semisimple n-cat (trivial?? not trivial?) |
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\item T.O.C. |
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\end{itemize} |
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\item medium priority |
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\begin{itemize} |
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\item $n=2$ examples |
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\item dimension $n+1$ (generalized Deligne conjecture?) |
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\item should be clear about PL vs Diff; probably PL is better |
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(or maybe not) |
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\item say what we mean by $n$-category, $A_\infty$ or $E_\infty$ $n$-category |
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\item something about higher derived coend things (derived 2-coend, e.g.) |
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\item shuffle product vs gluing product (?) |
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\item commutative algebra results |
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\item $A_\infty$ blob complex |
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\item connection between $A_\infty$ operad and topological $A_\infty$ cat defs |
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\end{itemize} |
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\item lower priority |
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\begin{itemize} |
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\item Derive Hochschild standard results from blob point of view? |
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\item Kh |
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\item Mention somewhere \cite{MR1624157} ``Skein homology''; it's not directly related, but has similar motivations. |
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\end{itemize} |
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\end{itemize} |
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\section{Introduction} |
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[Outline for intro] |
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\begin{itemize} |
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\item Starting point: TQFTs via fields and local relations. |
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This gives a satisfactory treatment for semisimple TQFTs |
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(i.e.\ TQFTs for which the cylinder 1-category associated to an |
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$n{-}1$-manifold $Y$ is semisimple for all $Y$). |
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\item For non-semiemple TQFTs, this approach is less satisfactory. |
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Our main motivating example (though we will not develop it in this paper) |
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is the $4{+}1$-dimensional TQFT associated to Khovanov homology. |
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It associates a bigraded vector space $A_{Kh}(W^4, L)$ to a 4-manifold $W$ together |
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with a link $L \subset \bd W$. |
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The original Khovanov homology of a link in $S^3$ is recovered as $A_{Kh}(B^4, L)$. |
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\item How would we go about computing $A_{Kh}(W^4, L)$? |
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For $A_{Kh}(B^4, L)$, the main tool is the exact triangle (long exact sequence) |
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\nn{... $L_1, L_2, L_3$}. |
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Unfortunately, the exactness breaks if we glue $B^4$ to itself and attempt |
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to compute $A_{Kh}(S^1\times B^3, L)$. |
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According to the gluing theorem for TQFTs-via-fields, gluing along $B^3 \subset \bd B^4$ |
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corresponds to taking a coend (self tensor product) over the cylinder category |
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associated to $B^3$ (with appropriate boundary conditions). |
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The coend is not an exact functor, so the exactness of the triangle breaks. |
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\item The obvious solution to this problem is to replace the coend with its derived counterpart. |
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This presumably works fine for $S^1\times B^3$ (the answer being to Hochschild homology |
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of an appropriate bimodule), but for more complicated 4-manifolds this leaves much to be desired. |
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If we build our manifold up via a handle decomposition, the computation |
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would be a sequence of derived coends. |
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A different handle decomposition of the same manifold would yield a different |
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sequence of derived coends. |
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To show that our definition in terms of derived coends is well-defined, we |
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would need to show that the above two sequences of derived coends yield the same answer. |
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This is probably not easy to do. |
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\item Instead, we would prefer a definition for a derived version of $A_{Kh}(W^4, L)$ |
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which is manifestly invariant. |
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(That is, a definition that does not |
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involve choosing a decomposition of $W$. |
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After all, one of the virtues of our starting point --- TQFTs via field and local relations --- |
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is that it has just this sort of manifest invariance.) |
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\item The solution is to replace $A_{Kh}(W^4, L)$, which is a quotient |
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\[ |
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\text{linear combinations of fields} \;\big/\; \text{local relations} , |
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\] |
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with an appropriately free resolution (the ``blob complex") |
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\[ |
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\cdots\to \bc_2(W, L) \to \bc_1(W, L) \to \bc_0(W, L) . |
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\] |
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Here $\bc_0$ is linear combinations of fields on $W$, |
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$\bc_1$ is linear combinations of local relations on $W$, |
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$\bc_2$ is linear combinations of relations amongst relations on $W$, |
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and so on. |
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\item None of the above ideas depend on the details of the Khovanov homology example, |
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so we develop the general theory in the paper and postpone specific applications |
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to later papers. |
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\item The blob complex enjoys the following nice properties \nn{...} |
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\end{itemize} |
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\bigskip |
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\hrule |
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\bigskip |
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|
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We then show that blob homology enjoys the following |
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\ref{property:gluing} properties. |
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|
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\begin{property}[Functoriality] |
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\label{property:functoriality}% |
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Blob homology is functorial with respect to diffeomorphisms. That is, fixing an $n$-dimensional system of fields $\cF$ and local relations $\cU$, the association |
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\begin{equation*} |
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X \mapsto \bc_*^{\cF,\cU}(X) |
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\end{equation*} |
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is a functor from $n$-manifolds and diffeomorphisms between them to chain complexes and isomorphisms between them. |
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\end{property} |
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192 |
|
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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 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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|
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\begin{property}[A map for gluing] |
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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 $\bdy X_2$, |
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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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\end{property} |
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209 |
|
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\begin{property}[Contractibility] |
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\label{property:contractibility}% |
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\todo{Err, requires a splitting?} |
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The blob complex for an $n$-category on an $n$-ball is quasi-isomorphic to its $0$-th homology. |
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\begin{equation} |
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\xymatrix{\bc_*^{\cC}(B^n) \ar[r]^{\iso}_{\text{qi}} & H_0(\bc_*^{\cC}(B^n))} |
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\end{equation} |
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\todo{Say that this is just the original $n$-category?} |
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\end{property} |
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219 |
|
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\begin{property}[Skein modules] |
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\label{property:skein-modules}% |
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The $0$-th blob homology of $X$ is the usual skein module associated to $X$. (See \S \ref{sec:local-relations}.) |
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\begin{equation*} |
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H_0(\bc_*^{\cF,\cU}(X)) \iso A^{\cF,\cU}(X) |
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225 |
\end{equation*} |
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\end{property} |
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\begin{property}[Hochschild homology when $X=S^1$] |
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\label{property:hochschild}% |
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230 |
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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232 |
\begin{equation*} |
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\xymatrix{\bc_*^{\cC}(S^1) \ar[r]^{\iso}_{\text{qi}} & HC_*(\cC)} |
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234 |
\end{equation*} |
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\end{property} |
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236 |
|
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237 |
\begin{property}[Evaluation map] |
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\label{property:evaluation}% |
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239 |
There is an `evaluation' chain map |
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240 |
\begin{equation*} |
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241 |
\ev_X: \CD{X} \tensor \bc_*(X) \to \bc_*(X). |
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242 |
\end{equation*} |
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(Here $\CD{X}$ is the singular chain complex of the space of diffeomorphisms of $X$, fixed on $\bdy X$.) |
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244 |
|
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Restricted to $C_0(\Diff(X))$ this is just the action of diffeomorphisms described in Property \ref{property:functoriality}. Further, for |
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any codimension $1$-submanifold $Y \subset X$ dividing $X$ into $X_1 \cup_Y X_2$, the following diagram |
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(using the gluing maps described in Property \ref{property:gluing-map}) commutes. |
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248 |
\begin{equation*} |
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249 |
\xymatrix{ |
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\CD{X} \otimes \bc_*(X) \ar[r]^{\ev_X} & \bc_*(X) \\ |
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\CD{X_1} \otimes \CD{X_2} \otimes \bc_*(X_1) \otimes \bc_*(X_2) |
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\ar@/_4ex/[r]_{\ev_{X_1} \otimes \ev_{X_2}} \ar[u]^{\gl^{\Diff}_Y \otimes \gl_Y} & |
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\bc_*(X_1) \otimes \bc_*(X_2) \ar[u]_{\gl_Y} |
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254 |
} |
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\end{equation*} |
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256 |
\end{property} |
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257 |
|
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258 |
\begin{property}[Gluing formula] |
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\label{property:gluing}% |
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\mbox{}% <-- gets the indenting right |
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\begin{itemize} |
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\item For any $(n-1)$-manifold $Y$, the blob homology of $Y \times I$ is |
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naturally an $A_\infty$ category. % We'll write $\bc_*(Y)$ for $\bc_*(Y \times I)$ below. |
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|
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\item For any $n$-manifold $X$, with $Y$ a codimension $0$-submanifold of its boundary, the blob homology of $X$ is naturally an |
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$A_\infty$ module for $\bc_*(Y \times I)$. |
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267 |
|
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\item For any $n$-manifold $X$, with $Y \cup Y^{\text{op}}$ a codimension |
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$0$-submanifold of its boundary, the blob homology of $X'$, obtained from |
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$X$ by gluing along $Y$, is the $A_\infty$ self-tensor product of |
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$\bc_*(X)$ as an $\bc_*(Y \times I)$-bimodule. |
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\begin{equation*} |
54 | 273 |
\bc_*(X') \iso \bc_*(X) \Tensor^{A_\infty}_{\mathclap{\bc_*(Y \times I)}} \!\!\!\!\!\!\xymatrix{ \ar@(ru,rd)@<-1ex>[]} |
22
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\end{equation*} |
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\end{itemize} |
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\end{property} |
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|
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Properties \ref{property:functoriality}, \ref{property:gluing-map} and \ref{property:skein-modules} will be immediate from the definition given in |
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\S \ref{sec:blob-definition}, and we'll recall them at the appropriate points there. \todo{Make sure this gets done.} |
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Properties \ref{property:disjoint-union} and \ref{property:contractibility} are established in \S \ref{sec:basic-properties}. |
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Property \ref{property:hochschild} is established in \S \ref{sec:hochschild}, Property \ref{property:evaluation} in \S \ref{sec:evaluation}, |
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and Property \ref{property:gluing} in \S \ref{sec:gluing}. |
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283 |
|
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284 |
\section{Definitions} |
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\label{sec:definitions} |
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|
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287 |
\subsection{Systems of fields} |
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\label{sec:fields} |
0 | 289 |
|
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Let $\cM_k$ denote the category (groupoid, in fact) with objects |
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291 |
oriented PL manifolds of dimension |
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$k$ and morphisms homeomorphisms. |
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(We could equally well work with a different category of manifolds --- |
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unoriented, topological, smooth, spin, etc. --- but for definiteness we |
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will stick with oriented PL.) |
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|
67 | 297 |
Fix a top dimension $n$, and a symmetric monoidal category $\cS$ whose objects are sets. While reading the definition, you should just think about the case $\cS = \Set$ with cartesian product, until you reach the discussion of a \emph{linear system of fields} later in this section, where $\cS = \Vect$, and \S \ref{sec:homological-fields}, where $\cS = \Kom$. |
0 | 298 |
|
67 | 299 |
A $n$-dimensional {\it system of fields} in $\cS$ |
300 |
is a collection of functors $\cC_k : \cM_k \to \Set$ for $0 \leq k \leq n$ |
|
59
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together with some additional data and satisfying some additional conditions, all specified below. |
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302 |
|
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\nn{refer somewhere to my TQFT notes \cite{kw:tqft}, and possibly also to paper with Chris} |
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304 |
|
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Before finishing the definition of fields, we give two motivating examples |
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306 |
(actually, families of examples) of systems of fields. |
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307 |
|
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The first examples: Fix a target space $B$, and let $\cC(X)$ be the set of continuous maps |
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from X to $B$. |
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310 |
|
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The second examples: Fix an $n$-category $C$, and let $\cC(X)$ be |
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312 |
the set of sub-cell-complexes of $X$ with codimension-$j$ cells labeled by |
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$j$-morphisms of $C$. |
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314 |
One can think of such sub-cell-complexes as dual to pasting diagrams for $C$. |
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315 |
This is described in more detail below. |
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316 |
|
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317 |
Now for the rest of the definition of system of fields. |
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318 |
\begin{enumerate} |
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\item There are boundary restriction maps $\cC_k(X) \to \cC_{k-1}(\bd X)$, |
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and these maps are a natural |
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321 |
transformation between the functors $\cC_k$ and $\cC_{k-1}\circ\bd$. |
60
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322 |
For $c \in \cC_{k-1}(\bd X)$, we will denote by $\cC_k(X; c)$ the subset of |
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323 |
$\cC(X)$ which restricts to $c$. |
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324 |
In this context, we will call $c$ a boundary condition. |
67 | 325 |
\item The subset $\cC_n(X;c)$ of top fields with a given boundary condition is an object in our symmetric monoidal category $\cS$. (This condition is of course trivial when $\cS = \Set$.) If the objects are sets with extra structure (e.g. $\cS = \Vect$ or $\Kom$), then this extra structure is considered part of the definition of $\cC_n$. Any maps mentioned below between top level fields must be morphisms in $\cS$. |
59
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326 |
\item There are orientation reversal maps $\cC_k(X) \to \cC_k(-X)$, and these maps |
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327 |
again comprise a natural transformation of functors. |
60
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328 |
In addition, the orientation reversal maps are compatible with the boundary restriction maps. |
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329 |
\item $\cC_k$ is compatible with the symmetric monoidal |
67 | 330 |
structures on $\cM_k$, $\Set$ and $\cS$: $\cC_k(X \du W) \cong \cC_k(X)\times \cC_k(W)$, |
59
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331 |
compatibly with homeomorphisms, restriction to boundary, and orientation reversal. |
62
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332 |
We will call the projections $\cC(X_1 \du X_2) \to \cC(X_i)$ |
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333 |
restriction maps. |
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334 |
\item Gluing without corners. |
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335 |
Let $\bd X = Y \du -Y \du W$, where $Y$ and $W$ are closed $k{-}1$-manifolds. |
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336 |
Let $X\sgl$ denote $X$ glued to itself along $\pm Y$. |
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337 |
Using the boundary restriction, disjoint union, and (in one case) orientation reversal |
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338 |
maps, we get two maps $\cC_k(X) \to \cC(Y)$, corresponding to the two |
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339 |
copies of $Y$ in $\bd X$. |
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340 |
Let $\Eq_Y(\cC_k(X))$ denote the equalizer of these two maps. |
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341 |
Then (here's the axiom/definition part) there is an injective ``gluing" map |
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342 |
\[ |
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343 |
\Eq_Y(\cC_k(X)) \hookrightarrow \cC_k(X\sgl) , |
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344 |
\] |
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345 |
and this gluing map is compatible with all of the above structure (actions |
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346 |
of homeomorphisms, boundary restrictions, orientation reversal, disjoint union). |
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347 |
Furthermore, up to homeomorphisms of $X\sgl$ isotopic to the identity, |
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the gluing map is surjective. |
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349 |
From the point of view of $X\sgl$ and the image $Y \subset X\sgl$ of the |
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350 |
gluing surface, we say that fields in the image of the gluing map |
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351 |
are transverse to $Y$ or cuttable along $Y$. |
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352 |
\item Gluing with corners. |
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353 |
Let $\bd X = Y \cup -Y \cup W$, where $\pm Y$ and $W$ might intersect along their boundaries. |
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354 |
Let $X\sgl$ denote $X$ glued to itself along $\pm Y$. |
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355 |
Note that $\bd X\sgl = W\sgl$, where $W\sgl$ denotes $W$ glued to itself |
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356 |
(without corners) along two copies of $\bd Y$. |
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357 |
Let $c\sgl \in \cC_{k-1}(W\sgl)$ be a be a cuttable field on $W\sgl$ and let |
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358 |
$c \in \cC_{k-1}(W)$ be the cut open version of $c\sgl$. |
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359 |
Let $\cC^c_k(X)$ denote the subset of $\cC(X)$ which restricts to $c$ on $W$. |
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|
360 |
(This restriction map uses the gluing without corners map above.) |
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|
361 |
Using the boundary restriction, gluing without corners, and (in one case) orientation reversal |
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|
362 |
maps, we get two maps $\cC^c_k(X) \to \cC(Y)$, corresponding to the two |
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|
363 |
copies of $Y$ in $\bd X$. |
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|
364 |
Let $\Eq^c_Y(\cC_k(X))$ denote the equalizer of these two maps. |
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|
365 |
Then (here's the axiom/definition part) there is an injective ``gluing" map |
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|
366 |
\[ |
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|
367 |
\Eq^c_Y(\cC_k(X)) \hookrightarrow \cC_k(X\sgl, c\sgl) , |
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|
368 |
\] |
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|
369 |
and this gluing map is compatible with all of the above structure (actions |
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|
370 |
of homeomorphisms, boundary restrictions, orientation reversal, disjoint union). |
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|
371 |
Furthermore, up to homeomorphisms of $X\sgl$ isotopic to the identity, |
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|
372 |
the gluing map is surjective. |
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|
373 |
From the point of view of $X\sgl$ and the image $Y \subset X\sgl$ of the |
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|
374 |
gluing surface, we say that fields in the image of the gluing map |
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|
375 |
are transverse to $Y$ or cuttable along $Y$. |
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|
376 |
\item There are maps $\cC_{k-1}(Y) \to \cC_k(Y \times I)$, denoted |
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|
377 |
$c \mapsto c\times I$. |
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|
378 |
These maps comprise a natural transformation of functors, and commute appropriately |
62
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|
379 |
with all the structure maps above (disjoint union, boundary restriction, etc.). |
60
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|
380 |
Furthermore, if $f: Y\times I \to Y\times I$ is a fiber-preserving homeomorphism |
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|
381 |
covering $\bar{f}:Y\to Y$, then $f(c\times I) = \bar{f}(c)\times I$. |
59
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|
382 |
\end{enumerate} |
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|
383 |
|
62
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|
384 |
\nn{need to introduce two notations for glued fields --- $x\bullet y$ and $x\sgl$} |
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|
385 |
|
61
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local relation def; extended isotopy def
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|
386 |
\bigskip |
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local relation def; extended isotopy def
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|
387 |
Using the functoriality and $\bullet\times I$ properties above, together |
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local relation def; extended isotopy def
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|
388 |
with boundary collar homeomorphisms of manifolds, we can define the notion of |
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|
389 |
{\it extended isotopy}. |
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|
390 |
Let $M$ be an $n$-manifold and $Y \subset \bd M$ be a codimension zero submanifold |
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|
391 |
of $\bd M$. |
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|
392 |
Let $x \in \cC(M)$ be a field on $M$ and such that $\bd x$ is cuttable along $\bd Y$. |
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|
393 |
Let $c$ be $x$ restricted to $Y$. |
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|
394 |
Let $M \cup (Y\times I)$ denote $M$ glued to $Y\times I$ along $Y$. |
62
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|
395 |
Then we have the glued field $x \bullet (c\times I)$ on $M \cup (Y\times I)$. |
61
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|
396 |
Let $f: M \cup (Y\times I) \to M$ be a collaring homeomorphism. |
62
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|
397 |
Then we say that $x$ is {\it extended isotopic} to $f(x \bullet (c\times I))$. |
61
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|
398 |
More generally, we define extended isotopy to be the equivalence relation on fields |
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|
399 |
on $M$ generated by isotopy plus all instance of the above construction |
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|
400 |
(for all appropriate $Y$ and $x$). |
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|
401 |
|
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|
402 |
\nn{should also say something about pseudo-isotopy} |
59
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|
403 |
|
67 | 404 |
%\bigskip |
405 |
%\hrule |
|
406 |
%\bigskip |
|
407 |
% |
|
408 |
%\input{text/fields.tex} |
|
409 |
% |
|
410 |
% |
|
411 |
%\bigskip |
|
412 |
%\hrule |
|
413 |
%\bigskip |
|
60
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|
414 |
|
8 | 415 |
\nn{note: probably will suppress from notation the distinction |
0 | 416 |
between fields and their (orientation-reversal) duals} |
417 |
||
418 |
\nn{remark that if top dimensional fields are not already linear |
|
419 |
then we will soon linearize them(?)} |
|
420 |
||
60
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|
421 |
We now describe in more detail systems of fields coming from sub-cell-complexes labeled |
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|
422 |
by $n$-category morphisms. |
0 | 423 |
|
8 | 424 |
Given an $n$-category $C$ with the right sort of duality |
425 |
(e.g. pivotal 2-category, 1-category with duals, star 1-category, disklike $n$-category), |
|
0 | 426 |
we can construct a system of fields as follows. |
427 |
Roughly speaking, $\cC(X)$ will the set of all embedded cell complexes in $X$ |
|
428 |
with codimension $i$ cells labeled by $i$-morphisms of $C$. |
|
429 |
We'll spell this out for $n=1,2$ and then describe the general case. |
|
430 |
||
431 |
If $X$ has boundary, we require that the cell decompositions are in general |
|
432 |
position with respect to the boundary --- the boundary intersects each cell |
|
433 |
transversely, so cells meeting the boundary are mere half-cells. |
|
434 |
||
435 |
Put another way, the cell decompositions we consider are dual to standard cell |
|
436 |
decompositions of $X$. |
|
437 |
||
438 |
We will always assume that our $n$-categories have linear $n$-morphisms. |
|
439 |
||
440 |
For $n=1$, a field on a 0-manifold $P$ is a labeling of each point of $P$ with |
|
441 |
an object (0-morphism) of the 1-category $C$. |
|
442 |
A field on a 1-manifold $S$ consists of |
|
443 |
\begin{itemize} |
|
8 | 444 |
\item A cell decomposition of $S$ (equivalently, a finite collection |
0 | 445 |
of points in the interior of $S$); |
8 | 446 |
\item a labeling of each 1-cell (and each half 1-cell adjacent to $\bd S$) |
0 | 447 |
by an object (0-morphism) of $C$; |
8 | 448 |
\item a transverse orientation of each 0-cell, thought of as a choice of |
0 | 449 |
``domain" and ``range" for the two adjacent 1-cells; and |
8 | 450 |
\item a labeling of each 0-cell by a morphism (1-morphism) of $C$, with |
0 | 451 |
domain and range determined by the transverse orientation and the labelings of the 1-cells. |
452 |
\end{itemize} |
|
453 |
||
454 |
If $C$ is an algebra (i.e. if $C$ has only one 0-morphism) we can ignore the labels |
|
8 | 455 |
of 1-cells, so a field on a 1-manifold $S$ is a finite collection of points in the |
0 | 456 |
interior of $S$, each transversely oriented and each labeled by an element (1-morphism) |
457 |
of the algebra. |
|
458 |
||
4
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|
459 |
\medskip |
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|
460 |
|
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|
461 |
For $n=2$, fields are just the sort of pictures based on 2-categories (e.g.\ tensor categories) |
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|
462 |
that are common in the literature. |
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|
463 |
We describe these carefully here. |
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|
464 |
|
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|
465 |
A field on a 0-manifold $P$ is a labeling of each point of $P$ with |
0 | 466 |
an object of the 2-category $C$. |
467 |
A field of a 1-manifold is defined as in the $n=1$ case, using the 0- and 1-morphisms of $C$. |
|
468 |
A field on a 2-manifold $Y$ consists of |
|
469 |
\begin{itemize} |
|
8 | 470 |
\item A cell decomposition of $Y$ (equivalently, a graph embedded in $Y$ such |
0 | 471 |
that each component of the complement is homeomorphic to a disk); |
8 | 472 |
\item a labeling of each 2-cell (and each partial 2-cell adjacent to $\bd Y$) |
0 | 473 |
by a 0-morphism of $C$; |
8 | 474 |
\item a transverse orientation of each 1-cell, thought of as a choice of |
0 | 475 |
``domain" and ``range" for the two adjacent 2-cells; |
8 | 476 |
\item a labeling of each 1-cell by a 1-morphism of $C$, with |
477 |
domain and range determined by the transverse orientation of the 1-cell |
|
0 | 478 |
and the labelings of the 2-cells; |
8 | 479 |
\item for each 0-cell, a homeomorphism of the boundary $R$ of a small neighborhood |
0 | 480 |
of the 0-cell to $S^1$ such that the intersections of the 1-cells with $R$ are not mapped |
481 |
to $\pm 1 \in S^1$; and |
|
8 | 482 |
\item a labeling of each 0-cell by a 2-morphism of $C$, with domain and range |
0 | 483 |
determined by the labelings of the 1-cells and the parameterizations of the previous |
484 |
bullet. |
|
485 |
\end{itemize} |
|
486 |
\nn{need to say this better; don't try to fit everything into the bulleted list} |
|
487 |
||
488 |
For general $n$, a field on a $k$-manifold $X^k$ consists of |
|
489 |
\begin{itemize} |
|
8 | 490 |
\item A cell decomposition of $X$; |
491 |
\item an explicit general position homeomorphism from the link of each $j$-cell |
|
0 | 492 |
to the boundary of the standard $(k-j)$-dimensional bihedron; and |
8 | 493 |
\item a labeling of each $j$-cell by a $(k-j)$-dimensional morphism of $C$, with |
0 | 494 |
domain and range determined by the labelings of the link of $j$-cell. |
495 |
\end{itemize} |
|
496 |
||
4
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|
497 |
%\nn{next definition might need some work; I think linearity relations should |
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|
498 |
%be treated differently (segregated) from other local relations, but I'm not sure |
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|
499 |
%the next definition is the best way to do it} |
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|
500 |
|
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|
501 |
\medskip |
0 | 502 |
|
8 | 503 |
For top dimensional ($n$-dimensional) manifolds, we're actually interested |
0 | 504 |
in the linearized space of fields. |
62
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|
505 |
By default, define $\lf(X) = \c[\cC(X)]$; that is, $\lf(X)$ is |
8 | 506 |
the vector space of finite |
0 | 507 |
linear combinations of fields on $X$. |
62
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|
508 |
If $X$ has boundary, we of course fix a boundary condition: $\lf(X; a) = \c[\cC(X; a)]$. |
0 | 509 |
Thus the restriction (to boundary) maps are well defined because we never |
510 |
take linear combinations of fields with differing boundary conditions. |
|
511 |
||
512 |
In some cases we don't linearize the default way; instead we take the |
|
62
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|
513 |
spaces $\lf(X; a)$ to be part of the data for the system of fields. |
0 | 514 |
In particular, for fields based on linear $n$-category pictures we linearize as follows. |
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|
515 |
Define $\lf(X; a) = \c[\cC(X; a)]/K$, where $K$ is the space generated by |
0 | 516 |
obvious relations on 0-cell labels. |
8 | 517 |
More specifically, let $L$ be a cell decomposition of $X$ |
0 | 518 |
and let $p$ be a 0-cell of $L$. |
519 |
Let $\alpha_c$ and $\alpha_d$ be two labelings of $L$ which are identical except that |
|
520 |
$\alpha_c$ labels $p$ by $c$ and $\alpha_d$ labels $p$ by $d$. |
|
521 |
Then the subspace $K$ is generated by things of the form |
|
522 |
$\lambda \alpha_c + \alpha_d - \alpha_{\lambda c + d}$, where we leave it to the reader |
|
523 |
to infer the meaning of $\alpha_{\lambda c + d}$. |
|
524 |
Note that we are still assuming that $n$-categories have linear spaces of $n$-morphisms. |
|
525 |
||
8 | 526 |
\nn{Maybe comment further: if there's a natural basis of morphisms, then no need; |
0 | 527 |
will do something similar below; in general, whenever a label lives in a linear |
8 | 528 |
space we do something like this; ? say something about tensor |
0 | 529 |
product of all the linear label spaces? Yes:} |
530 |
||
531 |
For top dimensional ($n$-dimensional) manifolds, we linearize as follows. |
|
532 |
Define an ``almost-field" to be a field without labels on the 0-cells. |
|
533 |
(Recall that 0-cells are labeled by $n$-morphisms.) |
|
534 |
To each unlabeled 0-cell in an almost field there corresponds a (linear) $n$-morphism |
|
535 |
space determined by the labeling of the link of the 0-cell. |
|
536 |
(If the 0-cell were labeled, the label would live in this space.) |
|
537 |
We associate to each almost-labeling the tensor product of these spaces (one for each 0-cell). |
|
62
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|
538 |
We now define $\lf(X; a)$ to be the direct sum over all almost labelings of the |
0 | 539 |
above tensor products. |
540 |
||
541 |
||
542 |
||
543 |
\subsection{Local relations} |
|
22
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rearranging; stating all the "properties" up front
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|
544 |
\label{sec:local-relations} |
0 | 545 |
|
60
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|
546 |
|
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|
547 |
A {\it local relation} is a collection subspaces $U(B; c) \sub \lf(B; c)$, |
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|
548 |
for all $n$-manifolds $B$ which are |
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|
549 |
homeomorphic to the standard $n$-ball and all $c \in \cC(\bd B)$, |
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|
550 |
satisfying the following properties. |
60
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|
551 |
\begin{enumerate} |
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|
552 |
\item functoriality: |
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local relation def; extended isotopy def
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553 |
$f(U(B; c)) = U(B', f(c))$ for all homeomorphisms $f: B \to B'$ |
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554 |
\item local relations imply extended isotopy: |
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555 |
if $x, y \in \cC(B; c)$ and $x$ is extended isotopic |
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|
556 |
to $y$, then $x-y \in U(B; c)$. |
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|
557 |
\item ideal with respect to gluing: |
62
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|
558 |
if $B = B' \cup B''$, $x\in U(B')$, and $c\in \cC(B'')$, then $x\bullet r \in U(B)$ |
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|
559 |
\end{enumerate} |
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|
560 |
See \cite{kw:tqft} for details. |
0 | 561 |
|
562 |
||
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|
563 |
For maps into spaces, $U(B; c)$ is generated by things of the form $a-b \in \lf(B; c)$, |
0 | 564 |
where $a$ and $b$ are maps (fields) which are homotopic rel boundary. |
565 |
||
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|
566 |
For $n$-category pictures, $U(B; c)$ is equal to the kernel of the evaluation map |
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|
567 |
$\lf(B; c) \to \mor(c', c'')$, where $(c', c'')$ is some (any) division of $c$ into |
0 | 568 |
domain and range. |
569 |
||
570 |
\nn{maybe examples of local relations before general def?} |
|
571 |
||
572 |
Given a system of fields and local relations, we define the skein space |
|
573 |
$A(Y^n; c)$ to be the space of all finite linear combinations of fields on |
|
574 |
the $n$-manifold $Y$ modulo local relations. |
|
575 |
The Hilbert space $Z(Y; c)$ for the TQFT based on the fields and local relations |
|
576 |
is defined to be the dual of $A(Y; c)$. |
|
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|
577 |
(See \cite{kw:tqft} or xxxx for details.) |
0 | 578 |
|
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|
579 |
\nn{should expand above paragraph} |
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|
580 |
|
0 | 581 |
The blob complex is in some sense the derived version of $A(Y; c)$. |
582 |
||
583 |
||
584 |
||
585 |
\subsection{The blob complex} |
|
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|
586 |
\label{sec:blob-definition} |
0 | 587 |
|
588 |
Let $X$ be an $n$-manifold. |
|
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|
589 |
Assume a fixed system of fields and local relations. |
0 | 590 |
In this section we will usually suppress boundary conditions on $X$ from the notation |
62
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|
591 |
(e.g. write $\lf(X)$ instead of $\lf(X; c)$). |
0 | 592 |
|
8 | 593 |
We only consider compact manifolds, so if $Y \sub X$ is a closed codimension 0 |
0 | 594 |
submanifold of $X$, then $X \setmin Y$ implicitly means the closure |
595 |
$\overline{X \setmin Y}$. |
|
596 |
||
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|
597 |
We will define $\bc_0(X)$, $\bc_1(X)$ and $\bc_2(X)$, then give the general case $\bc_k(X)$. |
0 | 598 |
|
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|
599 |
Define $\bc_0(X) = \lf(X)$. |
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|
600 |
(If $X$ has nonempty boundary, instead define $\bc_0(X; c) = \lf(X; c)$. |
0 | 601 |
We'll omit this sort of detail in the rest of this section.) |
602 |
In other words, $\bc_0(X)$ is just the space of all linearized fields on $X$. |
|
603 |
||
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|
604 |
$\bc_1(X)$ is, roughly, the space of all local relations that can be imposed on $\bc_0(X)$. |
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|
605 |
Less roughly (but still not the official definition), $\bc_1(X)$ is finite linear |
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|
606 |
combinations of 1-blob diagrams, where a 1-blob diagram to consists of |
0 | 607 |
\begin{itemize} |
608 |
\item An embedded closed ball (``blob") $B \sub X$. |
|
609 |
\item A field $r \in \cC(X \setmin B; c)$ |
|
610 |
(for some $c \in \cC(\bd B) = \cC(\bd(X \setmin B))$). |
|
611 |
\item A local relation field $u \in U(B; c)$ |
|
612 |
(same $c$ as previous bullet). |
|
613 |
\end{itemize} |
|
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|
614 |
In order to get the linear structure correct, we (officially) define |
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|
615 |
\[ |
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|
616 |
\bc_1(X) \deq \bigoplus_B \bigoplus_c U(B; c) \otimes \lf(X \setmin B; c) . |
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|
617 |
\] |
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|
618 |
The first direct sum is indexed by all blobs $B\subset X$, and the second |
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|
619 |
by all boundary conditions $c \in \cC(\bd B)$. |
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|
620 |
Note that $\bc_1(X)$ is spanned by 1-blob diagrams $(B, u, r)$. |
0 | 621 |
|
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|
622 |
Define the boundary map $\bd : \bc_1(X) \to \bc_0(X)$ by |
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|
623 |
\[ |
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|
624 |
(B, u, r) \mapsto u\bullet r, |
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|
625 |
\] |
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|
626 |
where $u\bullet r$ denotes the linear |
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|
627 |
combination of fields on $X$ obtained by gluing $u$ to $r$. |
8 | 628 |
In other words $\bd : \bc_1(X) \to \bc_0(X)$ is given by |
0 | 629 |
just erasing the blob from the picture |
630 |
(but keeping the blob label $u$). |
|
631 |
||
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|
632 |
Note that the skein space $A(X)$ |
0 | 633 |
is naturally isomorphic to $\bc_0(X)/\bd(\bc_1(X))) = H_0(\bc_*(X))$. |
634 |
||
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|
635 |
$\bc_2(X)$ is, roughly, the space of all relations (redundancies) among the |
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|
636 |
local relations encoded in $\bc_1(X)$. |
8 | 637 |
More specifically, $\bc_2(X)$ is the space of all finite linear combinations of |
63
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|
638 |
2-blob diagrams, of which there are two types, disjoint and nested. |
0 | 639 |
|
640 |
A disjoint 2-blob diagram consists of |
|
641 |
\begin{itemize} |
|
63
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|
642 |
\item A pair of closed balls (blobs) $B_0, B_1 \sub X$ with disjoint interiors. |
0 | 643 |
\item A field $r \in \cC(X \setmin (B_0 \cup B_1); c_0, c_1)$ |
644 |
(where $c_i \in \cC(\bd B_i)$). |
|
63
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|
645 |
\item Local relation fields $u_i \in U(B_i; c_i)$, $i=1,2$. |
0 | 646 |
\end{itemize} |
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|
647 |
We also identify $(B_0, B_1, u_0, u_1, r)$ with $-(B_1, B_0, u_1, u_0, r)$; |
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|
648 |
reversing the order of the blobs changes the sign. |
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|
649 |
Define $\bd(B_0, B_1, u_0, u_1, r) = |
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|
650 |
(B_1, u_1, u_0\bullet r) - (B_0, u_0, u_1\bullet r) \in \bc_1(X)$. |
0 | 651 |
In other words, the boundary of a disjoint 2-blob diagram |
652 |
is the sum (with alternating signs) |
|
653 |
of the two ways of erasing one of the blobs. |
|
654 |
It's easy to check that $\bd^2 = 0$. |
|
655 |
||
656 |
A nested 2-blob diagram consists of |
|
657 |
\begin{itemize} |
|
658 |
\item A pair of nested balls (blobs) $B_0 \sub B_1 \sub X$. |
|
659 |
\item A field $r \in \cC(X \setmin B_0; c_0)$ |
|
63
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|
660 |
(for some $c_0 \in \cC(\bd B_0)$), which is cuttable along $\bd B_1$. |
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|
661 |
\item A local relation field $u_0 \in U(B_0; c_0)$. |
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|
662 |
\end{itemize} |
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|
663 |
Let $r = r_1 \bullet r'$, where $r_1 \in \cC(B_1 \setmin B_0; c_0, c_1)$ |
0 | 664 |
(for some $c_1 \in \cC(B_1)$) and |
665 |
$r' \in \cC(X \setmin B_1; c_1)$. |
|
63
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|
666 |
Define $\bd(B_0, B_1, u_0, r) = (B_1, u_0\bullet r_1, r') - (B_0, u_0, r)$. |
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|
667 |
Note that the requirement that |
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|
668 |
local relations are an ideal with respect to gluing guarantees that $u_0\bullet r_1 \in U(B_1)$. |
0 | 669 |
As in the disjoint 2-blob case, the boundary of a nested 2-blob is the alternating |
670 |
sum of the two ways of erasing one of the blobs. |
|
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|
671 |
If we erase the inner blob, the outer blob inherits the label $u_0\bullet r_1$. |
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|
672 |
It is again easy to check that $\bd^2 = 0$. |
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|
673 |
|
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|
674 |
\nn{should draw figures for 1, 2 and $k$-blob diagrams} |
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|
675 |
|
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|
676 |
As with the 1-blob diagrams, in order to get the linear structure correct it is better to define |
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|
677 |
(officially) |
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|
678 |
\begin{eqnarray*} |
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|
679 |
\bc_2(X) & \deq & |
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|
680 |
\left( |
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|
681 |
\bigoplus_{B_0, B_1 \text{disjoint}} \bigoplus_{c_0, c_1} |
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|
682 |
U(B_0; c_0) \otimes U(B_1; c_1) \otimes \lf(X\setmin (B_0\cup B_1); c_0, c_1) |
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|
683 |
\right) \\ |
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|
684 |
&& \bigoplus \left( |
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|
685 |
\bigoplus_{B_0 \subset B_1} \bigoplus_{c_0} |
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|
686 |
U(B_0; c_0) \otimes \lf(X\setmin B_0; c_0) |
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|
687 |
\right) . |
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|
688 |
\end{eqnarray*} |
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|
689 |
The final $\lf(X\setmin B_0; c_0)$ above really means fields cuttable along $\bd B_1$, |
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690 |
but we didn't feel like introducing a notation for that. |
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|
691 |
For the disjoint blobs, reversing the ordering of $B_0$ and $B_1$ introduces a minus sign |
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|
692 |
(rather than a new, linearly independent 2-blob diagram). |
0 | 693 |
|
694 |
Now for the general case. |
|
695 |
A $k$-blob diagram consists of |
|
696 |
\begin{itemize} |
|
697 |
\item A collection of blobs $B_i \sub X$, $i = 0, \ldots, k-1$. |
|
63
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|
698 |
For each $i$ and $j$, we require that either $B_i$ and $B_j$have disjoint interiors or |
0 | 699 |
$B_i \sub B_j$ or $B_j \sub B_i$. |
700 |
(The case $B_i = B_j$ is allowed. |
|
701 |
If $B_i \sub B_j$ the boundaries of $B_i$ and $B_j$ are allowed to intersect.) |
|
702 |
If a blob has no other blobs strictly contained in it, we call it a twig blob. |
|
63
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703 |
\item Fields (boundary conditions) $c_i \in \cC(\bd B_i)$. |
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|
704 |
(These are implied by the data in the next bullets, so we usually |
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|
705 |
suppress them from the notation.) |
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|
706 |
$c_i$ and $c_j$ must have identical restrictions to $\bd B_i \cap \bd B_j$ |
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|
707 |
if the latter space is not empty. |
0 | 708 |
\item A field $r \in \cC(X \setmin B^t; c^t)$, |
63
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|
709 |
where $B^t$ is the union of all the twig blobs and $c^t \in \cC(\bd B^t)$ |
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|
710 |
is determined by the $c_i$'s. |
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|
711 |
$r$ is required to be cuttable along the boundaries of all blobs, twigs or not. |
0 | 712 |
\item For each twig blob $B_j$ a local relation field $u_j \in U(B_j; c_j)$, |
713 |
where $c_j$ is the restriction of $c^t$ to $\bd B_j$. |
|
714 |
If $B_i = B_j$ then $u_i = u_j$. |
|
715 |
\end{itemize} |
|
716 |
||
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|
717 |
If two blob diagrams $D_1$ and $D_2$ |
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|
718 |
differ only by a reordering of the blobs, then we identify |
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|
719 |
$D_1 = \pm D_2$, where the sign is the sign of the permutation relating $D_1$ and $D_2$. |
0 | 720 |
|
63
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|
721 |
$\bc_k(X)$ is, roughly, all finite linear combinations of $k$-blob diagrams. |
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|
722 |
As before, the official definition is in terms of direct sums |
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|
723 |
of tensor products: |
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|
724 |
\[ |
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|
725 |
\bc_k(X) \deq \bigoplus_{\overline{B}} \bigoplus_{\overline{c}} |
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|
726 |
\left( \otimes_j U(B_j; c_j)\right) \otimes \lf(X \setmin B^t; c^t) . |
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|
727 |
\] |
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|
728 |
Here $\overline{B}$ runs over all configurations of blobs, satisfying the conditions above. |
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|
729 |
$\overline{c}$ runs over all boundary conditions, again as described above. |
67 | 730 |
$j$ runs over all indices of twig blobs. The final $\lf(X \setmin B^t; c^t)$ must be interpreted as fields which are cuttable along all of the blobs in $\overline{B}$. |
0 | 731 |
|
732 |
The boundary map $\bd : \bc_k(X) \to \bc_{k-1}(X)$ is defined as follows. |
|
63
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|
733 |
Let $b = (\{B_i\}, \{u_j\}, r)$ be a $k$-blob diagram. |
0 | 734 |
Let $E_j(b)$ denote the result of erasing the $j$-th blob. |
735 |
If $B_j$ is not a twig blob, this involves only decrementing |
|
736 |
the indices of blobs $B_{j+1},\ldots,B_{k-1}$. |
|
737 |
If $B_j$ is a twig blob, we have to assign new local relation labels |
|
738 |
if removing $B_j$ creates new twig blobs. |
|
63
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|
739 |
If $B_l$ becomes a twig after removing $B_j$, then set $u_l = u_j\bullet r_l$, |
0 | 740 |
where $r_l$ is the restriction of $r$ to $B_l \setmin B_j$. |
741 |
Finally, define |
|
742 |
\eq{ |
|
8 | 743 |
\bd(b) = \sum_{j=0}^{k-1} (-1)^j E_j(b). |
0 | 744 |
} |
745 |
The $(-1)^j$ factors imply that the terms of $\bd^2(b)$ all cancel. |
|
746 |
Thus we have a chain complex. |
|
747 |
||
748 |
\nn{?? say something about the ``shape" of tree? (incl = cone, disj = product)} |
|
749 |
||
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|
750 |
\nn{?? remark about dendroidal sets} |
0 | 751 |
|
752 |
||
753 |
||
754 |
\section{Basic properties of the blob complex} |
|
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|
755 |
\label{sec:basic-properties} |
0 | 756 |
|
757 |
\begin{prop} \label{disjunion} |
|
758 |
There is a natural isomorphism $\bc_*(X \du Y) \cong \bc_*(X) \otimes \bc_*(Y)$. |
|
759 |
\end{prop} |
|
760 |
\begin{proof} |
|
761 |
Given blob diagrams $b_1$ on $X$ and $b_2$ on $Y$, we can combine them |
|
8 | 762 |
(putting the $b_1$ blobs before the $b_2$ blobs in the ordering) to get a |
0 | 763 |
blob diagram $(b_1, b_2)$ on $X \du Y$. |
764 |
Because of the blob reordering relations, all blob diagrams on $X \du Y$ arise this way. |
|
765 |
In the other direction, any blob diagram on $X\du Y$ is equal (up to sign) |
|
766 |
to one that puts $X$ blobs before $Y$ blobs in the ordering, and so determines |
|
767 |
a pair of blob diagrams on $X$ and $Y$. |
|
65
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|
768 |
These two maps are compatible with our sign conventions. |
0 | 769 |
The two maps are inverses of each other. |
770 |
\nn{should probably say something about sign conventions for the differential |
|
771 |
in a tensor product of chain complexes; ask Scott} |
|
772 |
\end{proof} |
|
773 |
||
774 |
For the next proposition we will temporarily restore $n$-manifold boundary |
|
775 |
conditions to the notation. |
|
776 |
||
8 | 777 |
Suppose that for all $c \in \cC(\bd B^n)$ |
778 |
we have a splitting $s: H_0(\bc_*(B^n, c)) \to \bc_0(B^n; c)$ |
|
0 | 779 |
of the quotient map |
780 |
$p: \bc_0(B^n; c) \to H_0(\bc_*(B^n, c))$. |
|
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|
781 |
For example, this is always the case if you coefficient ring is a field. |
0 | 782 |
Then |
783 |
\begin{prop} \label{bcontract} |
|
784 |
For all $c \in \cC(\bd B^n)$ the natural map $p: \bc_*(B^n, c) \to H_0(\bc_*(B^n, c))$ |
|
785 |
is a chain homotopy equivalence |
|
786 |
with inverse $s: H_0(\bc_*(B^n, c)) \to \bc_*(B^n; c)$. |
|
787 |
Here we think of $H_0(\bc_*(B^n, c))$ as a 1-step complex concentrated in degree 0. |
|
788 |
\end{prop} |
|
789 |
\begin{proof} |
|
790 |
By assumption $p\circ s = \id$, so all that remains is to find a degree 1 map |
|
791 |
$h : \bc_*(B^n; c) \to \bc_*(B^n; c)$ such that $\bd h + h\bd = \id - s \circ p$. |
|
792 |
For $i \ge 1$, define $h_i : \bc_i(B^n; c) \to \bc_{i+1}(B^n; c)$ by adding |
|
793 |
an $(i{+}1)$-st blob equal to all of $B^n$. |
|
794 |
In other words, add a new outermost blob which encloses all of the others. |
|
795 |
Define $h_0 : \bc_0(B^n; c) \to \bc_1(B^n; c)$ by setting $h_0(x)$ equal to |
|
796 |
the 1-blob with blob $B^n$ and label $x - s(p(x)) \in U(B^n; c)$. |
|
797 |
\end{proof} |
|
798 |
||
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|
799 |
Note that if there is no splitting $s$, we can let $h_0 = 0$ and get a homotopy |
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|
800 |
equivalence to the 2-step complex $U(B^n; c) \to \cC(B^n; c)$. |
0 | 801 |
|
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|
802 |
For fields based on $n$-categories, $H_0(\bc_*(B^n; c)) \cong \mor(c', c'')$, |
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|
803 |
where $(c', c'')$ is some (any) splitting of $c$ into domain and range. |
0 | 804 |
|
805 |
\medskip |
|
806 |
||
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|
807 |
\nn{Maybe there is no longer a need to repeat the next couple of props here, since we also state them in the introduction. |
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|
808 |
But I think it's worth saying that the Diff actions will be enhanced later. |
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|
809 |
Maybe put that in the intro too.} |
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|
810 |
|
0 | 811 |
As we noted above, |
812 |
\begin{prop} |
|
813 |
There is a natural isomorphism $H_0(\bc_*(X)) \cong A(X)$. |
|
814 |
\qed |
|
815 |
\end{prop} |
|
816 |
||
817 |
||
818 |
\begin{prop} |
|
819 |
For fixed fields ($n$-cat), $\bc_*$ is a functor from the category |
|
8 | 820 |
of $n$-manifolds and diffeomorphisms to the category of chain complexes and |
0 | 821 |
(chain map) isomorphisms. |
822 |
\qed |
|
823 |
\end{prop} |
|
824 |
||
825 |
In particular, |
|
826 |
\begin{prop} \label{diff0prop} |
|
827 |
There is an action of $\Diff(X)$ on $\bc_*(X)$. |
|
828 |
\qed |
|
829 |
\end{prop} |
|
830 |
||
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|
831 |
The above will be greatly strengthened in Section \ref{sec:evaluation}. |
0 | 832 |
|
833 |
\medskip |
|
834 |
||
835 |
For the next proposition we will temporarily restore $n$-manifold boundary |
|
836 |
conditions to the notation. |
|
837 |
||
838 |
Let $X$ be an $n$-manifold, $\bd X = Y \cup (-Y) \cup Z$. |
|
839 |
Gluing the two copies of $Y$ together yields an $n$-manifold $X\sgl$ |
|
840 |
with boundary $Z\sgl$. |
|
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|
841 |
Given compatible fields (boundary conditions) $a$, $b$ and $c$ on $Y$, $-Y$ and $Z$, |
0 | 842 |
we have the blob complex $\bc_*(X; a, b, c)$. |
843 |
If $b = -a$ (the orientation reversal of $a$), then we can glue up blob diagrams on |
|
844 |
$X$ to get blob diagrams on $X\sgl$: |
|
845 |
||
846 |
\begin{prop} |
|
847 |
There is a natural chain map |
|
848 |
\eq{ |
|
8 | 849 |
\gl: \bigoplus_a \bc_*(X; a, -a, c) \to \bc_*(X\sgl; c\sgl). |
0 | 850 |
} |
8 | 851 |
The sum is over all fields $a$ on $Y$ compatible at their |
0 | 852 |
($n{-}2$-dimensional) boundaries with $c$. |
853 |
`Natural' means natural with respect to the actions of diffeomorphisms. |
|
854 |
\qed |
|
855 |
\end{prop} |
|
856 |
||
857 |
The above map is very far from being an isomorphism, even on homology. |
|
22
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|
858 |
This will be fixed in Section \ref{sec:gluing} below. |
0 | 859 |
|
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|
860 |
\nn{Next para not need, since we already use bullet = gluing notation above(?)} |
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|
861 |
|
0 | 862 |
An instance of gluing we will encounter frequently below is where $X = X_1 \du X_2$ |
863 |
and $X\sgl = X_1 \cup_Y X_2$. |
|
864 |
(Typically one of $X_1$ or $X_2$ is a disjoint union of balls.) |
|
865 |
For $x_i \in \bc_*(X_i)$, we introduce the notation |
|
866 |
\eq{ |
|
8 | 867 |
x_1 \bullet x_2 \deq \gl(x_1 \otimes x_2) . |
0 | 868 |
} |
869 |
Note that we have resumed our habit of omitting boundary labels from the notation. |
|
870 |
||
871 |
||
872 |
||
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|
873 |
|
0 | 874 |
|
15
7340ab80db25
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|
875 |
\section{Hochschild homology when $n=1$} |
7340ab80db25
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|
876 |
\label{sec:hochschild} |
7340ab80db25
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|
877 |
\input{text/hochschild} |
7
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|
878 |
|
22
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|
879 |
\section{Action of $\CD{X}$} |
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|
880 |
\label{sec:evaluation} |
0 | 881 |
|
69 | 882 |
Let $CD_*(X, Y)$ denote $C_*(\Diff(X \to Y))$, the singular chain complex of |
0 | 883 |
the space of diffeomorphisms |
69 | 884 |
\nn{or homeomorphisms} |
885 |
between the $n$-manifolds $X$ and $Y$ (extending a fixed diffeomorphism $\bd X \to \bd Y$). |
|
886 |
For convenience, we will permit the singular cells generating $CD_*(X, Y)$ to be more general |
|
0 | 887 |
than simplices --- they can be based on any linear polyhedron. |
888 |
\nn{be more restrictive here? does more need to be said?} |
|
69 | 889 |
We also will use the abbreviated notation $CD_*(X) \deq CD_*(X, X)$. |
0 | 890 |
|
891 |
\begin{prop} \label{CDprop} |
|
892 |
For each $n$-manifold $X$ there is a chain map |
|
893 |
\eq{ |
|
69 | 894 |
e_X : CD_*(X, Y) \otimes \bc_*(X) \to \bc_*(Y) . |
0 | 895 |
} |
69 | 896 |
On $CD_0(X, Y) \otimes \bc_*(X)$ it agrees with the obvious action of $\Diff(X, Y)$ on $\bc_*(X)$ |
0 | 897 |
(Proposition (\ref{diff0prop})). |
69 | 898 |
For any splittings $X = X_1 \cup X_2$ and $Y = Y_1 \cup Y_2$, |
899 |
the following diagram commutes up to homotopy |
|
0 | 900 |
\eq{ \xymatrix{ |
69 | 901 |
CD_*(X, Y) \otimes \bc_*(X) \ar[r]^{e_X} & \bc_*(Y) \\ |
902 |
CD_*(X_1, Y_1) \otimes CD_*(X_2, Y_2) \otimes \bc_*(X_1) \otimes \bc_*(X_2) |
|
8 | 903 |
\ar@/_4ex/[r]_{e_{X_1} \otimes e_{X_2}} \ar[u]^{\gl \otimes \gl} & |
69 | 904 |
\bc_*(Y_1) \otimes \bc_*(Y_2) \ar[u]_{\gl} |
0 | 905 |
} } |
906 |
Any other map satisfying the above two properties is homotopic to $e_X$. |
|
907 |
\end{prop} |
|
908 |
||
18
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|
909 |
\nn{need to rewrite for self-gluing instead of gluing two pieces together} |
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|
910 |
|
16
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|
911 |
\nn{Should say something stronger about uniqueness. |
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|
912 |
Something like: there is |
19 | 913 |
a contractible subcomplex of the complex of chain maps |
16
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|
914 |
$CD_*(X) \otimes \bc_*(X) \to \bc_*(X)$ (0-cells are the maps, 1-cells are homotopies, etc.), |
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|
915 |
and all choices in the construction lie in the 0-cells of this |
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|
916 |
contractible subcomplex. |
19 | 917 |
Or maybe better to say any two choices are homotopic, and |
16
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|
918 |
any two homotopies and second order homotopic, and so on.} |
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|
919 |
|
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|
920 |
\nn{Also need to say something about associativity. |
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|
921 |
Put it in the above prop or make it a separate prop? |
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|
922 |
I lean toward the latter.} |
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|
923 |
\medskip |
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|
924 |
|
0 | 925 |
The proof will occupy the remainder of this section. |
18
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|
926 |
\nn{unless we put associativity prop at end} |
0 | 927 |
|
69 | 928 |
Without loss of generality, we will assume $X = Y$. |
929 |
||
0 | 930 |
\medskip |
931 |
||
932 |
Let $f: P \times X \to X$ be a family of diffeomorphisms and $S \sub X$. |
|
933 |
We say that {\it $f$ is supported on $S$} if $f(p, x) = f(q, x)$ for all |
|
42
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|
934 |
$x \notin S$ and $p, q \in P$. Equivalently, $f$ is supported on $S$ if there is a family of diffeomorphisms $f' : P \times S \to S$ and a `background' |
40
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|
935 |
diffeomorphism $f_0 : X \to X$ so that |
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|
936 |
\begin{align} |
42
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|
937 |
f(p,s) & = f_0(f'(p,s)) \;\;\;\; \mbox{for}\; (p, s) \in P\times S \\ |
40
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|
938 |
\intertext{and} |
42
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|
939 |
f(p,x) & = f_0(x) \;\;\;\; \mbox{for}\; (p, x) \in {P \times (X \setmin S)}. |
40
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|
940 |
\end{align} |
0 | 941 |
Note that if $f$ is supported on $S$ then it is also supported on any $R \sup S$. |
942 |
||
943 |
Let $\cU = \{U_\alpha\}$ be an open cover of $X$. |
|
944 |
A $k$-parameter family of diffeomorphisms $f: P \times X \to X$ is |
|
945 |
{\it adapted to $\cU$} if there is a factorization |
|
946 |
\eq{ |
|
8 | 947 |
P = P_1 \times \cdots \times P_m |
0 | 948 |
} |
949 |
(for some $m \le k$) |
|
950 |
and families of diffeomorphisms |
|
951 |
\eq{ |
|
8 | 952 |
f_i : P_i \times X \to X |
0 | 953 |
} |
8 | 954 |
such that |
0 | 955 |
\begin{itemize} |
42
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|
956 |
\item each $f_i$ is supported on some connected $V_i \sub X$; |
40
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|
957 |
\item the sets $V_i$ are mutually disjoint; |
8 | 958 |
\item each $V_i$ is the union of at most $k_i$ of the $U_\alpha$'s, |
0 | 959 |
where $k_i = \dim(P_i)$; and |
42
9744833c9b90
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|
960 |
\item $f(p, \cdot) = g \circ f_1(p_1, \cdot) \circ \cdots \circ f_m(p_m, \cdot)$ |
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|
961 |
for all $p = (p_1, \ldots, p_m)$, for some fixed $g \in \Diff(X)$. |
0 | 962 |
\end{itemize} |
69 | 963 |
A chain $x \in CD_k(X)$ is (by definition) adapted to $\cU$ if it is the sum |
0 | 964 |
of singular cells, each of which is adapted to $\cU$. |
965 |
||
69 | 966 |
(Actually, in this section we will only need families of diffeomorphisms to be |
967 |
{\it weakly adapted} to $\cU$, meaning that the support of $f$ is contained in the union |
|
968 |
of at most $k$ of the $U_\alpha$'s.) |
|
969 |
||
0 | 970 |
\begin{lemma} \label{extension_lemma} |
971 |
Let $x \in CD_k(X)$ be a singular chain such that $\bd x$ is adapted to $\cU$. |
|
972 |
Then $x$ is homotopic (rel boundary) to some $x' \in CD_k(X)$ which is adapted to $\cU$. |
|
18
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|
973 |
Furthermore, one can choose the homotopy so that its support is equal to the support of $x$. |
0 | 974 |
\end{lemma} |
975 |
||
22
ada83e7228eb
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|
976 |
The proof will be given in Section \ref{sec:localising}. |
0 | 977 |
|
978 |
\medskip |
|
979 |
||
69 | 980 |
Before diving into the details, we outline our strategy for the proof of Proposition \ref{CDprop}. |
16
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|
981 |
|
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|
982 |
Let $p$ be a singular cell in $CD_*(X)$ and $b$ be a blob diagram in $\bc_*(X)$. |
69 | 983 |
Suppose that there exists $V \sub X$ such that |
984 |
\begin{enumerate} |
|
985 |
\item $V$ is homeomorphic to a disjoint union of balls, and |
|
986 |
\item $\supp(p) \cup \supp(b) \sub V$. |
|
987 |
\end{enumerate} |
|
0 | 988 |
|
7
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|
989 |
\medskip |
0 | 990 |
|
69 | 991 |
\nn{to be continued....} |
992 |
||
993 |
||
994 |
%\nn{say something about associativity here} |
|
18
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|
995 |
|
55 | 996 |
\input{text/A-infty.tex} |
34 | 997 |
|
55 | 998 |
\input{text/gluing.tex} |
51
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|
999 |
|
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|
1000 |
|
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|
1001 |
|
47
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|
1002 |
\section{Commutative algebras as $n$-categories} |
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|
1003 |
|
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|
1004 |
\nn{this should probably not be a section by itself. i'm just trying to write down the outline |
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|
1005 |
while it's still fresh in my mind.} |
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|
1006 |
|
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|
1007 |
If $C$ is a commutative algebra it |
48
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|
1008 |
can (and will) also be thought of as an $n$-category whose $j$-morphisms are trivial for |
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|
1009 |
$j<n$ and whose $n$-morphisms are $C$. |
47
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|
1010 |
The goal of this \nn{subsection?} is to compute |
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|
1011 |
$\bc_*(M^n, C)$ for various commutative algebras $C$. |
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|
1012 |
|
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|
1013 |
Let $k[t]$ denote the ring of polynomials in $t$ with coefficients in $k$. |
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|
1014 |
|
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|
1015 |
Let $\Sigma^i(M)$ denote the $i$-th symmetric power of $M$, the configuration space of $i$ |
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|
1016 |
unlabeled points in $M$. |
48
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|
1017 |
Note that $\Sigma^0(M)$ is a point. |
47
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|
1018 |
Let $\Sigma^\infty(M) = \coprod_{i=0}^\infty \Sigma^i(M)$. |
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|
1019 |
|
49 | 1020 |
Let $C_*(X, k)$ denote the singular chain complex of the space $X$ with coefficients in $k$. |
47
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|
1021 |
|
51
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|
1022 |
\begin{prop} \label{sympowerprop} |
52 | 1023 |
$\bc_*(M, k[t])$ is homotopy equivalent to $C_*(\Sigma^\infty(M), k)$. |
47
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|
1024 |
\end{prop} |
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|
1025 |
|
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|
1026 |
\begin{proof} |
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|
1027 |
To define the chain maps between the two complexes we will use the following lemma: |
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|
1028 |
|
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|
1029 |
\begin{lemma} |
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|
1030 |
Let $A_*$ and $B_*$ be chain complexes, and assume $A_*$ is equipped with |
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|
1031 |
a basis (e.g.\ blob diagrams or singular simplices). |
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|
1032 |
For each basis element $c \in A_*$ assume given a contractible subcomplex $R(c)_* \sub B_*$ |
48
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|
1033 |
such that $R(c')_* \sub R(c)_*$ whenever $c'$ is a basis element which is part of $\bd c$. |
47
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|
1034 |
Then the complex of chain maps (and (iterated) homotopies) $f:A_*\to B_*$ such that |
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|
1035 |
$f(c) \in R(c)_*$ for all $c$ is contractible (and in particular non-empty). |
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|
1036 |
\end{lemma} |
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|
1037 |
|
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|
1038 |
\begin{proof} |
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|
1039 |
\nn{easy, but should probably write the details eventually} |
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|
1040 |
\end{proof} |
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|
1041 |
|
48
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|
1042 |
Our first task: For each blob diagram $b$ define a subcomplex $R(b)_* \sub C_*(\Sigma^\infty(M))$ |
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|
1043 |
satisfying the conditions of the above lemma. |
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|
1044 |
If $b$ is a 0-blob diagram, then it is just a $k[t]$ field on $M$, which is a |
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|
1045 |
finite unordered collection of points of $M$ with multiplicities, which is |
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|
1046 |
a point in $\Sigma^\infty(M)$. |
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|
1047 |
Define $R(b)_*$ to be the singular chain complex of this point. |
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|
1048 |
If $(B, u, r)$ is an $i$-blob diagram, let $D\sub M$ be its support (the union of the blobs). |
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|
1049 |
The path components of $\Sigma^\infty(D)$ are contractible, and these components are indexed |
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|
1050 |
by the numbers of points in each component of $D$. |
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|
1051 |
We may assume that the blob labels $u$ have homogeneous $t$ degree in $k[t]$, and so |
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|
1052 |
$u$ picks out a component $X \sub \Sigma^\infty(D)$. |
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|
1053 |
The field $r$ on $M\setminus D$ can be thought of as a point in $\Sigma^\infty(M\setminus D)$, |
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|
1054 |
and using this point we can embed $X$ in $\Sigma^\infty(M)$. |
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|
1055 |
Define $R(B, u, r)_*$ to be the singular chain complex of $X$, thought of as a |
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|
1056 |
subspace of $\Sigma^\infty(M)$. |
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|
1057 |
It is easy to see that $R(\cdot)_*$ satisfies the condition on boundaries from the above lemma. |
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|
1058 |
Thus we have defined (up to homotopy) a map from |
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|
1059 |
$\bc_*(M^n, k[t])$ to $C_*(\Sigma^\infty(M))$. |
47
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|
1060 |
|
48
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|
1061 |
Next we define, for each simplex $c$ of $C_*(\Sigma^\infty(M))$, a contractible subspace |
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|
1062 |
$R(c)_* \sub \bc_*(M^n, k[t])$. |
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|
1063 |
If $c$ is a 0-simplex we use the identification of the fields $\cC(M)$ and |
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|
1064 |
$\Sigma^\infty(M)$ described above. |
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|
1065 |
Now let $c$ be an $i$-simplex of $\Sigma^j(M)$. |
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|
1066 |
Choose a metric on $M$, which induces a metric on $\Sigma^j(M)$. |
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|
1067 |
We may assume that the diameter of $c$ is small --- that is, $C_*(\Sigma^j(M))$ |
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|
1068 |
is homotopy equivalent to the subcomplex of small simplices. |
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|
1069 |
How small? $(2r)/3j$, where $r$ is the radius of injectivity of the metric. |
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|
1070 |
Let $T\sub M$ be the ``track" of $c$ in $M$. |
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|
1071 |
\nn{do we need to define this precisely?} |
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|
1072 |
Choose a neighborhood $D$ of $T$ which is a disjoint union of balls of small diameter. |
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|
1073 |
\nn{need to say more precisely how small} |
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|
1074 |
Define $R(c)_*$ to be $\bc_*(D, k[t]) \sub \bc_*(M^n, k[t])$. |
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|
1075 |
This is contractible by \ref{bcontract}. |
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|
1076 |
We can arrange that the boundary/inclusion condition is satisfied if we start with |
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47
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|
1077 |
low-dimensional simplices and work our way up. |
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|
1078 |
\nn{need to be more precise} |
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|
1079 |
|
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|
1080 |
\nn{still to do: show indep of choice of metric; show compositions are homotopic to the identity |
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|
1081 |
(for this, might need a lemma that says we can assume that blob diameters are small)} |
47
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|
1082 |
\end{proof} |
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|
1083 |
|
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|
1084 |
|
50 | 1085 |
\begin{prop} \label{ktcdprop} |
49 | 1086 |
The above maps are compatible with the evaluation map actions of $C_*(\Diff(M))$. |
1087 |
\end{prop} |
|
1088 |
||
1089 |
\begin{proof} |
|
1090 |
The actions agree in degree 0, and both are compatible with gluing. |
|
1091 |
(cf. uniqueness statement in \ref{CDprop}.) |
|
1092 |
\end{proof} |
|
1093 |
||
1094 |
\medskip |
|
1095 |
||
1096 |
In view of \ref{hochthm}, we have proved that $HH_*(k[t]) \cong C_*(\Sigma^\infty(S^1), k)$, |
|
1097 |
and that the cyclic homology of $k[t]$ is related to the action of rotations |
|
1098 |
on $C_*(\Sigma^\infty(S^1), k)$. |
|
1099 |
\nn{probably should put a more precise statement about cyclic homology and $S^1$ actions in the Hochschild section} |
|
1100 |
Let us check this directly. |
|
1101 |
||
50 | 1102 |
According to \nn{Loday, 3.2.2}, $HH_i(k[t]) \cong k[t]$ for $i=0,1$ and is zero for $i\ge 2$. |
1103 |
\nn{say something about $t$-degree? is this in [Loday]?} |
|
1104 |
||
49 | 1105 |
We can define a flow on $\Sigma^j(S^1)$ by having the points repel each other. |
1106 |
The fixed points of this flow are the equally spaced configurations. |
|
50 | 1107 |
This defines a map from $\Sigma^j(S^1)$ to $S^1/j$ ($S^1$ modulo a $2\pi/j$ rotation). |
49 | 1108 |
The fiber of this map is $\Delta^{j-1}$, the $(j-1)$-simplex, |
1109 |
and the holonomy of the $\Delta^{j-1}$ bundle |
|
50 | 1110 |
over $S^1/j$ is induced by the cyclic permutation of its $j$ vertices. |
1111 |
||
1112 |
In particular, $\Sigma^j(S^1)$ is homotopy equivalent to a circle for $j>0$, and |
|
1113 |
of course $\Sigma^0(S^1)$ is a point. |
|
1114 |
Thus the singular homology $H_i(\Sigma^\infty(S^1))$ has infinitely many generators for $i=0,1$ |
|
1115 |
and is zero for $i\ge 2$. |
|
1116 |
\nn{say something about $t$-degrees also matching up?} |
|
1117 |
||
1118 |
By xxxx and \ref{ktcdprop}, |
|
1119 |
the cyclic homology of $k[t]$ is the $S^1$-equivariant homology of $\Sigma^\infty(S^1)$. |
|
1120 |
Up to homotopy, $S^1$ acts by $j$-fold rotation on $\Sigma^j(S^1) \simeq S^1/j$. |
|
51
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|
1121 |
If $k = \z$, $\Sigma^j(S^1)$ contributes the homology of an infinite lens space: $\z$ in degree |
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|
1122 |
0, $\z/j \z$ in odd degrees, and 0 in positive even degrees. |
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|
1123 |
The point $\Sigma^0(S^1)$ contributes the homology of $BS^1$ which is $\z$ in even |
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|
1124 |
degrees and 0 in odd degrees. |
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|
1125 |
This agrees with the calculation in \nn{Loday, 3.1.7}. |
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|
1126 |
|
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|
1127 |
\medskip |
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|
1128 |
|
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|
1129 |
Next we consider the case $C = k[t_1, \ldots, t_m]$, commutative polynomials in $m$ variables. |
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|
1130 |
Let $\Sigma_m^\infty(M)$ be the $m$-colored infinite symmetric power of $M$, that is, configurations |
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diff
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|
1131 |
of points on $M$ which can have any of $m$ distinct colors but are otherwise indistinguishable. |
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|
1132 |
The components of $\Sigma_m^\infty(M)$ are indexed by $m$-tuples of natural numbers |
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|
1133 |
corresponding to the number of points of each color of a configuration. |
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|
1134 |
A proof similar to that of \ref{sympowerprop} shows that |
195a0a91e062
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|
1135 |
|
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|
1136 |
\begin{prop} |
52 | 1137 |
$\bc_*(M, k[t_1, \ldots, t_m])$ is homotopy equivalent to $C_*(\Sigma_m^\infty(M), k)$. |
51
195a0a91e062
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|
1138 |
\end{prop} |
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|
1139 |
|
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|
1140 |
According to \nn{Loday, 3.2.2}, |
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|
1141 |
\[ |
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|
1142 |
HH_n(k[t_1, \ldots, t_m]) \cong \Lambda^n(k^m) \otimes k[t_1, \ldots, t_m] . |
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|
1143 |
\] |
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|
1144 |
Let us check that this is also the singular homology of $\Sigma_m^\infty(S^1)$. |
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|
1145 |
We will content ourselves with the case $k = \z$. |
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|
1146 |
One can define a flow on $\Sigma_m^\infty(S^1)$ where points of the same color repel each other and points of different colors do not interact. |
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|
1147 |
This shows that a component $X$ of $\Sigma_m^\infty(S^1)$ is homotopy equivalent |
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|
1148 |
to the torus $(S^1)^l$, where $l$ is the number of non-zero entries in the $m$-tuple |
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|
1149 |
corresponding to $X$. |
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|
1150 |
The homology calculation we desire follows easily from this. |
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|
1151 |
|
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|
1152 |
\nn{say something about cyclic homology in this case? probably not necessary.} |
49 | 1153 |
|
52 | 1154 |
\medskip |
49 | 1155 |
|
52 | 1156 |
Next we consider the case $C = k[t]/t^l$ --- polynomials in $t$ with $t^l = 0$. |
1157 |
Define $\Delta_l \sub \Sigma^\infty(M)$ to be those configurations of points with $l$ or |
|
1158 |
more points coinciding. |
|
49 | 1159 |
|
52 | 1160 |
\begin{prop} |
1161 |
$\bc_*(M, k[t]/t^l)$ is homotopy equivalent to $C_*(\Sigma^\infty(M), \Delta_l, k)$ |
|
1162 |
(relative singular chains with coefficients in $k$). |
|
1163 |
\end{prop} |
|
49 | 1164 |
|
52 | 1165 |
\begin{proof} |
1166 |
\nn{...} |
|
1167 |
\end{proof} |
|
49 | 1168 |
|
1169 |
\nn{...} |
|
1170 |
||
47
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|
1171 |
|
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|
1172 |
|
22
ada83e7228eb
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diff
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|
1173 |
|
ada83e7228eb
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diff
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|
1174 |
\appendix |
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|
1175 |
|
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diff
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|
1176 |
\section{Families of Diffeomorphisms} \label{sec:localising} |
0 | 1177 |
|
1178 |
||
1179 |
Lo, the proof of Lemma (\ref{extension_lemma}): |
|
1180 |
||
1181 |
\nn{should this be an appendix instead?} |
|
1182 |
||
1183 |
\nn{for pedagogical reasons, should do $k=1,2$ cases first; probably do this in |
|
1184 |
later draft} |
|
1185 |
||
1186 |
\nn{not sure what the best way to deal with boundary is; for now just give main argument, worry |
|
1187 |
about boundary later} |
|
1188 |
||
8 | 1189 |
Recall that we are given |
0 | 1190 |
an open cover $\cU = \{U_\alpha\}$ and an |
1191 |
$x \in CD_k(X)$ such that $\bd x$ is adapted to $\cU$. |
|
1192 |
We must find a homotopy of $x$ (rel boundary) to some $x' \in CD_k(X)$ which is adapted to $\cU$. |
|
1193 |
||
1194 |
Let $\{r_\alpha : X \to [0,1]\}$ be a partition of unity for $\cU$. |
|
1195 |
||
1196 |
As a first approximation to the argument we will eventually make, let's replace $x$ |
|
8 | 1197 |
with a single singular cell |
0 | 1198 |
\eq{ |
8 | 1199 |
f: P \times X \to X . |
0 | 1200 |
} |
1201 |
Also, we'll ignore for now issues around $\bd P$. |
|
1202 |
||
1203 |
Our homotopy will have the form |
|
1204 |
\eqar{ |
|
8 | 1205 |
F: I \times P \times X &\to& X \\ |
1206 |
(t, p, x) &\mapsto& f(u(t, p, x), x) |
|
0 | 1207 |
} |
1208 |
for some function |
|
1209 |
\eq{ |
|
8 | 1210 |
u : I \times P \times X \to P . |
0 | 1211 |
} |
1212 |
First we describe $u$, then we argue that it does what we want it to do. |
|
1213 |
||
1214 |
For each cover index $\alpha$ choose a cell decomposition $K_\alpha$ of $P$. |
|
1215 |
The various $K_\alpha$ should be in general position with respect to each other. |
|
1216 |
We will see below that the $K_\alpha$'s need to be sufficiently fine in order |
|
1217 |
to insure that $F$ above is a homotopy through diffeomorphisms of $X$ and not |
|
1218 |
merely a homotopy through maps $X\to X$. |
|
1219 |
||
1220 |
Let $L$ be the union of all the $K_\alpha$'s. |
|
1221 |
$L$ is itself a cell decomposition of $P$. |
|
1222 |
\nn{next two sentences not needed?} |
|
1223 |
To each cell $a$ of $L$ we associate the tuple $(c_\alpha)$, |
|
1224 |
where $c_\alpha$ is the codimension of the cell of $K_\alpha$ which contains $c$. |
|
1225 |
Since the $K_\alpha$'s are in general position, we have $\sum c_\alpha \le k$. |
|
1226 |
||
1227 |
Let $J$ denote the handle decomposition of $P$ corresponding to $L$. |
|
1228 |
Each $i$-handle $C$ of $J$ has an $i$-dimensional tangential coordinate and, |
|
1229 |
more importantly, a $k{-}i$-dimensional normal coordinate. |
|
1230 |
||
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|
1231 |
For each (top-dimensional) $k$-cell $c$ of each $K_\alpha$, choose a point $p_c \in c \sub P$. |
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|
1232 |
Let $D$ be a $k$-handle of $J$, and let $D$ also denote the corresponding |
0 | 1233 |
$k$-cell of $L$. |
1234 |
To $D$ we associate the tuple $(c_\alpha)$ of $k$-cells of the $K_\alpha$'s |
|
1235 |
which contain $d$, and also the corresponding tuple $(p_{c_\alpha})$ of points in $P$. |
|
1236 |
||
1237 |
For $p \in D$ we define |
|
1238 |
\eq{ |
|
8 | 1239 |
u(t, p, x) = (1-t)p + t \sum_\alpha r_\alpha(x) p_{c_\alpha} . |
0 | 1240 |
} |
1241 |
(Recall that $P$ is a single linear cell, so the weighted average of points of $P$ |
|
1242 |
makes sense.) |
|
1243 |
||
1244 |
So far we have defined $u(t, p, x)$ when $p$ lies in a $k$-handle of $J$. |
|
8 | 1245 |
For handles of $J$ of index less than $k$, we will define $u$ to |
0 | 1246 |
interpolate between the values on $k$-handles defined above. |
1247 |
||
8 | 1248 |
If $p$ lies in a $k{-}1$-handle $E$, let $\eta : E \to [0,1]$ be the normal coordinate |
0 | 1249 |
of $E$. |
1250 |
In particular, $\eta$ is equal to 0 or 1 only at the intersection of $E$ |
|
1251 |
with a $k$-handle. |
|
1252 |
Let $\beta$ be the index of the $K_\beta$ containing the $k{-}1$-cell |
|
1253 |
corresponding to $E$. |
|
1254 |
Let $q_0, q_1 \in P$ be the points associated to the two $k$-cells of $K_\beta$ |
|
1255 |
adjacent to the $k{-}1$-cell corresponding to $E$. |
|
1256 |
For $p \in E$, define |
|
1257 |
\eq{ |
|
8 | 1258 |
u(t, p, x) = (1-t)p + t \left( \sum_{\alpha \ne \beta} r_\alpha(x) p_{c_\alpha} |
1259 |
+ r_\beta(x) (\eta(p) q_1 + (1-\eta(p)) q_0) \right) . |
|
0 | 1260 |
} |
1261 |
||
1262 |
In general, for $E$ a $k{-}j$-handle, there is a normal coordinate |
|
1263 |
$\eta: E \to R$, where $R$ is some $j$-dimensional polyhedron. |
|
1264 |
The vertices of $R$ are associated to $k$-cells of the $K_\alpha$, and thence to points of $P$. |
|
1265 |
If we triangulate $R$ (without introducing new vertices), we can linearly extend |
|
1
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|
1266 |
a map from the vertices of $R$ into $P$ to a map of all of $R$ into $P$. |
0 | 1267 |
Let $\cN$ be the set of all $\beta$ for which $K_\beta$ has a $k$-cell whose boundary meets |
1268 |
the $k{-}j$-cell corresponding to $E$. |
|
1269 |
For each $\beta \in \cN$, let $\{q_{\beta i}\}$ be the set of points in $P$ associated to the aforementioned $k$-cells. |
|
1270 |
Now define, for $p \in E$, |
|
1271 |
\eq{ |
|
8 | 1272 |
u(t, p, x) = (1-t)p + t \left( |
1273 |
\sum_{\alpha \notin \cN} r_\alpha(x) p_{c_\alpha} |
|
1274 |
+ \sum_{\beta \in \cN} r_\beta(x) \left( \sum_i \eta_{\beta i}(p) \cdot q_{\beta i} \right) |
|
1275 |
\right) . |
|
0 | 1276 |
} |
1277 |
Here $\eta_{\beta i}(p)$ is the weight given to $q_{\beta i}$ by the linear extension |
|
1278 |
mentioned above. |
|
1279 |
||
1280 |
This completes the definition of $u: I \times P \times X \to P$. |
|
1281 |
||
1282 |
\medskip |
|
1283 |
||
1284 |
Next we verify that $u$ has the desired properties. |
|
1285 |
||
1286 |
Since $u(0, p, x) = p$ for all $p\in P$ and $x\in X$, $F(0, p, x) = f(p, x)$ for all $p$ and $x$. |
|
1287 |
Therefore $F$ is a homotopy from $f$ to something. |
|
1288 |
||
7
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|
1289 |
Next we show that if the $K_\alpha$'s are sufficiently fine cell decompositions, |
0 | 1290 |
then $F$ is a homotopy through diffeomorphisms. |
1291 |
We must show that the derivative $\pd{F}{x}(t, p, x)$ is non-singular for all $(t, p, x)$. |
|
1292 |
We have |
|
1293 |
\eq{ |
|
8 | 1294 |
% \pd{F}{x}(t, p, x) = \pd{f}{x}(u(t, p, x), x) + \pd{f}{p}(u(t, p, x), x) \pd{u}{x}(t, p, x) . |
1295 |
\pd{F}{x} = \pd{f}{x} + \pd{f}{p} \pd{u}{x} . |
|
0 | 1296 |
} |
1297 |
Since $f$ is a family of diffeomorphisms, $\pd{f}{x}$ is non-singular and |
|
1298 |
\nn{bounded away from zero, or something like that}. |
|
1299 |
(Recall that $X$ and $P$ are compact.) |
|
1300 |
Also, $\pd{f}{p}$ is bounded. |
|
1301 |
So if we can insure that $\pd{u}{x}$ is sufficiently small, we are done. |
|
1302 |
It follows from Equation xxxx above that $\pd{u}{x}$ depends on $\pd{r_\alpha}{x}$ |
|
7
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|
1303 |
(which is bounded) |
0 | 1304 |
and the differences amongst the various $p_{c_\alpha}$'s and $q_{\beta i}$'s. |
1305 |
These differences are small if the cell decompositions $K_\alpha$ are sufficiently fine. |
|
1306 |
This completes the proof that $F$ is a homotopy through diffeomorphisms. |
|
1307 |
||
1308 |
\medskip |
|
1309 |
||
1310 |
Next we show that for each handle $D \sub P$, $F(1, \cdot, \cdot) : D\times X \to X$ |
|
1311 |
is a singular cell adapted to $\cU$. |
|
1312 |
This will complete the proof of the lemma. |
|
1313 |
\nn{except for boundary issues and the `$P$ is a cell' assumption} |
|
1314 |
||
8 | 1315 |
Let $j$ be the codimension of $D$. |
0 | 1316 |
(Or rather, the codimension of its corresponding cell. From now on we will not make a distinction |
1317 |
between handle and corresponding cell.) |
|
1318 |
Then $j = j_1 + \cdots + j_m$, $0 \le m \le k$, |
|
1319 |
where the $j_i$'s are the codimensions of the $K_\alpha$ |
|
1320 |
cells of codimension greater than 0 which intersect to form $D$. |
|
1321 |
We will show that |
|
1322 |
if the relevant $U_\alpha$'s are disjoint, then |
|
1323 |
$F(1, \cdot, \cdot) : D\times X \to X$ |
|
1324 |
is a product of singular cells of dimensions $j_1, \ldots, j_m$. |
|
1325 |
If some of the relevant $U_\alpha$'s intersect, then we will get a product of singular |
|
1326 |
cells whose dimensions correspond to a partition of the $j_i$'s. |
|
1327 |
We will consider some simple special cases first, then do the general case. |
|
1328 |
||
1329 |
First consider the case $j=0$ (and $m=0$). |
|
1330 |
A quick look at Equation xxxx above shows that $u(1, p, x)$, and hence $F(1, p, x)$, |
|
1331 |
is independent of $p \in P$. |
|
1332 |
So the corresponding map $D \to \Diff(X)$ is constant. |
|
1333 |
||
1334 |
Next consider the case $j = 1$ (and $m=1$, $j_1=1$). |
|
1335 |
Now Equation yyyy applies. |
|
1336 |
We can write $D = D'\times I$, where the normal coordinate $\eta$ is constant on $D'$. |
|
1337 |
It follows that the singular cell $D \to \Diff(X)$ can be written as a product |
|
1338 |
of a constant map $D' \to \Diff(X)$ and a singular 1-cell $I \to \Diff(X)$. |
|
1339 |
The singular 1-cell is supported on $U_\beta$, since $r_\beta = 0$ outside of this set. |
|
1340 |
||
1341 |
Next case: $j=2$, $m=1$, $j_1 = 2$. |
|
8 | 1342 |
This is similar to the previous case, except that the normal bundle is 2-dimensional instead of |
0 | 1343 |
1-dimensional. |
1344 |
We have that $D \to \Diff(X)$ is a product of a constant singular $k{-}2$-cell |
|
1345 |
and a 2-cell with support $U_\beta$. |
|
1346 |
||
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|
1347 |
Next case: $j=2$, $m=2$, $j_1 = j_2 = 1$. |
0 | 1348 |
In this case the codimension 2 cell $D$ is the intersection of two |
1349 |
codimension 1 cells, from $K_\beta$ and $K_\gamma$. |
|
1350 |
We can write $D = D' \times I \times I$, where the normal coordinates are constant |
|
1351 |
on $D'$, and the two $I$ factors correspond to $\beta$ and $\gamma$. |
|
1352 |
If $U_\beta$ and $U_\gamma$ are disjoint, then we can factor $D$ into a constant $k{-}2$-cell and |
|
1353 |
two 1-cells, supported on $U_\beta$ and $U_\gamma$ respectively. |
|
1354 |
If $U_\beta$ and $U_\gamma$ intersect, then we can factor $D$ into a constant $k{-}2$-cell and |
|
1355 |
a 2-cell supported on $U_\beta \cup U_\gamma$. |
|
1356 |
\nn{need to check that this is true} |
|
1357 |
||
1358 |
\nn{finally, general case...} |
|
1359 |
||
1360 |
\nn{this completes proof} |
|
1361 |
||
13 | 1362 |
\input{text/explicit.tex} |
0 | 1363 |
|
55 | 1364 |
\input{text/obsolete.tex} |
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|
1365 |
|
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|
1366 |
% ---------------------------------------------------------------- |
23
7b0a43bdd3c4
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|
1367 |
%\newcommand{\urlprefix}{} |
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|
1368 |
\bibliographystyle{plain} |
22
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|
1369 |
%Included for winedt: |
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|
1370 |
%input "bibliography/bibliography.bib" |
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|
1371 |
\bibliography{bibliography/bibliography} |
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|
1372 |
% ---------------------------------------------------------------- |
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|
1373 |
|
22
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|
1374 |
This paper is available online at \arxiv{?????}, and at |
47
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diff
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|
1375 |
\url{http://tqft.net/blobs}, |
939a4a5b1d80
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|
1376 |
and at \url{http://canyon23.net/math/}. |
7
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|
1377 |
|
22
ada83e7228eb
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21
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|
1378 |
% A GTART necessity: |
ada83e7228eb
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|
1379 |
% \Addresses |
ada83e7228eb
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|
1380 |
% ---------------------------------------------------------------- |
ada83e7228eb
rearranging; stating all the "properties" up front
scott@6e1638ff-ae45-0410-89bd-df963105f760
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|
1381 |
\end{document} |
ada83e7228eb
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|
1382 |
% ---------------------------------------------------------------- |
0 | 1383 |
|
1384 |
||
1385 |
||
1386 |
||
1387 |
%Recall that for $n$-category picture fields there is an evaluation map |
|
1388 |
%$m: \bc_0(B^n; c, c') \to \mor(c, c')$. |
|
1389 |
%If we regard $\mor(c, c')$ as a complex concentrated in degree 0, then this becomes a chain |
|
1390 |
%map $m: \bc_*(B^n; c, c') \to \mor(c, c')$. |